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Algebraic Connectivity and Fiedler Vector Explained

Added: 2026-08-07

Deep Dive

Basic Graph Theory concepts, including vertices, edges, degree matrices, and adjacency matrices.

Graphe non orienté, orienté, sommet adjacent à un sommet dans un graphe, degré d'un sommet

Graphe non orienté, orienté, sommet adjacent à un sommet dans un graphe, degré d'un sommet

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A graph consists of vertices (nodes) and edges connecting them. The number of vertices is called the order of the graph. Graphs are classified as undirected (edges traversable in both directions) or directed (edges traversable in one direction, called arcs). Adjacency describes when two vertices are connected by an edge. In undirected graphs, adjacency is symmetric; in directed graphs, it is directional. The degree of a vertex in an undirected graph equals the number of incident edges (or neighbors). In directed graphs, we distinguish out-degree (edges originating from the vertex) and in-degree (edges arriving at the vertex). A fundamental property: the sum of all vertex degrees in an undirected graph equals twice the number of edges, since each edge contributes to two vertices' degrees.

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Discrete Maths Unit - 5

Discrete Maths Unit - 5

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This comprehensive section covers the foundational concepts of graph theory. A graph consists of vertices (nodes/points) and edges (lines connecting vertices). Adjacent vertices are connected by an edge, while isolated vertices have no connections. Graphs can be undirected (no direction) or directed (edges have direction). Simple graphs have exactly one edge between any two vertices, while multigraphs allow multiple edges. Loops connect a vertex to itself. Weighted graphs assign costs to edges. The degree of a vertex is the number of edges incident to it, with self-loops counting twice. An isolated vertex has degree 0, while a pendant vertex has degree 1. The Handshaking Theorem states that the sum of all vertex degrees equals twice the number of edges, as each edge contributes to exactly two vertices. This theorem also implies that the number of odd-degree vertices must be even. Special graph types include complete graphs (n(n-1)/2 edges), regular graphs (all vertices same degree), and bipartite graphs (n²/4 maximum edges). Matrix representations enable algorithmic analysis: adjacency matrices use rows and columns for vertices with entries indicating edge presence, while incidence matrices use rows for vertices and columns for edges. For directed graphs, adjacency matrices capture edge direction, and incidence matrices use +1 for outgoing edges and -1 for incoming edges.

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CCSU BCA 1st Year Maths-1 | Unit 3 Elementary Graph Theory | Complete Explanation

CCSU BCA 1st Year Maths-1 | Unit 3 Elementary Graph Theory | Complete Explanation

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A graph G = (V, E) consists of vertices (nodes) and edges connecting them. Key concepts include: self-loop (edge connecting a vertex to itself), parallel edges (multiple edges between same vertices), simple graph (no self-loops or parallel edges), multigraph (allows multiple edges and self-loops), finite/infinite graphs (based on vertex/edge count), incident (edge connected to vertex), adjacent (vertices connected by edge), degree (number of incident edges), isolated vertex (degree zero), pendant vertex (degree one), and the Handshaking Theorem (sum of degrees equals twice the number of edges). Directed graphs (digraphs) have ordered pairs indicating direction, with in-degree and out-degree. Matrix representations are essential for computer applications: the incidence matrix (n×m) has 1 if edge j is incident on vertex i, 0 otherwise; the adjacency matrix (n×n) has 1 if edge exists between vertices i and j, 0 otherwise. For directed graphs, incidence matrices use +1 for outgoing edges and -1 for incoming edges.

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IB MAI HL - 13.01.0 Graph Theory Introduction - Definitions

IB MAI HL - 13.01.0 Graph Theory Introduction - Definitions

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Graph theory studies interconnected networks represented by vertices (points/nodes) connected by edges (lines/connections). Two vertices are adjacent if directly connected. Graphs are defined by their connections, not visual layout—different diagrams can represent equivalent graphs. An adjacency matrix uses rows and columns for vertices, with 1s indicating connections and 0s indicating none. The degree of a vertex equals the number of edges connected to it, found by summing row/column entries. Vertices are classified as even (degree divisible by 2) or odd based on their degree count.

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MT131 - Ch6 Exam Questions & Answers (Dr.Khalid Khashan)

MT131 - Ch6 Exam Questions & Answers (Dr.Khalid Khashan)

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This section covers fundamental graph theory concepts: (1) The sum of all vertex degrees equals twice the number of edges, so edges = total_degree_sum / 2; (2) In complete bipartite graphs K(m,n), vertices = m + n and edges = m × n; (3) Converting adjacency matrices to graphs by identifying vertices from row/column labels and drawing edges where matrix entries are 1; (4) The degree of a vertex equals the sum of entries in its row, with loops counting twice. These concepts form the foundation for understanding graph structure and properties.

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Deep Dive

Fundamental Linear Algebra, specifically eigenvalues, eigenvectors, and symmetric matrices.

ゼロからの線形代数【統計検定®準１級準備講座②】

ゼロからの線形代数【統計検定®準１級準備講座②】

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This section covers eigenvalues and eigenvectors, fundamental concepts in linear algebra. For an n×n matrix A, a scalar λ is an eigenvalue if there exists a non-zero vector v such that Av = λv. The equation (A - λI)v = 0 must have a non-zero solution, which requires det(A - λI) = 0. The eigenvalues are the roots of the characteristic equation. For a 2×2 matrix [[a, b], [c, d]], the characteristic equation is λ² - (a+d)λ + (ad-bc) = 0. The spectral theorem states that any real symmetric matrix can be diagonalized by an orthogonal matrix: A = QDQᵀ, where Q is orthogonal and D is diagonal. This is important because symmetric matrices have real eigenvalues and orthogonal eigenvectors.

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Symmetric Matrices & Positive Definiteness | MIT 18.06

Symmetric Matrices & Positive Definiteness | MIT 18.06

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Symmetric matrices (A = A^T) possess two essential properties: all eigenvalues are real numbers, and eigenvectors corresponding to different eigenvalues are orthogonal. When eigenvalues are repeated, eigenvectors can still be chosen to be perpendicular. This orthogonality enables a complete set of orthonormal eigenvectors.

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Mátrix sajátértéke és sajátvektora, példával

Mátrix sajátértéke és sajátvektora, példával

51.5K

Eigenvalues and eigenvectors are fundamental concepts in linear algebra where a non-zero vector v satisfies Av = λv for an n×n matrix A and scalar λ; the characteristic polynomial det(A - λI) = 0 determines the eigenvalues, and for symmetric matrices, corresponding eigenvectors are orthogonal to each other.

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Part 4: Eigenvalues and Eigenvectors

Part 4: Eigenvalues and Eigenvectors

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Eigenvalues and eigenvectors are special vectors and scalars associated with a matrix where multiplying the matrix by the eigenvector yields a scalar multiple of the eigenvector (SX = λX), representing one-dimensional problems within the matrix; for symmetric matrices, these eigenvectors are orthogonal, enabling the spectral theorem decomposition S = QΛQ^T, which breaks down the matrix into simpler components, while rectangular matrices use singular values instead of eigenvalues for similar purposes.

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A ESTRANHA Conexão entre FIBONACCI e a RAZÃO ÁUREA

A ESTRANHA Conexão entre FIBONACCI e a RAZÃO ÁUREA

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Eigenvalues and eigenvectors are fundamental concepts in linear algebra. An eigenvector w of a matrix M satisfies Mw = λw for some scalar λ (the eigenvalue), meaning the transformation maps w to a scalar multiple of itself. For the Fibonacci matrix M = [[1, 1], [1, 0]], eigenvalues are found by solving det(M - λI) = 0, yielding λ² - λ - 1 = 0. The solutions are φ = (1 + √5)/2 and ψ = (1 - √5)/2, which are exactly the Golden Ratio and its conjugate. The corresponding eigenvectors are w = [φ, 1]ᵀ and v = [-1, φ]ᵀ. This demonstrates how the Golden Ratio naturally emerges from the mathematical structure of the Fibonacci sequence through linear algebra.

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Deep Dive

The Graph Laplacian Matrix (L = D - A) and its fundamental properties, such as being positive semi-definite.

(Spectral; Fall 23) 14 - Coalescing; introduction to Laplacian

(Spectral; Fall 23) 14 - Coalescing; introduction to Laplacian

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The Laplacian matrix L = D - A combines the degree matrix D with the adjacency matrix A. It is always symmetric. The all-ones vector is always an eigenvector with eigenvalue zero. The quadratic form x^T L x equals Σ(x_i - x_j)^2 over all edges, proving the Laplacian is positive semi-definite (all eigenvalues ≥ 0). This fundamental property connects graph structure to matrix analysis and underlies many spectral graph theory applications.

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(Spectral; Fall 23) 15 - Properties of Laplacian, including joins

(Spectral; Fall 23) 15 - Properties of Laplacian, including joins

225

The Laplacian matrix L = D - A combines the degree matrix D and adjacency matrix A. Its quadratic form X^T L X equals the sum of squared differences (x_i - x_j)^2 over all edges, proving it's positive semi-definite with non-negative eigenvalues. The all-ones vector is always an eigenvector with eigenvalue 0. Crucially, the multiplicity of the zero eigenvalue equals the number of connected components in the graph, as eigenvectors must be constant on each component.

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Russell Lyons: Monotonicity for Continuous-Time Random Walks

Russell Lyons: Monotonicity for Continuous-Time Random Walks

161

The graph Laplacian matrix is positive semi-definite. This is because each edge contributes a rank-two matrix that is positive semi-definite (specifically, a two-by-two matrix with off-diagonal entries -r and diagonal entries r), and the Laplacian is a sum of such matrices. Therefore, by definition, the entire Laplacian matrix is positive semi-definite.

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The Laplace Matrix and its Positivity Semi Definite Property

The Laplace Matrix and its Positivity Semi Definite Property

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The Laplace Matrix of a graph is defined as L = D - A, where D is the diagonal matrix of vertex degrees and A is the adjacency matrix. This matrix is always positive semi-definite because for any vector x, x^T L x equals the sum of squared differences (x_i - x_j)^2 over all edges, which is non-negative. Additionally, the vector of all ones is always in the kernel of L, as L applied to this vector yields zero due to the definition of vertex degrees.

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Lecture 12b of kernel methods: Kernels on graphs

Lecture 12b of kernel methods: Kernels on graphs

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The graph Laplacian L = D - A is defined using the degree matrix D (diagonal with node degrees) and adjacency matrix A. Key properties include: (1) L is symmetric and positive semi-definite; (2) Zero is an eigenvalue with multiplicity one, corresponding to the constant eigenvector; (3) The image of L consists of vectors with zero mean. These properties make the Laplacian fundamental for defining smoothness on graphs.

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Deep Dive

Introductory Spectral Graph Theory, connecting graph topology with matrix representations.

Spectral Graph Theory I: Introduction to Spectral Graph Theory

Spectral Graph Theory I: Introduction to Spectral Graph Theory

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The normalized Laplacian connects algebraic properties to combinatorial graph topology. The smallest eigenvalue λ₁ ≥ 0, with λ₁ = 0 if and only if the graph is disconnected. The second smallest eigenvalue λ₂ characterizes the number of connected components: λ₂ = 0 if and only if the graph has at least two components, and the multiplicity of zero equals the number of connected components. This establishes fundamental connections between matrix eigenvalues and graph connectivity, enabling algebraic determination of graph structure.

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(Spectral; Fall 23) 01 - Brief overview of spectral graph theory; review of some linear algebra

(Spectral; Fall 23) 01 - Brief overview of spectral graph theory; review of some linear algebra

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Spectral graph theory studies how the eigenvalues (spectra) of matrices associated with graphs reveal structural information about those graphs, connecting graph theory with linear algebra; a graph consists of vertices (objects) and edges (connections), which can be represented as matrices where the spectra capture essential properties like number of edges, components, and information propagation rates, though not all structural details.

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Computing eigenvalues and eigenvectors

Computing eigenvalues and eigenvectors

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Spectral graph theory studies relationships between graph properties and the spectra (eigenvalues) of their adjacency matrices. Key results include: (1) The minimum number of colors needed to properly color a graph is bounded by the largest eigenvalue plus one; (2) The total number of edges in any graph equals half the sum of the squares of all eigenvalues of the adjacency matrix. These connections enable analysis of graph structure through linear algebraic methods, with applications ranging from network analysis to algorithm design.

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An Introduction to Spectral Graph Theory

An Introduction to Spectral Graph Theory

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This comprehensive section covers the foundational framework of spectral graph theory. It begins with the adjacency matrix A, a square matrix where A_uv = 1 if vertices u and v are adjacent. Since A is real symmetric, it possesses real eigenvalues μ₁ ≥ μ₂ ≥ ... ≥ μₙ. Key insights emerge through matrix powers: A^k's (u,v) entry counts walks of length k from u to v, while trace(A^k) equals closed walks of length k. The sum of squared eigenvalues equals twice the edge count. The section proves that graphs are bipartite if and only if their spectra are symmetric about zero. It concludes with eigenvalue bounds: average degree ≤ μ₁ ≤ maximum degree, and introduces the sensitivity conjecture about hypercube induced subgraphs, resolved by Huang using twisted adjacency matrices with concentrated eigenvalues.

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Spectral Graph Theory Explained: Laplacian, Eigenvalues, and Clustering

Spectral Graph Theory Explained: Laplacian, Eigenvalues, and Clustering

Spectral graph theory studies the eigenvalues and eigenvectors of graph matrices (particularly the Laplacian matrix L = D - A, where D is the degree matrix and A is the adjacency matrix) to analyze graph properties; the Laplacian matrix is called so because it measures the discrepancy between a vertex's value and the average of its neighbors, similar to the continuous Laplacian operator, and its second smallest eigenvalue (the Fiedler value) and corresponding eigenvector (the Fiedler vector) reveal natural partitions or communities within a graph, making spectral embedding and spectral clustering powerful tools for graph partitioning and machine learning applications.

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Algebraic Connectivity

Fiedler Vector Properties

Proving Connected Subgraphs

Eigenvalue Contradiction

Conclusion and Proof

Algebraic Connectivity

0:01

Playing Section

1
Defines algebraic connectivity as the second smallest eigenvalue of the Laplacian matrix.
2
Explains that a zero value for this eigenvalue indicates a disconnected graph.
3
Introduces the Fiedler vector as the eigenvector corresponding to this eigenvalue.

Deep Dive

Spectral Clustering algorithms and their applications in image segmentation and data machine learning.

Stanford CS229M - Lecture 19: Mixture of Gaussians, spectral clustering

Stanford CS229M - Lecture 19: Mixture of Gaussians, spectral clustering

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Spectral clustering leverages eigenvalues and eigenvectors of similarity matrices to partition data into clusters. Given n data points and a similarity matrix G where G_ij captures pairwise relationships, the goal is to group points based on their connections. This framework applies broadly: social network community detection (identifying user groups from friendship graphs), image segmentation (grouping visually similar regions), and topic modeling (clustering documents by shared topics). The core insight is that spectral properties of the similarity matrix reveal hidden structure in the data, transforming combinatorial clustering problems into algebraic eigenvalue problems amenable to linear algebraic techniques.

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Laplacian #2 - spectral clustering

Laplacian #2 - spectral clustering

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Spectral clustering can be applied to image segmentation by treating image pixels as graph nodes. Adjacent pixels with similar colors are connected with high weights in the adjacency matrix. The Laplacian eigenvectors then provide a natural partition of the image into regions with similar colors. This application demonstrates how spectral clustering can identify meaningful structures in complex data by leveraging the graph's structural properties.

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W5_L3: Spectral clustering - mincut

W5_L3: Spectral clustering - mincut

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Spectral clustering is a two-stage algorithm: (1) Solve the relaxed problem to obtain a real-valued score vector f*, where each component represents a continuous mapping of nodes to real numbers; (2) Apply k-means clustering on the mapped space (the f* values) to obtain the final discrete clustering. The hope is that the relaxed solution produces values close to 0 and 1, which k-means can properly cluster. This approach enables efficient clustering of non-convex data structures by leveraging the spectral properties of the graph Laplacian.

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Remote Sensing Image Analysis and Interpretation: Feature extraction and image segmentation

Remote Sensing Image Analysis and Interpretation: Feature extraction and image segmentation

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Spectral indices combine multiple bands into interpretable metrics exploiting biological principles: healthy vegetation absorbs in red/blue but reflects in green/infrared. NDVI = (NIR - Red)/(NIR + Red) ranges from -1 to 1. Bispectral plots track plant development stages. While effective for presence/absence detection, NDVI struggles with complex parameters like biomass where relationships become nonlinear. Image segmentation groups similar pixels into homogeneous regions before classification, reducing computational burden. Clustering algorithms like k-means partition data into k clusters by minimizing squared distances to centroids through iterative assignment and recomputation. SLIC extends k-means by incorporating spatial information alongside spectral features, producing compact superpixels of approximately equal size through regular grid initialization and gradient-based centroid shifting.

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Algorithms Festival @ CSA, IISc

Algorithms Festival @ CSA, IISc

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Spectral clustering uses eigenvectors of the similarity matrix for dimensionality reduction before clustering. The algorithm computes a similarity matrix, then the eigenvectors of the Laplacian matrix, and applies k-means on these features. This approach finds non-convex clusters that traditional methods like k-means cannot find, making it valuable for image segmentation and social network analysis. The computational complexity is O(n³) for eigendecomposition, making it expensive for large datasets. Limitations include sensitivity to outliers and the need to specify the number of clusters. Despite these limitations, spectral clustering is widely used in practice for its ability to discover complex cluster structures.

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Deep Dive

Graph Partitioning and Cheeger's Inequality, which relates the bottleneck of a graph to the second smallest eigenvalue.

Cheeger's Inequality || @ CMU || Lecture 15d of CS Theory Toolkit

Cheeger's Inequality || @ CMU || Lecture 15d of CS Theory Toolkit

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Cheeger's Inequality establishes that the minimum conductance (bottleneck) of a graph is bounded by the square root of the second-smallest eigenvalue of the graph's Laplacian matrix, providing a constructive method to convert any real-valued function with small Rayleigh quotient into a subset with small conductance by examining superlevel or sublevel sets.

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Spectral Graph Theory Explained: Laplacian, Eigenvalues, and Clustering

Spectral Graph Theory Explained: Laplacian, Eigenvalues, and Clustering

The second smallest eigenvalue (Fiedler eigenvalue λ₂) and its eigenvector (Fiedler vector f₂) provide powerful tools for graph partitioning. Named after Miroslav Fiedler, this pair identifies weak connections or bottlenecks between communities. The Fiedler vector minimizes the Rayleigh quotient subject to being orthogonal to the constant eigenvector, naturally finding the weakest connections. By examining the sign pattern of f₂, one can partition the graph into communities where vertices within each community share similar values while vertices across communities have opposite signs. This technique reveals natural divisions in connected graphs without requiring prior knowledge of community structure.

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The Unreasonable Effectiveness of Spectral Graph Theory: A Confluence of Algorithms, Geometry & ...

The Unreasonable Effectiveness of Spectral Graph Theory: A Confluence of Algorithms, Geometry & ...

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This section develops the theory of graph expansion and bottlenecks, connecting geometric intuition to spectral analysis. It defines expansion φ(S) as the ratio of cut edges to subset volume, characterizing how well-connected a graph is. The minimal expansion φ*(G) identifies the worst bottleneck. The Discrete Cheeger Inequality establishes that λ₂/2 ≤ φ*(G) ≤ √(4λ₂), showing bottlenecks correspond to small eigenvalues. This provides a spectral algorithm for finding bottlenecks by examining the second eigenvector. The lecture emphasizes that bottlenecks are computationally important—they represent communities in social networks and enable divide-and-conquer algorithms.

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Daniel Spielman “Miracles of Algebraic Graph Theory”

Daniel Spielman “Miracles of Algebraic Graph Theory”

58K

Spectral graph drawing uses the second and third Laplacian eigenvectors as coordinate axes to visualize graphs, successfully reconstructing geometric structure from combinatorial data. This technique produces clean layouts for triangulations and meshes but fails for random regular graphs. Cheeger's inequality bounds graph conductance (partition quality) by λ₂/2 ≤ φ(G) ≤ √(2Dλ₂), relating spectral properties to partition quality. Large λ₂ implies high conductance (no good partitions exist), while small λ₂ guarantees good partitions. This enables spectral clustering by finding level sets of the second eigenvector. The complete binary tree achieves conductance ~2/n with λ₂ ~1/n, while path graphs show quadratic discrepancies. This framework underlies parallel computing partitioning and community detection algorithms.

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CSE202, Lec 16: The conductance of graphs, the normalized Laplacian and the spectral gap

CSE202, Lec 16: The conductance of graphs, the normalized Laplacian and the spectral gap

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This comprehensive section develops the spectral theory underlying Cheeger's inequality. It begins by establishing that the normalized laplacian has eigenvalues in [0, 2] with the all-ones vector as the first eigenvector. The section then applies the Courant-Fischer minimax principle, demonstrating that the minimum rayleigh quotient among unit vectors orthogonal to the first eigenvector equals the second eigenvalue λ₂. This theoretical foundation enables the derivation of Cheeger's inequality, showing how the hard-to-compute graph partitioning problem reduces to efficiently computable eigenvalues. The section emphasizes that symmetric matrices have orthogonal eigenvectors forming a basis, making spectral analysis mathematically rigorous.

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Deep Dive

Consensus Protocols and synchronization in multi-agent systems, where algebraic connectivity dictates convergence speed.

CAM Visit Day Colloquium - Alex Townsend (3/26/21)

CAM Visit Day Colloquium - Alex Townsend (3/26/21)

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Algebraic connectivity (second smallest Laplacian eigenvalue) measures graph connectivity. Intuitively, higher algebraic connectivity should lead to faster synchronization. However, razor's edge examples show graphs with high algebraic connectivity can still be weakly unstable. Adding edges can both stabilize existing equilibria and create new ones, making analysis complex. The region between known bounds is filled with examples where the Jacobian has multiple zero eigenvalues, requiring sophisticated analysis.

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10 BeFEMTO-Distributed Interference Management in Femtocell Networks

10 BeFEMTO-Distributed Interference Management in Femtocell Networks

282

Network connectivity in cooperative systems is modeled using random geometric graphs: (1) Nodes are distributed randomly in space with given density; (2) An adjacency matrix indicates whether pairs of nodes are within communication range; (3) Degree matrix and Laplacian matrix describe connectivity properties; (4) Algebraic connectivity (second smallest eigenvalue of Laplacian) determines convergence speed of consensus algorithms.

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O fascinante encontro do caos e da sincronia: como a ordem emerge em sistemas imprevisíveis?

O fascinante encontro do caos e da sincronia: como a ordem emerge em sistemas imprevisíveis?

153

For networks of n oscillators interacting through coupling strength alpha, the Laplacian matrix describes the network structure. The second smallest eigenvalue (lambda_2), called algebraic connectivity, measures how well-connected the network is—a network is truly connected only if lambda_2 > 0. For simple linear coupling, global synchronization occurs when alpha exceeds a critical threshold alpha_C, which is inversely proportional to lambda_2. Networks with higher algebraic connectivity require less coupling strength to synchronize.

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CSL Emerging Topics 2011 - Controls of Networked Multi-Robot Systems - Magnus Egerstedt

CSL Emerging Topics 2011 - Controls of Networked Multi-Robot Systems - Magnus Egerstedt

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This section develops the consensus framework as the canonical approach to multi-robot coordination. Starting with the rendezvous problem—robots meeting at a common location without knowing absolute positions—the solution emerges naturally: each robot moves toward the center of mass of its neighbors. This simple update rule achieves consensus (all agents converge to the same value) when the communication graph is connected. Algebraic graph theory provides the mathematical foundation: the graph Laplacian L = D - A (degree minus adjacency matrix) governs system dynamics. The Laplacian is positive semi-definite with exactly one zero eigenvalue if and only if the graph is connected, and the null space spans the vector of all ones—explaining why consensus emerges. The second smallest eigenvalue (algebraic connectivity) determines convergence rate, creating a fundamental trade-off: complete graphs offer fastest convergence but require maximum communication, while line graphs minimize communication but slow convergence significantly. The framework extends to dynamic environments through switched graph analysis, requiring only that connectivity is maintained over time.

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Consensus Dynamics (Lectures on Multiagent Systems)

Consensus Dynamics (Lectures on Multiagent Systems)

212

This section analyzes the convergence behavior of consensus protocols in undirected connected graphs. It establishes that all agents converge to the same value as time approaches infinity, with the final convergence point being the average of initial conditions. The analysis demonstrates how to calculate convergence time using eigenvalues of the system matrix, specifically the second smallest eigenvalue (algebraic connectivity). The settling time is approximately 4 divided by the absolute value of this eigenvalue, providing a quantitative measure of system performance.

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Deep Dive

Community Detection in social networks and complex biological network analysis.

A gentle introduction to network science: Dr Renaud Lambiotte, University of Oxford

A gentle introduction to network science: Dr Renaud Lambiotte, University of Oxford

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Community detection identifies densely connected groups within networks, revealing modular organization present across diverse systems from social networks to biological networks. The goal is to partition nodes into groups where internal connections dominate over external ones. This addresses the challenge that large networks become visually unmanageable—visualization loses meaning when too many nodes are displayed. Communities enable abstraction: instead of visualizing millions of individual nodes, one can represent groups as super-nodes, enabling navigation at multiple resolutions. The karate club network (Zachary's data) serves as a classic benchmark for testing community detection algorithms, having known ground truth communities from a real-world social split.

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Barabási Albert-László: Behálózva — a hálózatok csodálatos világa (Atomcsill, 2012.11.22.)

Barabási Albert-László: Behálózva — a hálózatok csodálatos világa (Atomcsill, 2012.11.22.)

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Real networks contain communities—groups of nodes more densely connected to each other than to nodes outside. In social networks, these are like friend groups where everyone knows everyone else but has fewer connections to people outside. Community structure appears not only in social networks but also in biological networks (proteins with similar functions cluster together) and the World Wide Web (pages with similar content cluster together). Community detection algorithms analyze network structure to identify these communities by coloring nodes with different colors. In a study of mobile phone call networks, communities were identified based on call patterns—one community made most calls at night while another made most calls during the day, reflecting real-world social and behavioral patterns.

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Community Detection

Community Detection

335

Community detection identifies tightly knit groups of nodes that are highly interconnected internally while loosely connected externally. These communities, also called clusters or modules, represent subgraphs with high clustering coefficients and more intra-community edges than inter-community edges. This modular organization is fundamental in biological systems like metabolic and protein interaction networks, as well as engineered systems. The Zachary's Karate Club case study (1977) demonstrated this concept: Wayne Zachary predicted how a karate club would split into two factions based solely on the existing social network structure, establishing a gold standard for validating community detection algorithms.

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Community Detection for Extremely Large Networks - Aidan Lakshman

Community Detection for Extremely Large Networks - Aidan Lakshman

184

Networks are fundamental data structures where nodes represent entities and edges represent relationships. Communities are densely connected groups within networks that share stronger internal connections than external ones. In genomics, communities represent gene families across organisms, identified through sequence similarity comparisons. Community detection algorithms take graphs as input and produce node labelings where same-labeled nodes belong to the same community. These algorithms are categorized by optimization objectives: modularity optimization, mathematical/physical models, and random walk approaches. Different algorithms optimize different criteria, producing varying community structures on the same network. Traditional benchmarks using synthetic graphs with uniform node degrees poorly represent real biological networks. A more realistic biological benchmark incorporates power-law distributed node degrees, non-uniform edge weight distributions, and collapsing weak edges to zero. Testing on these refined benchmarks revealed that label propagation and Louvain algorithms perform best, while MCL remains a mainstay in biological network analysis.

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Methods in building and analysing biological networks

Methods in building and analysing biological networks

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Communities are densely connected areas within a network. Algorithms exist to identify these communities, which are important because they associate with specific biological processes, pathways, or well-characterized protein complexes. For example, components of the proteasome or nuclear pore complex tend to cluster together in protein-protein interaction networks.

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Deep Dive

Expander Graphs and network robustness, evaluating how well-connected a network remains under node or edge failures.

La théorie des graphes ou comment ne pas prendre la grosse tête | Voyages au pays des maths | ARTE

La théorie des graphes ou comment ne pas prendre la grosse tête | Voyages au pays des maths | ARTE

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A finite graph is an expander if, when its vertices are divided into two roughly equal groups, a large number of edges connect the two groups. The quality of expansion is measured by the ratio of connecting edges to total vertices. Expander graphs represent robust networks because cutting some edges cannot easily disconnect the network. Linear graphs and disconnected graphs are poor expanders.

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Network Robustness - Applied Social Network Analysis in Python

Network Robustness - Applied Social Network Analysis in Python

296

Network robustness refers to a network's ability to maintain its connectivity and general functions when nodes or edges are removed due to failures or attacks. Node connectivity measures the minimum number of nodes that must be removed to disconnect a graph, while edge connectivity measures the minimum number of edges required for disconnection. Networks with higher node and edge connectivity values are more robust, as they require more removals to be disconnected. These concepts are essential for designing resilient networks in applications like air transportation, internet infrastructure, and power grids.

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Ramanujan graphs Part 1: Alain Valette

Ramanujan graphs Part 1: Alain Valette

225

Expanders are families of finite connected K-regular graphs with increasing size that satisfy uniform expansion properties. They must be robustly connected (hard to disconnect) while having few edges (linear in number of vertices, not quadratic like complete graphs). The Cheeger constant h(X) measures this robustness: h(X) = min{|∂S|/|S| : ∅ ≠ S ⊂ V, |S| ≤ |V|/2}, where ∂S is the edge boundary of S. Complete graphs have h ≈ 1, while cycles have h ≈ 4/n (going to zero).

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Jak zobaczyć i zrozumieć wyższe wymiary? - prof. Dawid Kielak

Jak zobaczyć i zrozumieć wyższe wymiary? - prof. Dawid Kielak

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Expander graphs solve practical network design problems: connect servers with minimal cable while ensuring the network stays connected even if any single cable is cut. Russian mathematicians in the 1960s first proved such graphs exist, and later mathematicians showed how to construct them using symmetry groups with non-Euclidean geometry. These graphs are now used in everyday applications like sorting algorithms on social media platforms and in building quantum computer architectures for error correction. Mathematical research often has practical applications that may not be apparent at the time of discovery, demonstrating the long-term value of mathematical research.

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Expander Pruning

Expander Pruning

208

Expander graphs are robustly connected graphs where every subset S has cut size proportional to its volume. Formally, a graph is an ε-expander if cut edges ≥ ε × volume(S) for all subsets S. Expanders resist disconnection under edge deletions: non-expanders have sparse cuts that disconnect with few deletions, while expanders ensure deleting k edges disconnects only a small portion proportional to k. Expander pruning maintains a pruning set P such that the complement remains an expander despite adversarial updates. The one-shot version handles batch deletions: given an expander G and deleted edges D, find P with |P| proportional to |D| so that G \ D \ P is a 5/6 expander. This enables dynamic algorithms for fundamental graph problems.

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Algebraic Connectivity

Fiedler Vector Properties

Proving Connected Subgraphs

Eigenvalue Contradiction

Conclusion and Proof

Algebraic Connectivity

0:01

Playing Section

1
Defines algebraic connectivity as the second smallest eigenvalue of the Laplacian matrix.
2
Explains that a zero value for this eigenvalue indicates a disconnected graph.
3
Introduces the Fiedler vector as the eigenvector corresponding to this eigenvalue.

163 views323:49@nptel-nociitm9240Original Release: 2025-11-13

Algebraic connectivity (the second smallest eigenvalue of the Laplacian matrix) measures how well-connected a graph is, where a higher value indicates better connectivity; the corresponding eigenvector, called the Fiedler vector, has the property that the vertex sets with positive and negative entries each induce connected subgraphs in any connected graph.

[00:00:01][Music] Hello everyone and welcome to this lecture. In this lecture we will start a new topic and this new topic is algebraic connectivity. A very very famous topic. Okay. In order to define what algebraic connectivity we have to go back to our famous matrix and this matrix is lelassian matrix. Okay. So what was a lelassian matrix? So remember if a uh we have a graph G okay the lelassian matrix is nothing but we denote it by LG. Okay was nothing but degree matrix minus our adjacency matrix. And we have seen many many beautiful properties of this lelassin matrix. The one one thing that was very very crucial is this matrix is actually so I will denote it by simply by L okay simply by L is positive semidefinite matrix that means all the on values of L are always non- negative okay and there was one more important thing is that zero will always be an igon value and why it is so if you see the structure of L okay okay on the dina there are degrees and of dino are minus one.

[00:01:25]Whenever there is two vertices I and J have an edge between them then it will be minus one otherwise zero. So if you add all the entries of a row it will give you zero. Okay. So that means all one vector is an igon vector corresponding to zero on value. Okay.

[00:01:45]Now what about the second smallest value? The first smallest value is by default zero. What about the second smallest value? Let us denote the second smallest igon value by mu. Okay. So second is smallest smallest on value of l. Okay remember if this mu is also zero that means what the graph is disconnected. G is disconnected. And why it happens? Why it happens? We have already proven that the multiplicity of zero as an igon value tells that the same number of connected components are there. Okay. So if the multiplicity is k that means there are k connected components are there. So if mu is also zero that mean there are more than one connected component as the graph is disconnected. And by the way what about the uh the mu of k? Okay. The what about for suppose complete graph on n vertices complete graph what will be its mu? If you check this the mu is nothing but n the best possible the best possible.

[00:02:53]Okay. So in a way this mu tells how well connected a graph is and it was introduced by fiddler. Okay. And due to it the igon vector or on vector corresponding to mu okay is known as a fiddler vector. Okay. So fiddler vector is actually so let me now write it fiddler vector uh uh an igon vector anon vector uh corresponding to corresponding to mu.

[00:03:25]Okay corresponding to to mu. Okay, we will see very very beautiful results uh just by looking at mu. Okay, in this lecture, okay, but there is one more important thing. Okay, uh I mean if you see that this suppose X is an F vector. Suppose X is a F vector.

[00:03:43]Okay, this has to be orthogonal to all one vector. This is coming from just a spectral theorem or just a really quotient. Okay, uh because this all the vector because it's a symmetry matrix.

[00:03:55]So we can always take the igon vectors to be orthogonal. One is already there.

[00:03:59]So x is has to be perpendicular to or orthogonal to one. That means in x there will be some positive entries. There will be some negative entries and maybe some zero entries. Okay. So now let us move forward for one beautiful um uh theorem based on the structure of x. Okay. So suppose so I told you that there will be some positive entries, some negative entries, some zero entries. So let me define u two vortex at using x. Okay. So using x. Okay. So suppose I denote a vortex at v plus. Now I'm talking about the vortex of the graph g is actually all the indices.

[00:04:37]Okay. All the indices such that or all the vertex i such that vortex i such that uh the corresponding entry in your part vector is positive or greater. So such that x i is greater or equal to zero. Okay. And similarly suppose V minus a set of vertices I where all these I are less or equal to zero. Okay.

[00:05:01]So these are basically I very very simple uh notations and definitions of this V plus and V minus. And by the way both of these will be suppose if graph is connected. If the graph is connected uh then the both has to be non- empty.

[00:05:16]Okay. So if uh g is connected if g is let me write is connected then then v plus and v minus uh both uh both are both are non- empty or why it is so why it is so non empty yeah it's due to this it's due to this fact okay very very good so now let me introduce or give a very very nice uh theorem on uh test structure of basically the fid vector.

[00:05:49]Okay. So let g be a connected graph.

[00:05:52]Then the theorem is then the theorem is v + and I will write similarly v minus v minus similarly v minus okay so this is a vortex set right this is a subset of a vortex set okay induced induced induce rather than so maybe I will this spelling induce induce a connected connected subgraph. Okay. So basically what you do okay if you pick all the vortex vertices in V plus okay and think about the induced subgraph on it. This will be connected subgraph. This will be connected. Similarly if you pick vertices in V minus and see the induced subgraph on V minus this will be again uh connected subgraph. Okay. So how to prove this? How to prove this beautiful fact? Okay. So but before that I need I need oh theorem okay theorem this is theorem okay I need to prove uh actually I have to make use of a very very small result very very a crisp result okay so let me first write that result and then we will uh head towards proving this beautiful theorem okay so the result is this okay so I need let me call it actually because I will be using it for proving this uh nice result okay so suppose B is a symmetry positive semi suppose let B B positive semidefinite matrix okay of order n cross n of order n cross n okay then then our claim is then xrpose for any vector x okay so maybe I go for any vector x for any vector x okay xrpose bx equal to zero zero if and only if if and only if bx is equal to zero. How to prove this? This is very very simple one. So we know that prove let us first prove it. Okay, we know that if b is positive seminent as b is uh positive semideent matrix that means there exists a matrix c there exists a matrix such that b equal to c transpose. This also we have seen many times. Okay. Now you see that what will be this if you do xrpose bx. Okay. So this is nothing but xrpose cpose uh cx right. This is this is this. So that means this is nothing but uh this you can write actually as cx transpose cx.

[00:08:29]Now this is xrpose bx imply uh zero implies that c transpose x is actually zero right this is coming from here. So now so cx is zero that means what if you do this this further implies that crpose cs is equal to zero. Okay so we have actually proven that one part that this will imply that bx is equal to zero and the converse part is actually very very obvious. So converse part is obvious. So if bx is equal to zero that means this is uh x transform bx. So converse part is straightforward. Okay converse uh converse is obvious. Okay, very good of this. Very good. So we will make use of this result uh to prove uh our uh the main theorem for today. Okay, so the theorem is V plus induce a connected subgraph. Remember what was B plus? B plus were all the uh the vertices in G uh such that corresponding entries in your fular vectors are non negative.

[00:09:29]Okay. So let us prove the proof of theorem. Uh so let me write it here. the proof of the proof of the theorem. Okay. Okay.

[00:09:43]Very good. Okay. Since v plus is non- negative. Okay. So v plus is non- negative. We have seen that because x vector has to be perpendicular to so it's a sorry non- empty non-mpt.

[00:09:56]Okay. Actually uh what you can do is so suppose suppose v plus for the contradiction purpose uh assume that assume that v plus is not connected.

[00:10:07]Okay so this is plus assume that uh v plus is actually disconnected. Okay. So we will prove it using contradiction disconnected disconnected uh subgraph subgraph.

[00:10:22]Okay. Induce or maybe uh maybe induces induce.

[00:10:30]Okay. Great. So let us by default let us uh choose that out of v plus. Okay. So there are vertices suppose uh you can write from 1 to up to s. Right. So these are forming one connected component and then there are the vertices s plus so maybe write is this uh smaller one s + one up to r here. So basically your v plus is actually this okay so let me write v plus is equal to all the vertices from 1 to up to r. Okay. And out of this uh this these are suppose uh so this is a connected component uh a component component. Okay. And this is actually subgraph on uh rest of the vertices.

[00:11:13]Okay. Okay. So now uh a component uh in in v plus a component in v plus. Okay.

[00:11:21]Okay. So now let us see the uh the structure of the lelassian matrix. Okay.

[00:11:25]So it will be like something this. Okay.

[00:11:27]So llian matrix can be written as okay as this L11 this may be corresponding to uh this component okay then this uh component has not any with the rest of the graph in B plus I will write it zero okay and then it will have connection with uh the rest of the vertices or maybe this V minus okay V minus is there okay very good so now similarly here you can write L22 this will be L 23 and this will be L uh 31 L 32 and L 33. Okay. Now do what?

[00:12:07]Okay. So suppose uh you have a filter vector X. Okay. Now this break this filter vector in actually three parts which will I mean the size will be actually uh corresponding to the size of L11, L22 and L3. So let me write it as x1 x2 okay and x3 here. Okay. So these are actually the basically chunks of chunks I have taken from the filler vector x. Okay. So this is suppose this I call it x. x is filler vector. Okay.

[00:12:36]Fiddler uh vector. Very good. Now you see that okay so from here to here. So this constitute of what? V plus. Okay.

[00:12:46]And this you can just think about the remaining vertices. Okay. Now that means what? So x3 will be what? So all the entries in x3 will be negative. Okay.

[00:12:55]All the three will be negative because all the zeros or plus are consumed in in this part v plus. Okay. Now now what? So let us write the equation from here that uh uh this so this will be what? L11 * x1 and then uh this is zero. Okay. plus L13 * X3. Okay, this is actually giving you what? This will be giving you because this will be equal to mu * the same vector, right? Mu * the same vector.

[00:13:30]Okay, because your algebraic connectivity is mu here. So, this will be giving you this. Okay, so this is indeed equal to mu * x1. Okay, now what are the implications of it? What are the implications of it? If you see that this L13 L13 is always less or equal to zero right every entry is either 0 or minus1 okay and you know that and X3 is less than zero so this implies what this implies this right this implies let me write in the next page so this implies maybe maybe here I can write this this implies that L13 * L cq is always greater or equal to zero okay now the important thing is important thing is known where L1 has at least one non-zero entry. Why it is so?

[00:14:16]L13 has at least one non-zero entry. Why it is so? If you see that because this graph is connected. So this chunk this chunk will have some uh basically so this part must be have one edge with this right. Similarly this part I mean uh no need to take care of L L2 to write now. Okay. So you know that this component must have an edge with L13 because the graph is connected right?

[00:14:39]The graph is connected. So as the graph is edge edge G is connected as G is maybe I write down here connected okay L3 has nonzero entry non zero entry okay so that means this L13 is not equal to zero okay very good okay now how can we make use of these uh uh these observations so that from here okay so you can first thing is is This you can immediately write that L11 minus mu okay time x1 is less or equal to zero this coming from uh this fact okay now what now what and also due to this fact that this l11 minus mu x1 is not equal to zero okay due to this very very good okay uh these observations are super super easy now how to uh make use of this to prove that V+ actually induces a connected subgraph. Okay, what I need what I need.

[00:15:48]Okay, so up to the here the things are pretty much clear. Okay. Now let us uh claim that let us claim that uh and by the way suppose if you if you uh okay so yeah maybe if you if you do this uh this this operation what will be the quantity what will be the what we can say about this number remember x1 is always uh due to our definition is is uh always uh non- negative okay and this part is again uh uh less or equal to zero. So this will again give you uh less or equal to zero.

[00:16:26]Okay. Due to this, okay, because x1 is what? This is non- negative vector.

[00:16:30]Okay. And this is lesser or equal to zero. So that mean this this has to be less or equal to zero. Very good. Now we claim that let us make one claim. Claim is this. Okay. Claim is this L11. So this uh this matrix L11 minus mu uh is maybe I will write this. Okay. So now mu is a mu * identity matrix. In fact I have to actually make here identity wherever. So sorry about this identity matrix here. Okay. So maybe above also and yeah very good. So this very uh is not a not a uh positive semi-definite matrix. So let us uh let us actually prove it not a positive semi-definite matrix. Okay. So maybe we are claiming that this is not equal to zero. Not equal to zero. Okay. So is not a positive symmetry matrix. So suppose it's a positive semi-definite matrix.

[00:17:25]That means for every vector x. So uh so suppose suppose suppose uh uh l11 minus mu is is is positive semidefinite. This implies then then what will happen? Then we can write this right x1. So maybe actually I write this as one here. X1 transpose oh sorry this is transpose I'm making some mistakes here. Okay. So this will be transpose X1 transpose time L11 uh yeah minus mu identity uh matrix and times again x1 this has to be greater or equal to zero because we have assumed that it's a positive 7 matrix. Okay this implies uh by this let me call one.

[00:18:09]Okay. So by one so by one this implies that uh uh basically x1 transpose okay * l11 minus mu * identity matrix this will give you this will give you yeah x1 equal to zero okay this will be equal to z but remember remember if it is uh zero and l1 mu i is positive semi matrix we have just 2.1 lab in that case in that case what will So let me go to the lema.

[00:18:41]Okay. In that case what will happen? Uh in that case we will prove that this bx actually we have proven that this bx is equal to zero. Right. So that implies here is this. Let me scroll it bit faster. Okay. So this implies that so this further implies that l11 - mu * identity matrix * x1 is indeed zero which is a contradiction. This is a contradiction due to this. Okay. So contra so let me call it now two. I'm doing the reverse side uh reverse direction but anyway so using two this is a contradiction. Okay. So by two by two it is it is a contradiction. Okay.

[00:19:25]Contraction.

[00:19:27]Okay. This implies that basically this implies that L1 minus H is not a positive semideft matrix.

[00:19:40]This is not a positive seminar matrix.

[00:19:43]This implies further that there will be at least one on value negative. Okay, that in fact that means that which means that uh and this further means that uh an igon value an value an value of l11 okay is less than is is less than mu okay and now the similar thing we can say about l22 similarly similarly okay similarly similarly uh The an igon value similarly let me write with smaller a okay so an igon value on value of l22 is less than is less than okay so less than uh mu okay okay so that means what that means what if you talk about this matrix L 1 0 0 remember this is a principal submatrix right principal submatrix of bigger L. Okay. So there will be the second smallest value. So uh so this implies that uh so so suppose the second smallest value of this matrix is mu dash. Okay. So let uh let so let me call let the let the second uh smallest on value on value on value of this matrix is of is mu dash. Okay. So that means what this mu will be uh so mu dash will be always less than uh always less than this right because uh it has an igon value which is less than mu okay it has an value which is less than mu so the second smallest value of uh basically this matrix is actually so suppose this is mu dash mu dash is less than mu okay but by koi theorem But by cowi interace theorem. Okay, remember this is a principal submatrix of L.

[00:21:54]Okay, so this is a principle by KI uh interace. Okay, interace we know that the second smallest on value of the larger matrix okay which is in fact do must be less or equal to second smallest value of this. So there by Intel theorem mu is actually lesser or equal to mu dash. Okay. Which is again a contradiction. Which is again a contradiction. Okay. So is which is a contradiction. Which is a contradiction.

[00:22:30]Okay. Hence hence we claim that okay our our assumption was what? V + H are disconnected. Okay. So this claim is wrong. Hence V plus has to be connected.

[00:22:42]Hence, hence V+ is giving an an connected connected induced subgraph.

[00:22:56]induced subgraph. Okay. And similarly we can say about V minus. And similarly about and similarly L uh it holds for it holds for for V minus. Okay. So V plus is giving a connected induc subgraph. V minus is also connect giving an induced subgraph.

[00:23:19]This we can see just by looking at the structure of fiddler on vector. Okay. We will see some examples in our tutorial session or in assignment. Okay. So I hope you enjoyed this lecture uh the power of algebraic connectivity and uh the corresponding fiddler vector. Okay.

[00:23:37]So I'm stopping this lecture here. See you next time. Thank you very much.

[00:23:41]Yeah.

[00:23:45][Music]

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