标签: Graph Theory

Chapter 3 Matrix Tree Theorem

title: Chapter 3 Matrix Tree Theorem icon: file order: 3 author: Krigo category: - MATH tag: - MATH - Graph Theory footer: Thank's myself hardwork. copyrigh: 无版权 date: 2024-05-28

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Chapter 4 Laplacian Eigenpairs

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Chapter 5 Algebraic Connectivity

Definition 5.1

IMPORTANT

The algebraic connectivity of $G$ is the second smallest eigenvalue, $\lambda_2(G)$ , of the Laplacian matrix $L(G)$ . We denote the algebraic connectivty of $G$ by $a(G)$ .

$\lambda_2(G) = a(G)$
$\lambda_2(G)$ is the second smallest eigenvalues.

Krigo大约 3 分钟

Classwork - 1

In Class Test

Find the adjacency matrix, its squire and cube of the following graph $G$ .

How many walks from $v_1$ to $v_2$ of length $3$ ?

List all such walks one by one.

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Classwork - 18

In Class Test - 18

Calculate all Laplacian eigenpairs of the following graph $G$ .

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Classwork - 2

In Class Test

Let $G$ be the following graph. Find the characteristic polynomial of $A(G)$ . Verify Theorem 1.8.

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Classwork - 3

In Class Test

Find the characteristic polynomial of adjacent matrix of the following graph by Theorem 1.12, 1.14 and 1.17.

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Classwork - 4

In Class Test

Find the characteristic polynomial of adjacent matrix of the following graph by Theorem 1.7, 1.17 and 1.18.

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Classwork - 5

In Class Test

Consider the following graph G and Corollary 1.26. Suppose $q(\mu) = (\mu^2 - \alpha)(\mu^3 - 3)(\mu + b), a,b>0$ and $q(A(G)) = 24J$ . Find the spectrum of the graph.

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Classwork - 6

In Class Test

Consider the following graph $G$ . Find $q(\mu)$ in Corollary 1.26 and the spectrum of $G$ .

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