标签: MATH

Introduction

Krigo小于 1 分钟

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

Krigo小于 1 分钟

Chapter 4 Laplacian Eigenpairs

Krigo小于 1 分钟

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

Numerical Method for Linear equations

重要

Consider the time-dependent Cauchy probelm of scalar equation,

\begin{equation} \left\{ \begin{array}{lr} u_t + au_x = 0, \qquad -\infty < x < \infty, t \geq 0 \\ u(x,0) = u_0(x) \end{array} \right. \end{equation}

Define:

$h = \Delta x$
$k = \Delta t$
$x_j = jh, j = \cdots,-1,0,1,\cdots$
$t_n = nk, n = 0,1,2,\cdots$
$x_{j+\frac{1}{2}} = x_j + h/2 = (j+\frac{1}{2})h$
$\lambda = \frac{k}{h}$

Krigo大约 4 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.

Krigo小于 1 分钟

Classwork - 18

In Class Test - 18

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

Krigo小于 1 分钟

Classwork - 2

In Class Test

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

Krigo小于 1 分钟

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.

Krigo小于 1 分钟

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.

Krigo小于 1 分钟