Adjacency matrix

Builds on Graph

Definition

Let $G$ be a finite graph with vertex set $\{v_1, \dots, v_n\}$ . The adjacency matrix of $G$ is the $n \times n$ matrix $A = (a_{ij})$ with $a_{ij} = \begin{cases} 1 & \text{if } v_i \sim v_j, \\ 0 & \text{otherwise.} \end{cases}$

Remarks

The matrix $A$ is symmetric and has zeros on the diagonal. The sum of row $i$ is the degree of $v_i$ . The entry $(A^k)_{ij}$ is the number of walks of length $k$ from $v_i$ to $v_j$ . Another order of the vertices gives the matrix $P^{T} A P$ for a permutation matrix $P$ .

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Graph definition
Adjacency matrix definition

Where this leads

3 pages build on this one.

Tag	`0NZ8MDBN6Q`
URI	`m2://0NZ8MDBN6Q`
Kind	definition
Writing	W1 stated
MSC	05C50
Defines	Adjacency matrix
Uses	Graph

Adjacency matrix

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by