The following properties I am having difficulty with
Let G be a graph with vertices v 1, v 2 , ..., vn and let A = ai,j be the adjacency matrix of G.
The diagonal entries of A are all 0; that is, a i i = 0 for i = 1,...,n.
The adjacency matrix is symmetric, that is a i j = a j i for all i, j. degv i i is the number of 1’s in row i; this is also the number of 1’s in column i (row i and column i are the same)
I do not get these properties:
The $(i,j)$ entry of $A^2$ is the number of different walks of length $2$ from $v_i$ to $v_j$, in particular, the degree of $v_i$ is the ith main diagonal entry of $A^2$.
In general, for any $k\gt 1$, the $(i,j)$ entry of $A^k$ is the number of walks of length $k$ from $v_i$ to $v_j$.
so.. If I have a triangle graph with vertices = v1, v2 , v3
And Adjacency matrix $$A=\begin{pmatrix}0 & 1 & 1\\1 & 0 & 1\\ 1 & 1 & 0\end{pmatrix}\text{ then }A^2=\begin{pmatrix}2 & 1 & 1\\1 & 2 & 1 \\ 1 & 1 & 2\end{pmatrix}$$
then the property $(i,j)$ entry of $A^2$ is the number of different walks of length $2$ from $v_i$ to $v_j$, in particular, the degree of $v_i$ is the ith main diagonal entry of $A^2$.
means that from v1 to v1 the number of different walks with length two is 2?
Any help is appreciated thanks