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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

sam786
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    Can you write down the formula for matrix multiplication? – saulspatz Apr 13 '18 at 14:25
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    Please use MathJax to format your question. Note that you have to $ signs around the formulas for the formatting to show up. – saulspatz Apr 13 '18 at 14:28
  • What is $A$? What is Ak? Do you mean $A^k$? – Thern Apr 13 '18 at 14:34
  • yes I did sorry – sam786 Apr 13 '18 at 14:35
  • Ah, so $A_{ij}$ is the number of walks of length 1 from $v_i$ to $v_j$ (probably meaning that $v_i$ and $v_j$ are adjacent in the graph). Things are slowly making more sense. I assume that all edges in the graph have length 1? Please ask the question in a way so that the reader does not have to derive any important property on his/her own. – Thern Apr 13 '18 at 14:40
  • Also, I would assume that the $(i,j)$ entry of $A^k$ is the number of different walks of length $\leq k$, right? As saulspatz already suggested, the property will become clearer if you write down the formula for matrix multiplication. – Thern Apr 13 '18 at 14:51
  • Also related: https://math.stackexchange.com/questions/231841/powers-of-adjacency-matrix-determination-of-connection-in-graph/231854, https://math.stackexchange.com/questions/527204/if-a-is-the-adjacency-matrix-of-a-graph-why-does-the-i-j-entry-of-an-g – Misha Lavrov Apr 13 '18 at 15:25
  • Have you tried drawing your example graph and counting walks explicitly? – amd Apr 13 '18 at 18:42
  • Thanks everyone for answering, I get it now after reading it several times. It is basically just saying that k is equal to the length of walks. so if K = 2, then the length of walks is two, if K=3 then length of walks is 3 so length of walks can be less or greater than k. – sam786 Apr 16 '18 at 15:12

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