I am reading "Algebraic Graph Theory" by Biggs 1974. In the section symmetry and regularity of graphs in page 136, it is defined a set $\{A_0,A_1,...,A_d\}$ of $n\times n$ matrices as follows: $(A_h)_{rs}$ :when $d(v_r,v_s)=h$ entry $r,s$ in matrix $A_h$ is 1,otherwise it is zero. Then $A_0=I$, $A_1$ is the usual adjacency matrix $A$ of graph, and we notice that $A_0+A_1+...+A_d=J$. ($d$= diameter of graph and $J=[1]_{n\times n}$). I would like to know about eignvalue of matrix $A_h$. are there resources on the eigenvalues of this matrix? (Even for special cases) Thank you in advance.
eignvalues of hth distance matrix
3
$\begingroup$
combinatorics
eigenvalues-eigenvectors
algebraic-graph-theory
-
1The most studied case are the distance regular graphs (is this what is specifically discussed in Biggs?). In this case the eigenvalues are known (in terms of the parameters of the graph). You can find this in Godsil & Royle, or just google distance regular graphs eigenvalues to find some results. – 2017-02-14