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On a version of the spectral excess theorem (1906.01307v1)

Published 4 Jun 2019 in math.CO

Abstract: Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency spectrum of $\Gamma$ and the arithmetic (or harmonic) mean of the numbers of vertices at distance $\le D-1$ of every vertex. The same results is proved for any graph by using its Laplacian matrix $L$ and corresponding spectrum. When $D=d$ we reobtain the spectral excess theorem characterizing distance-regular graphs.

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