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The Laplacian spectral excess theorem for distance-regular graphs
The spectral excess theorem states that, in a regular graph G, the average
excess, which is the mean of the numbers of vertices at maximum distance from a
vertex, is bounded above by the spectral excess (a number that is computed by
using the adjacency spectrum of G), and G is distance-regular if and only if
equality holds. In this note we prove the corresponding result by using the
Laplacian spectrum without requiring regularity of G
A short proof of the odd-girth theorem
Recently, it has been shown that a connected graph with
distinct eigenvalues and odd-girth is distance-regular. The proof of
this result was based on the spectral excess theorem. In this note we present
an alternative and more direct proof which does not rely on the spectral excess
theorem, but on a known characterization of distance-regular graphs in terms of
the predistance polynomial of degree
Some spectral and quasi-spectral characterizations of distance-regular graphs
© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth. (C) 2016 Published by Elsevier Inc.Peer ReviewedPostprint (author's final draft
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