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Characterizing Distance-Regularity of Graphs by the Spectrum
AMS classifications: 05E30; 05B20distance-regular graphs;eigenvalues;cospectral graphs
On almost distance-regular graphs
Distance-regular graphs are a key concept in Algebraic Combinatorics and have
given rise to several generalizations, such as association schemes. Motivated
by spectral and other algebraic characterizations of distance-regular graphs,
we study `almost distance-regular graphs'. We use this name informally for
graphs that share some regularity properties that are related to distance in
the graph. For example, a known characterization of a distance-regular graph is
the invariance of the number of walks of given length between vertices at a
given distance, while a graph is called walk-regular if the number of closed
walks of given length rooted at any given vertex is a constant. One of the
concepts studied here is a generalization of both distance-regularity and
walk-regularity called -walk-regularity. Another studied concept is that of
-partial distance-regularity or, informally, distance-regularity up to
distance . Using eigenvalues of graphs and the predistance polynomials, we
discuss and relate these and other concepts of almost distance-regularity, such
as their common generalization of -walk-regularity. We introduce the
concepts of punctual distance-regularity and punctual walk-regularity as a
fundament upon which almost distance-regular graphs are built. We provide
examples that are mostly taken from the Foster census, a collection of
symmetric cubic graphs. Two problems are posed that are related to the question
of when almost distance-regular becomes whole distance-regular. We also give
several characterizations of punctually distance-regular graphs that are
generalizations of the spectral excess theorem
On Almost Distance-Regular Graphs
2010 Mathematics Subject Classification: 05E30, 05C50;distance-regular graph;walk-regular graph;eigenvalues;predistance polynomial
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
Some Spectral and Quasi-Spectral Characterizations of Distance-Regular Graphs
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
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