On Almost Distance-Regular Graphs
CentER Discussion Paper Series No. 2010-28
Posted: 24 Mar 2010 Last revised: 30 Mar 2010
Date Written: March 15, 2010
Abstract
Distance-regular graphs have been a key concept in Algebraic Combinatorics and have given place 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 m-walk-regularity. Another studied concept is that of m-partial distance-regularity, or informally, distance-regularity up to distance m. Using eigen- values of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of (l, m)-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.
Keywords: Distance-Regular Graph, Walk-Regular Graph, Eigenvalues, Predistance Polynomial
JEL Classification: C0
Suggested Citation: Suggested Citation