On almost distance-regular graphs

Abstract

Distance-regular graphs are a key concept in Algebraic Combinatorics\ud and have given rise to several generalizations, such as association\ud schemes. Motivated by spectral and other algebraic characterizations\ud of distance-regular graphs, we study ‘almost distanceregular\ud graphs’. We use this name informally for graphs that share\ud some regularity properties that are related to distance in the graph.\ud For example, a known characterization of a distance-regular graph\ud is the invariance of the number of walks of given length between\ud vertices at a given distance, while a graph is called walk-regular if\ud the number of closed walks of given length rooted at any given vertex\ud is a constant. One of the concepts studied here is a generalization\ud of both distance-regularity and walk-regularity called m-walkregularity.\ud Another studied concept is that of m-partial distanceregularity\ud or, informally, distance-regularity up to distance m. Using\ud eigenvalues of graphs and the predistance polynomials, we discuss\ud and relate these and other concepts of almost distance-regularity,\ud such as their common generalization of ( ,m)-walk-regularity. We\ud introduce the concepts of punctual distance-regularity and punctual\ud walk-regularity as a fundament upon which almost distanceregular\ud graphs are built. We provide examples that are mostly\ud taken from the Foster census, a collection of symmetric cubic\ud graphs. Two problems are posed that are related to the question of\ud when almost distance-regular becomes whole distance-regular. We\ud also give several characterizations of punctually distance-regular\ud graphs that are generalizations of the spectral excess theorem.Postprint (published version

The spectral excess theorem for distance-regular graphs having distance- d graph with fewer distinct eigenvalues

“…, d, in which case they turn out to be the distance polynomials. In fact, we have the following strongest proposition, which is a combination of results in [14,7]. …”

Partially metric association schemes with a multiplicity three

“…If the association scheme (X, R) is t-partially metric, then the corresponding scheme graph Γ of R 1 is called a t-partially metric scheme graph. This scheme graph is t-partially distance-regular in the sense of [7], and even stronger, it is twalk-regular. Thus, the intersection numbers of Γ are well-defined for 0 ≤ i ≤ t. In this case we have…”

Edge-distance-regular graphs

“…Moreover, Dalfó, Van Dam and Fiol [6] recently obtained some new characterizations of distanceregular graphs in terms of the cospectrality of their perturbed graphs. Also, distanceregular graphs have given rise to several generalizations, such as association schemes (see Brouwer and Haemers [3]) and almost distance-regular graphs [7]. When we look at the distance partition of the graph from each of its edges instead of its vertices, we arrive, in a natural way, to the concept of edge-distance-regularity.…”