Geometric aspects of 2-walk-regular graphs

Abstract

A $t$-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most $t$. Such graphs generalize distance-regular graphs and $t$-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance regular graphs. We will generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's multiplicity bound and Terwilliger's analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show

Partially metric association schemes with a multiplicity three

“…Yamazaki's lemma. The following result was shown by Yamazaki [23] and is analogous to the result that a cubic 1-walk-regular graph is 2-walk-regular [6]. For convenience and because the terminology in [23] is different, we give a proof of this result.…”

“…Also other kinds of product constructions for association schemes are possible, giving rise to many association schemes with a multiplicity three, and suggesting that classifying all association schemes with a multiplicity three may be impossible. Likewise, multiplicity two may be too hard, although in this case our result in [6,Prop. 6.5] should be useful.…”

“…is the valency of Γ. An illustrating example of a 3-partially metric scheme graph is given by the so-called flag graph of the 11-point biplane; see Figure 1 in [6] or [7] for the corresponding "relation-distribution diagram". Such a diagram is similar as the distance-distribution diagram of a distance-regular graph.…”

“…It is isomorphic to the generalized Petersen graph GP (24, 5) and has spectrum [22] and generates an association scheme with scheme graph having relation-distribution diagram as in Figure 6, where as before, the cosines for eigenvalue 1 are included; once more these cosines follow from the relation distribution diagram using (6).…”

“…If k = 3, then we have one of the cases (i)-(vii) by Theorem 4.3. If k > 3, then a 1 > 0 by [6,Lemma 6.7].…”