Geometric aspects of 2-walk-regular graphs

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

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Triangle-free distance-regular graphs with an eigenvalue multiplicity equal to their valency and diameter 3

AbstractIn this paper, triangle-free distance-regular graphs with diameter 3 and an eigenvalue θ with multiplicity equal to their valency are studied. Let Γ be such a graph. We first show that θ=−1 if and only if Γ is antipodal. Then we assume that the graph Γ is primitive. We show that it is formally self-dual (and hence Q-polynomial and 1-homogeneous), all its eigenvalues are integral, and the eigenvalue with multiplicity equal to the valency is either second largest or the smallest.Let x,y∈VΓ be two adjacent vertices, and z∈Γ2(x)∩Γ2(y). Then the intersection number τ2≔|Γ(z)∩Γ3(x)∩Γ3(y)| is independent of the choice of vertices x, y and z. In the case of the coset graph of the doubly truncated binary Golay code, we have b2=τ2. We classify all the graphs with b2=τ2 and establish that the just mentioned graph is the only example. In particular, we rule out an infinite family of otherwise feasible intersection arrays

Geometric aspects of 2-walk-regular graphs

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

Notes on simplicial rook graphs

The simplicial rook graph ${\rm SR}(m,n)$ is the graph of which the vertices are the sequences of nonnegative integers of length $m$ summing to $n$, where two such sequences are adjacent when they differ in precisely two places. We show that ${\rm SR}(m,n)$ has integral eigenvalues, and smallest eigenvalue $s = \max (-n, -{m \choose 2})$, and that this graph has a large part of its spectrum in common with the Johnson graph $J(m+n-1,n)$. We determine the automorphism group and several other properties