Strongly walk-regular graphs

Abstract

We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an $\ell >1$ such that the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly $\ell$-walk-regular for all $\ell$, whereas the graphs from the fourth family are $\ell$-walk-regular for every odd $\ell$. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly $\ell$-walk-regular for even $\ell$. We will characterize the case that regular four-eigenvalue graphs are strongly $\ell$-walk-regular for every odd $\ell$, in terms of the eigenvalues. There are several examples of infinite families of such graphs. We will show that every other regular four-eigenvalue graph can be strongly $\ell$-walk-regular for at most one $\ell$. There are several examples of infinite families of such graphs that are strongly 3-walk-regular. It however remains open whether there are any graphs that are strongly $\ell$-walk-regular for only one particular $\ell$ different from 3

Three-weight codes, triple sum sets, and strongly walk regular graphs

“…Since the seminal article of Delsarte [6], there is a well-known interplay between two-weight codes and strongly regular graphs (SRG) via the coset graph of the dual code [1,2,19]. Strongly walk-regular graphs (SWRG) were introduced in [20] as a generalization of strongly regular graphs. Instead of a regularity condition bearing on paths of length 2, the notion of SWRG demands a regularity on paths of length > 1.…”

“…The aim of this note is to connect these two notions by means of a third one: the coset graph of a projective code [1]. The graphical approach of [20] makes several results on multiple sum sets of [4,11,12] more transparent. Sometimes, new results are obtained (Theorem 6).…”

A Complete Characterization of Plateaued Boolean Functions in Terms of Their Cayley Graphs

“…Fiol and Garriga [9] introduced t-walk-regular graphs as a generalization of both distanceregular and walk-regular graphs. We call a graph Γ = (V, E) a t-walk-regular (assuming Γ has its diameter at least t) if the number of walks of every given length between two vertices x, y ∈ V depends only on the distance between x, y, provided it is ≤ t. In [8], van Dam and Omidi generalized this concept and called Γ a strongly -walk-regular with parameters (σ , µ , ν ) if there are σ , µ , ν walks of length between every two adjacent, every two non-adjacent, and every two identical vertices, respectively. Certainly, every strongly regular graph of parameters (v, r, e, d) is a strongly 2-walk-regular graph with parameters (e, d, r).…”

Directed strongly walk-regular graphs

“…Then ℓ is odd and it follows by Proposition 5.5 that Γ has four distinct eigenvalues k > θ 1 > θ 2 > θ 3 . The theory that was developed for strongly walk-regular undirected graphs with four eigenvalues in [6,Section 4] can almost literally be extended to this case. In particular, we obtain the following results.…”

“…In [6], we introduced the concept of "strongly walk-regular graphs" as a generalization of strongly regular graphs. Here we generalize this concept to directed graphs: a digraph is called strongly ℓ-walk-regular with ℓ > 1 if the number of walks of length ℓ from a vertex to another vertex depends only on whether the first vertex is the same as, adjacent to, or not adjacent to the second vertex.…”