An infinite series of regular edge‐ but not vertex‐transitive graphs

Abstract

Abstract. Let n be an integer and q be a prime power. Then for any 3 ≤ n ≤ q − 1, or n = 2 and q odd, we construct a connected q-regular edgebut not vertex-transitive graph of order 2q n+1 . This graph is defined via a system of equations over the finite field of q elements. For n = 2 and q = 3, our graph is isomorphic to the Gray graph.

Linearized Wenger graphs

“…The original definition was introduced by Wenger [21] for p-regular bipartite graphs and then was extended by Lazbnik and Ustimenko [11] for arbitrary prime power q. An equivalent representation of these graphs appeared later in Lazebnik and Viglione [13] and then a more general class of graphs was defined in [19], on which we concentrate in this paper.…”

“…The original definition was introduced by Wenger [21] for p-regular bipartite graphs and then was extended by Lazbnik and Ustimenko [11] for arbitrary prime power q. An equivalent representation of these graphs appeared later in Lazebnik and Viglione [13] and then a more general class of graphs was defined in [19], on which we concentrate in this paper.Let m ≥ 1 be a positive integer and g k (x, y) ∈ F q [x, y] for 2 ≤ k ≤ m + 1. Let P = F m+1 q and L = F m+1 q be two copies of the (m + 1)-dimensional vector space over F q , which are called the point set and the line set respectively.…”

Isomorphism criterion for monomial graphs

“…For the definition of these graphs, their origins and numerous applications, see [7] and references therein. For more recent applications, see [2]- [6] and [8]. Though most constructions in the papers mentioned above were motivated by particular applications, the study of general properties of these graphs, initiated in [7] and continued in [8], turned out to be an exciting area of research.…”

On the Diameter of Wenger Graphs

“…For prime powers q, they appeared in a paper by Lazebnik and Ustimenko [2], as an example of graphs based on root systems. It was later shown by Lazebnik and Viglione in [3] that W n (q) is a semisymmetric (i.e. edge-, but not vertex-transitive) graph for any n ≥ 3 and q ≥ 3, or n = 2 and q odd; indeed, the particular form of the equations above comes from that paper.…”

“…Viglione ( ) Department of Mathematics, Kean University, Union, NJ 07083, USA e-mail: rviglion@cougar.kean.edu When (p) is adjacent to [l], we write (p) ∼ [l]. The adjacency conditions above imply that a neighbor of a given point or line is uniquely determined by its first coordinate (see [3]). Hence in such cases we may denote (p 1 , p 2 , .…”