Linearized Wenger graphs

Abstract

Abstract. Motivated by recent extensive studies on Wenger graphs, we introduce a new infinite class of bipartite graphs of the similar type, called linearized Wenger graphs. The spectrum, diameter and girth of these linearized Wenger graphs are determined. IntroductionLet F q be a finite field of order q such that p is prime and q = p e a prime power. All graph theory notions can be found in Bollobás [2]. Recently, a class of bipartite graphs called Wenger graphs which are defined over F q has attracted a lot of attention because of their nice graphical properties [5,11,12,16,18,19,20,21]. For example, the number of edges of these graphs meets the lower bound of Turán number of the cycle with length 4, 6, 10 [21]. 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. Let G = G q (g 2 , · · · , g m+1 ) = (V, E) be the graph with vertex set V = P ∪ L and the edge set E is defined as follow: there is an edge from a pointdenoted by P ∼ L (we force G to be a undirected graph by removing the arrows), if the following m equalities hold:If g k (x, y), k = 2, · · · , m + 1, are all monomials, the graph is called a monomial graph; see [6]. If g k (x, y) = x k−1 y, k = 2, · · · , m + 1, then the graph is just the original Wenger graph in [5], also denoted by W m (q). It was shown in [11] that the automorphism group of W m (q) acts transitively on each of P and L, and on the set of edges of W m (q). In other words, the graphs W m (q) are point-, line-, and edge-transitive. It is also shown that, see [12], W 1 (q) is

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