On the spectrum of wenger graphs

Abstract

Let q=pe, where p is a prime and e≥1 is an integer. For m≥1, let P and L be two copies of the (m+1)-dimensional vector spaces over the finite field Fq. Consider the bipartite graph Wm(q) with partite sets P and L defined as follows: a point (p)=(p1, p2,..., pm+1)∈P is adjacent to a line [l]=[l1, l2,..., lm+1]∈L if and only if the following m equalities hold: li+1+pi+1=lip1 for i=1,..., m. We call the graphs Wm(q) Wenger graphs. In this paper, we determine all distinct eigenvalues of the adjacency matrix of Wm(q) and their multiplicities. We also survey results on Wenger graphs. © 2014 Elsevier Inc.