On the spectrum of Wenger graphs

Let q = p(e), 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 F-q. Consider the bipartite graph W-m(q) with partite sets P and L defined as follows: a point (p) = (p(1), p(2), ... , p(m+1)) is an element of P is adjacent to a line [l] = [l(1), l(2), ... , l(m+1)] is an element of L if and only if the following m equalities hold: l(i+1) + p(i+1) = l(i)p(1) for i = 1, ... , m. We call the graphs W-m(q) Wenger graphs. In this paper, we determine all distinct eigenvalues of the adjacency matrix of W-m(q) and their multiplicities. We also survey results on Wenger graphs. (C) 2014 Elsevier Inc. All rights reserved.