On the spectrum of Wenger graphs

Let q=peq=pe, where p   is a prime and e⩾1e⩾1 is an integer. For m⩾1m⩾1, let P and L   be two copies of the (m+1)(m+1)-dimensional vector spaces over the finite field FqFq. Consider the bipartite graph Wm(q)Wm(q) with partite sets P and L   defined as follows: a point (p)=(p1,p2,…,pm+1)∈P(p)=(p1,p2,…,pm+1)∈P is adjacent to a line [l]=[l1,l2,…,lm+1]∈L[l]=[l1,l2,…,lm+1]∈L if and only if the following m   equalities hold: li+1+pi+1=lip1li+1+pi+1=lip1 for i=1,…,mi=1,…,m. We call the graphs Wm(q)Wm(q) Wenger graphs. In this paper, we determine all distinct eigenvalues of the adjacency matrix of Wm(q)Wm(q) and their multiplicities. We also survey results on Wenger graphs.