A note on the spectrum of linearized Wenger graphs

Let F_q be a finite field of order q=p^e, where p is a positive prime. For m≥1, let P and L be two copies of F_q^m+1. To each m-tuple g=(g₂,…,g_m+1) of polynomials in F_q[x,y], we consider the bipartite graph W_q(g). The vertex set V of W_q(g) is P∪L. The edge set E of W_q(g) consists of (p,l)∈P×L satisfying p₂+l₂=g₂(p₁,l₁),p₃+l₃=g₃(p₁,l₁),…,p_m+1+l_m+1=g_m+1(p₁,l₁),where p=(p₁,p₂,…,p_m+1)∈P and l=(l₁,l₂,…,l_m+1)∈L. W_q(g) is called linearized Wenger graph when g=(xy,x^py,…,x^pm−1y). In this paper, we determine the eigenvalues of linearized Wenger graph and their multiplicities in the case of m<e, which is an open problem put forward by Cao et al. (2015).