A note on the spectrum of linearized Wenger graphs

Let Fq be a finite field of order q=pe, where p is a positive prime. For m≥1, let P and L be two copies of Fqm+1. To each m-tuple g=(g2,…,gm+1) of polynomials in Fq[x,y], we consider the bipartite graph Wq(g). The vertex set V of Wq(g) is P∪L. The edge set E of Wq(g) consists of (p,l)∈P×L satisfying p2+l2=g2(p1,l1),p3+l3=g3(p1,l1),…,pm+1+lm+1=gm+1(p1,l1),where p=(p1,p2,…,pm+1)∈P and l=(l1,l2,…,lm+1)∈L. Wq(g) is called linearized Wenger graph when g=(xy,xpy,…,xpm−1y). In this paper, we determine the eigenvalues of linearized Wenger graph and their multiplicities in the case of m