A Note on the Isomorphism Problem for Monomial Digraphs

Let p be a prime e be a positive integer, q=pe, and let Fq denote the finite field of q elements. Let m,n, 1≤m,n≤q−1, be integers. The monomial digraph D=D(q;m,n) is defined as follows: the vertex set of D is Fq2, and ((x1,x2),(y1,y2)) is an arc in D if x2+y2=x1my1n. In this note we study the question of isomorphism of monomial digraphs D(q;m1,n1) and D(q;m2,n2). Several necessary conditions and several sufficient conditions for the isomorphism are found. We conjecture that one simple sufficient condition is also a necessary one.