A New Family of Algebraically Defined Graphs with Small Automorphism Group

Let p be an odd prime, q = p, e ≥ 1, and F = Fq denote the finite field of q elements. Let f : F → F and g : F → F be functions, and let P and L be two copies of the 3-dimensional vector space F. Consider a bipartite graph ΓF(f, g) with vertex partitions P and L and with edges defined as follows: for every (p) = (p1, p2, p3) ∈ P and every [l] = [l1, l2, l3] ∈ L, {(p), [l]} = (p)[l] is an edge in ΓF(f, g) if p2 + l2 = f(p1, l1) and p3 + l3 = g(p1, p2, l1). The following question appeared in Nassau [9]: Given ΓF(f, g), is it always possible to find a function h : F → F such that the graph ΓF(f, h) with the same vertex set as ΓF(f, g) and with edges (p)[l] defined in a similar way by the system p2 + l2 = f(p1, l1) and p3 + l3 = h(p1, l1), is isomorphic to ΓF(f, g) for infinitely many q? In this paper we show that the answer to the question is negative and the graphs ΓFp(p1l1, p1l1p2(p1 + p2 + p1p2)) provide such an example for p ≡ 1 (mod 3). Our argument is based on proving that the automorphism group of these graphs has order p, which is the smallest possible order of the automorphism group of graphs of the form ΓF(f, g).