On the equational graphs over finite fields

In this paper, we generalize the notion of functional graph. Specifically, given an equation E(X, Y) = 0 with variables X and Y over a finite field F-q, of odd characteristic, we define a digraph by choosing the elements in F-q as vertices and drawing an edge from x to y if and only if E(x, y) = 0. We call this graph as equational graph. In this paper, we study the equational graph when choosing E(X, Y) = (Y-2-f (X))(lambda Y-2 - f (X)) with f (X) a polynomial over F-q and lambda a non-square element in F-q. We show that if f is a permutation polynomial over F-q, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials f of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected. (C) 2020 Elsevier Inc. All rights reserved.