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)=(Y²−f(X))(λY²−f(X)) with f(X) a polynomial over F_q and λ 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.