Graph representation functions computable by finite automata

We consider a simple model for representing a graph in computer memory in which every vertex is assigned a word in a finite alphabet - vertex code - and the adjacency of two vertices is a function Ψ of their codes. The function Ψ is called the representation function. We say that Ψ is universal if a Ψ-representation exists for every simple graph G. In this paper, we study representation functions computable by automata with two states. The main result is a criterion characterizing the universal functions. In the case of binary alphabet, we provide some bounds on the dimension of minimum Ψ-representation