Formal Expresions of Infinite Graphs and Their Families

We discuss some methods for rigorous expressions of infinite labeled directed graphs. In these methods, every" node of a graph is identified by a string of symbols from a finite alphabet, and edges are expressed as binary relations or functions among strings. We characterize these relations with some concepts in the theory of formal languages and automata, and from this characterization, we obtain various classes of graphs. Then inclusion relationships among these classes arc discussed. Although our methods originate in a ve~" naive idea, they appear to be suitable for both theoretical treatments and computer processings, so we ourselves regard this paper as a preliminary for various discussions that our formalization enables.