vertextovertex

Vertextovertex is a term used in graph theory and theoretical computer science to describe a constructed relation that links vertices via a rule or function rather than solely by adjacency. The name emphasizes its role as a relation that maps one vertex to another, potentially encoding a transformation or process.

In a directed graph G = (V, E), a vertextovertex relation can be understood in two common ways. One interpretation treats it as a subset R of V × V, where each pair (u, v) in R indicates that v is the result of applying a rule to u. A related interpretation views the relation as a function f: V → V (possibly partial or non-deterministic), in which the vertextovertex relation consists of the pairs (u, f(u)). Both viewpoints capture the idea that the connection from one vertex to another is determined by a rule rather than by a fixed edge alone.

Properties of vertextovertex relations vary. They may be deterministic or non-deterministic, total or partial, and they can form functional graphs when viewed as a function. Such relations are often used to model state transitions, automata, or graph rewriting rules, where the move from one vertex to another embodies a computational step or transformation.

Applications and examples. In a state machine, a vertextovertex may represent the transition function between states. In network dynamics, it can model propagation rules where each vertex maps to a successor. A simple example is a four-vertex set with a function f cycling three vertices and leaving one fixed, illustrating a basic functional graph.

See also: vertex, edge, graph, relation, functional graph.