Radio Labeling Associated with a Class of Commutative Rings Using Zero-Divisor Graph

Abstract

Graph labeling is useful in networks because each transmitter has a different transmission capacity to send or receive wired or wireless links. An interference of signals can occur when transmitters that are close together receive close frequencies. This problem has been modeled mathematically in the radio labeling problem on graphs, where vertices represent transmitters and edges indicate closeness of the transmitters. For this purpose, each vertex is labeled with a unique positive integer, and to minimize the interference, the difference between maximum and minimum used labels has to be minimized. A radio labeling for a graph is a function from the set of vertices to the set of positive integers satisfying the condition , where is the shortest distance between two distinct vertices , and is the diameter of the graph The minimum span of a radio labeling for is called the radio number of Because the problem of finding radio labeling appears to be difficult in general, many particular cases have been studied. Let be a commutative ring with nonzero identity, and its set of (nonzero) zero-divisors. The zero-divisor graph of a ring is the graph with vertex set and edge set . In this paper, we investigate the radio number for an associated zero-divisor graph, . The study provides some combinatorial properties associated with commutative rings and can be useful for the structures of network communication problems.