A Novel Radio Geometric Mean Algorithm for a Graph

Abstract

Radio antennas switch signals in the form of radio waves using different frequency bands of the electromagnetic spectrum. To avoid interruption, each radio station is assigned a unique number. The channel assignment problem refers to this application. A radio geometric mean labeling of a connected graph G is an injective function h from the vertex set, V(G) to the set of natural numbers N such that for any two distinct vertices x and y of G, h(x)·h(y)≥diam+1−d(x,y). The radio geometric mean number of h, rgmn(h), is the maximum number assigned to any vertex of G. The radio geometric mean number of G, rgmn(G) is the minimum value of rgmn(h), taken over all radio geometric mean labeling h of G. In this paper, we present two theorems for calculating the exact radio geometric mean number of paths and cycles. We also present a novel algorithm for determining the upper bound for the radio geometric mean number of a given graph. We verify that the upper bounds obtained from this algorithm coincide with the exact value of the radio geometric mean number for paths, cycles, stars, and bi-stars.