A Novel Radio Geometric Mean Algorithm for a Graph

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, $V\left(G\right)$ to the set of natural numbers $N$ such that for any two distinct vertices $x$ and $y$ of $G$, $\sqrt{h\left(x\right)·h\left(y\right)}\ge diam+1-d\left(x,y\right)$. The radio geometric mean number of $h$, $rgmn\left(h\right)$, is the maximum number assigned to any vertex of $G$. The radio geometric mean number of G, $rgmn\left(G\right)$ is the minimum value of $rgmn\left(h\right)$, 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.