$k$-geometric graphs

TL;DR

This paper introduces the concept of $k$-geometric mean graphs, explores new classes of such graphs, and investigates their properties and labelings, extending the existing theory of geometric mean graphs.

Contribution

It defines $k$-geometric mean labeling, proves certain classes of graphs are $k$-geometric mean graphs, and studies join operations preserving geometric mean labelings.

Findings

Identified new classes of $k$-geometric mean graphs.

Proved some graph classes admit $k$-geometric mean labelings.

Analyzed join operations maintaining geometric mean labelings.

Abstract

A finite, simple and undirected graph $G = (V, E)$ with $p$ vertices and $q$ edges is said to be a $k$-geometric mean graph for a positive integer $k$ if there is an injection $\psi :V(G)\to \{k,k+1,\dots,k+q\}$ such that, when each edge $uv\in E(G)$ is assigned the label $\lfloor\sqrt{\psi(u)\psi(v)}\rfloor$ or $\lceil\sqrt{\psi(u)\psi(v)}\rceil$, the resulting edge label set is $\{k,k+1,...,k+q-1\}$ and $\psi$ is called a \emph{$k$-geometric mean labeling} of $G$. The special case $k=1$, a $1$-geometric mean labeling is called a geometric mean labeling, and a $1$-geometric mean graph is called a geometric mean graph. In this paper, we present new classes of geometric mean graphs. Then we introduce $k$-geometric mean labeling and prove some classes of graphs are $k$-geometric mean. We also study some classes of finite join of graphs that admit geometric mean labeling.