Few Families of Harmonic Mean Graphs

Abstract

A graph $G=(V,E)$ with $p$ vertices and $q$ edges is called a Harmonic mean graph if it is possible to label the vertices $v\in V$ with distinct labels $f(v)$ from $1,2,\dots ,q+1$ in such a way that when each edge $e=uv$ is labeled with $f(e=uv)=\lceil \frac{2f(u)f(v)}{f(u)+f(v)}\rceil$ or $\lfloor \frac{2f(u)f(v)}{f(u)+f(v)}\rfloor$, then the edge labels are distinct. In this case, $f$ is called a Harmonic mean labeling of $G$. In this paper, we investigate some new families of Harmonic mean graphs.