Super geometric mean graphs

Abstract

Let $G$ be a graph and $f:V\left(G\right)\to \{1,2,3,\dots ,p+q\}$ be an injection. For each edge $uv$, the induced edge labeling $f^{*}$ is defined as $f^{*}\left(uv\right)=\left[\sqrt{f\left(u\right)f\left(v\right)}\right]$. Then $f$ is called a super geometric mean labeling if $f\left(V\left(G\right)\right){\cup }^{\text{}}\left\{f^{*}\left(uv\right);uv\in E\left(G\right)\right\}=\{1,2,3,\dots ,p+q\}$. A graph that admits a super geometric mean labeling is called a super geometric mean graph. In this paper, we discuss the super geometric meanness of union of any paths, union of any cycles of order $\ge 5$, the graph $P_n\odot S_m$ for $m\le 3$, square graph, total graph, the $H$-graph, the graph $G\odot S_1$ and $G\odot S_2$ for any $H$-graph $G$, subdivision of $K_{1,3}$ and some chain graphs.