F-geometric mean labeling of graphs obtained by duplicating any edge of some graphs

Abstract

A function $f$ is called an $F$-geometric mean labeling of a graph $G=(V,E)$ with $p$ vertices and $q$ edges if $f:V\to \{1,2,3,\dots ,q+1\}$ is injective and the induced function $f^{*}:E\to \{1,2,3,\dots ,q\}$ defined as

$$f^{*}(uv)=\lfloor \sqrt{f(u)f(v)}\rfloor ,\text{ for all }uv\in E,$$

is bijective. A graph that admits an $\text{F}$-geometric mean labeling is called an F-geometric mean graph. In this paper we discuss the $\text{F}$-geometric meanness of graphs obtained by duplicating any edge of some graphs.