Even vertex odd mean labeling of transformed trees

Abstract

A graph $G$ with $p$ vertices and $q$ edges is said to have an even vertex odd mean labeling if there exists an injective function $f:V\left(G\right)\to \{0,2,4,\dots ,2q-2,2q\}$ such that the induced map $f^{*}E\left(G\right)\to \{1,3,5,\dots ,2q-1\}$ defined by $f^{*}\left(uv\right)=\frac{f\left(u\right)+f(v)}{2}$ is a bijection. A graph that admits an even vertex odd mean labeling is called an even vertex odd mean graph. In this paper, we prove that every $T_p$-tree $T$, $T@P_n$, $T@2P_n$, $T\odot \overline{K_n}$, $T@c_n$ and $T\widehat{○}{C}_n$ are even vertex odd mean graphs.