On Construction of Even Vertex Odd Mean Graphs

Abstract

A graph $G$ with $p$ vertices and $q$ edges is said to have an \linebreak even vertex odd mean labeling if there exists an injective function $f:V(G)\rightarrow\{0,2,4,\dots,2q-2,2q\}$ such that the induced map $f^*:E(G)\rightarrow\{1,3,5,\dots, 2q-1\}$ defined by $f^*(uv)=\frac{f(u)+f(v)}{2}$ 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 discuss the construction of two kinds of even vertex odd mean graphs. Here we prove that $(P_n;S_1)$ for $n\geq 1,$ $(P_{2n};S_m)$ for $m\geq 2, n\geq 1,$ $(P_m;C_n)$ for $n\equiv 0(mod \ 4),$ $m\geq 1,$ $[P_m;C_n]$ for $n\equiv 0(mod \ 4),$ $m\geq 1$ and $[P_m;C_n^{(2)}]$ for $n\equiv 0(mod \ 4),$ $m\geq 1$ are even vertex odd mean graphs.