\(k\)-Odd Edge Mean Labeling of Some Basic Graphs

Abstract

A \(\left(p,q\right)\) graph \(G\) is said to have a \(k\)-odd edge mean labeling \((k\ge 1)\), if there exists an injection \(f\) from the edges of \(G\) to \(\{0,1,2,3,\ldots,2k+2p-3\}\) such that the induced map \(f^*\) defined on \(V\) by \(f^{*} (v)=\left\lceil \frac{\sum f(vu) }{\deg (v)} \right\rceil\) is a bijection from \(V\) to \(\{2k-1,2k+1,2k+3,\ldots, 2k+2p-3\}\). A graph that admits a \(k\)-odd edge mean labeling is called a \(k\)-odd edge mean graph. In this paper, we have introduced \(k\)-odd edge mean labeling and we have investigated the same labeling for basic graphs like path and star.  Also we have examined the existence and non existence of cycles.