PMC-graphs derived from cycles

Abstract

The graph $G=(V,E)$ consists of $p$ vertices and $q$ edges. Let \begin{align*} \rho =\left\{ \begin{array}{ccc} \frac {p} {2},&\mbox {\rm $p$ is even} \\ \frac {p-1}{2}, &\mbox{\rm $p$ is odd,}\end{array}\right. \end{align*} and $\Gamma = \{\pm1,\pm2,\dots, \pm \rho \}$.  Consider a function $\Lambda: V\rightarrow \Gamma$ that allocates unique labels from $\Gamma$ to the various vertices of $V$ when $p$ is even and allocates a unique labels in $\Gamma$ to  $p-1$ vertices of $V$, repeating a label for the remaining one vertex when $p$ is odd. Then the labeling as mentioned above is called a pair mean cordial labeling (PMC-labeling) if for every edge $uv$ of $G$, there is a labeling $\frac{\Lambda(u)+\Lambda(v)}{2}$ if $\Lambda(u)+\Lambda(v)$ is even and $\frac{\Lambda(u)+\Lambda(v)+1}{2}$ if $\Lambda(u)+\Lambda(v)$ is odd such that $|\bar {\mathbb{S}}_{\Lambda_{1}}-\bar{\mathbb{S}}_{\Lambda_{1}^{c}}|\leq 1$ where $\bar{\mathbb{S}}_{\Lambda_{1}}$ and $\bar{\mathbb{S}}_{\Lambda_{1}^{c}}$ are denoted the number of edges labelled with $1$ and the number of edges not labelled with $1$, respectively. A graph $G$ that has a pair mean cordial labeling is called a pair mean cordial graph (PMC-Graph). This research paper examines the PMC-labeling behaviour of some graphs, like the pagoda graph, antiweb-gear graph, spherical graph, alternate triangular cycle, alternate quadrilateral cycle, balloon of triangular snake and balloon of quadrilateral snake.