On pair mean cordial graphs

On pair mean cordial graphs

Let a graph $G=(V,E)$ be a $(p,q)$ graph. Define \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 $M=\{\pm 1,\pm 2,\dots \pm \rho\}$ called the set of labels. Consider a mapping $\lambda: V\rightarrow M $ by assigning different labels in $M$ to the different elements of $V$ when $p$ is even and different labels in $M$ to $p-1$ elements of $V$ and repeating a label for the remaining one vertex when $p$ is odd. The labeling as defined above is said to be a pair mean cordial labeling if for each edge $uv$ of $G$, there exists 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 $\bar{\mathbb{S}}_{\lambda_{1}^{c}}$ respectively denote the number of edges labeled with 1 and the number of edges not labeled with 1. A graph $G$ for which there exists a pair mean cordial labeling is called a pair mean cordial graph. In this paper, we investigate the pair mean cordial labeling behavior of few graphs including the closed helm graph, web graph, jewel graph, sunflower graph, flower graph, tadpole graph, dumbbell graph, umbrella graph, butterfly graph, jelly fish, triangular book graph, quadrilateral book graph.

Let a graph $G=(V,E)$ be a $(p,q)$ graph. Define \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 $M=\{\pm 1,\pm 2,\dots \pm \rho\}$ called the set of labels. Consider a mapping $\lambda: V\rightarrow M $ by assigning different labels in $M$ to the different elements of $V$ when $p$ is even and different labels in $M$ to $p-1$ elements of $V$ and repeating a label for the remaining one vertex when $p$ is odd. The labeling as defined above is said to be a pair mean cordial labeling if for each edge $uv$ of $G$, there exists 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 $\bar{\mathbb{S}}_{\lambda_{1}^{c}}$ respectively denote the number of edges labeled with 1 and the number of edges not labeled with 1. A graph $G$ for which there exists a pair mean cordial labeling is called a pair mean cordial graph. In this paper, we investigate the pair mean cordial labeling behavior of few graphs including the closed helm graph, web graph, jewel graph, sunflower graph, flower graph, tadpole graph, dumbbell graph, umbrella graph, butterfly graph, jelly fish, triangular book graph, quadrilateral book graph.