TOTAL MEAN CORDIAL LABELING OF SOME CYCLE RELATED GRAPHS

Ponraj, R.,Narayanan, S. Sathish The Korean Society for Computational and Applied M 2015 Journal of applied mathematics & informatics Vol.33 No.1

A Total Mean Cordial labeling of a graph G = (V, E) is a function $f:V(G){\rightarrow}\{0,1,2\}$ such that $f(xy)={\Large\lceil}\frac{f(x)+f(y)}{2}{\Large\rceil}$ where $x,y{\in}V(G)$, $xy{\in}E(G)$, and the total number of 0, 1 and 2 are balanced. That is ${\mid}ev_f(i)-ev_f(j){\mid}{\leq}1$, $i,j{\in}\{0,1,2\}$ where $ev_f(x)$ denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). If there is a total mean cordial labeling on a graph G, then we will call G is Total Mean Cordial. Here, We investigate the Total Mean Cordial labeling behaviour of prism, gear, helms.

PAIR MEAN CORDIAL LABELING OF GRAPHS OBTAINED FROM PATH AND CYCLE

PONRAJ, R.,PRABHU, S. The Korean Society for Computational and Applied M 2022 Journal of applied and pure mathematics Vol.4 No.3/4

Let a graph G = (V, E) be a (p, q) graph. Define $${\rho}\;=\;\{\array{{\frac{p}{2}}&p\text{ is even}\\{\frac{p-1}{2}}\;&p\text{ is odd,}}$$ and M = {±1, ±2, ⋯ ± 𝜌} called the set of labels. Consider a mapping λ : V → 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 λ(u) + λ(v) is even and $\frac{{\lambda}(u)+{\lambda}(v)+1}{2}$ if λ(u) + λ(v) is odd such that ${\mid}\bar{\mathbb{S}}_{{\lambda}_1}-\bar{\mathbb{S}}_{{\lambda}^c_1}{\mid}{\leq}1$ where $\bar{\mathbb{S}}_{{\lambda}_1}$ and $\bar{\mathbb{S}}_{{\lambda}^c_1}$ 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 of graphs which are obtained from path and cycle.

PAIR DIFFERENCE CORDIAL LABELING OF PETERSEN GRAPHS P(n, k)

R. PONRAJ,A. GAYATHRI,S. SOMASUNDARAM The Korean Society for Computational and Applied M 2023 Journal of applied and pure mathematics Vol.5 No.1/2

Let G = (V, E) be a (p, q) graph. Define $${\rho}=\{{\frac{2}{p}},\;{\text{{\qquad} if p is even}}\\{\frac{2}{p-1}},\;{{\text{if p is odd}}$$ and L = {±1, ±2, ±3, … , ±ρ} called the set of labels. Consider a mapping f : V ⟶ L by assigning different labels in L to the different elements of V when p is even and different labels in L 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 difference cordial labeling if for each edge uv of G there exists a labeling |f(u) - f(v)| such that ${\mid}{\Delta}_{f_1}-{\Delta}_{f^c_1}{\mid}{\leq}1$, where ${\Delta}_{f_1}$ and ${\Delta}_{f^c_1}$ respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate pair difference cordial labeling behaviour of Petersen graphs P(n, k) like P(n, 2), P(n, 3), P(n, 4).

4-TOTAL DIFFERENCE CORDIAL LABELING OF SOME SPECIAL GRAPHS

PONRAJ, R.,PHILIP, S. YESU DOSS,KALA, R. The Korean Society for Computational and Applied M 2022 Journal of applied and pure mathematics Vol.4 No.1/2

Let G be a graph. Let f : V (G) → {0, 1, 2, …, k-1} be a map where k ∈ ℕ and k > 1. For each edge uv, assign the label |f(u) - f(v)|. f is called k-total difference cordial labeling of G if |t_df (i) - t_df (j) | ≤ 1, i, j ∈ {0, 1, 2, …, k - 1} where t_df (x) denotes the total number of vertices and the edges labeled with x. A graph with admits a k-total difference cordial labeling is called k-total difference cordial graphs. In this paper we investigate the 4-total difference cordial labeling behaviour of shell butterfly graph, Lilly graph, Shackle graphs etc..

Pair difference cordial labeling of Petersen graphs P(n,k)

R. Ponraj,A. Gayathri,S. Somasundaram 한국전산응용수학회 2023 Journal of Applied and Pure Mathematics Vol.5 No.1

Let $G = (V, E)$ be a $(p,q)$ graph. Define \begin{equation*} \rho = \begin{cases} \frac{p}{2} ,& \text{if $p$ is even}\\ \frac{p-1}{2} ,& \text{if $p$ is odd}\\ \end{cases} \end{equation*}\\ and $L = \{\pm1 ,\pm2, \pm3 , \cdots ,\pm\rho\}$ called the set of labels.\\ \noindent Consider a mapping $f : V \longrightarrow L$ by assigning different labels in L to the different elements of V when p is even and different labels in L 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 difference cordial labeling if for each edge $uv$ of $G$ there exists a labeling $\left|f(u) - f(v)\right|$ such that $\left|\Delta_{f_1} - \Delta_{f_1^c}\right| \leq 1$, where $\Delta_{f_1}$ and $\Delta_{f_1^c}$ respectively denote the number of edges labeled with $1$ and number of edges not labeled with $1$. A graph $G$ for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate pair difference cordial labeling behaviour of Petersen graphs $P(n,k)$ like $P(n,2), P(n,3),P(n,4)$.

On pair mean cordial graphs

R. Ponraj,S. Prabhu 한국전산응용수학회 2023 Journal of Applied and Pure Mathematics Vol.5 No.3

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.

ON PAIR MEAN CORDIAL GRAPHS

R. PONRAJ,S. PRABHU The Korean Society for Computational and Applied M 2023 Journal of applied and pure mathematics Vol.5 No.3

Let a graph G = (V, E) be a (p, q) graph. Define $${\rho}=\{\array{{\frac{p}{2}} & \;\;p\text{ is even} \\ {\frac{p-1}{2}} & \;\;p\text{ is odd,}$$ and M = {±1, ±2, … ± ρ} called the set of labels. Consider a mapping λ : V → 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 λ(u) + λ(v) is even and ${\frac{{\lambda}(u)+{\lambda}(v)+1}{2}}$ if λ(u) + λ(v) is odd such that ${\mid}{\bar{{\mathbb{S}}}}_{\lambda}{_1}-{\bar{{\mathbb{S}}}}_{{\lambda}^c_1}{\mid}{\leq}1$ where ${\bar{{\mathbb{S}}}}_{\lambda}{_1}$ and ${\bar{{\mathbb{S}}}}_{{\lambda}^c_1}$ 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.

4-total mean cordial labeling of arrow graphs and shell graphs

R. Ponraj,S. Subbulakshmi 한국전산응용수학회 2023 Journal of Applied and Pure Mathematics Vol.5 No.5

In this paper we investigate the $4$-total mean cordial labeling behavior of arrow graphs, shell-Butterfly graph and graphs obtained by joining two copies of shell graphs by a path.

4-TOTAL MEAN CORDIAL LABELING OF ARROW GRAPHS AND SHELL GRAPHS

R. PONRAJ,S. SUBBULAKSHMI The Korean Society for Computational and Applied M 2023 Journal of applied and pure mathematics Vol.5 No.5

In this paper we investigate the 4-total mean cordial labeling behavior of arrow graphs, shell-Butterfly graph and graphs obtained by joining two copies of shell graphs by a path.

4-total difference cordial labeling of some special graphs

R. Ponraj,S. Yesu Doss Philip,R. Kala 한국전산응용수학회 2022 Journal of Applied and Pure Mathematics Vol.4 No.1

Let G be a graph. Let f:V(G)\to\{0,1,2, \ldots, k-1\} be a map where k \in \mathbb{N} and k>1. For each edge uv, assign the label \left|f(u)-f(v)\right|. f is called k-total difference cordial labeling of G if \left|t_{df}(i)-t_{df}(j)\right|\leq 1, i,j \in \{0,1,2,\ldots,k-1\} where t_{df}(x) denotes the total number of vertices and the edges labeled with x. A graph with admits a k-total difference cordial labeling is called k-total difference cordial graphs. In this paper we investigate the 4-total difference cordial labeling behaviour of shell butterfly graph, Lilly graph, Shackle graphs etc..