STUDY OF SOME GRAPHICAL PARAMETERS OF SOME GRAPH STRUCTURE

Abstract

Let $G$ be a simple graph. $L(G)$ is the Laplacian matrix of $G$ and $a(G)$ is the algebraic connectivity of the graph $G$. Pawar and Joshi [9] defined a simple undirected graph $P G(R)$ on a ring $R$ having vertex set $R$ and any two distinct vertices $a, b$ are adjacent if and only if $a \cdot b=0$ or $b a=0$ or $a+b$ is a unit element of $R$. Satyanarayana et al. [11] defined prime graph $P G(R)$ by taking all elements of the ring $R$ as vertices and two distinct vertices $a, b$ are adjacent if and only if $a R b=0$ or $b R a=0$. Another simple undirected graph $\square_2(R)$ is defined by Gupta [10] whose vertices are all the non-zero elements of the ring $R$ and two distinct vertices $a, b$ are adjacent if and only if $a \cdot b=0$ or $b a=0$ or $a+b$ is a zero divisor (includingzero). In this paper, we prove that for any prime $p$, the graphs $P G_1\left(Z_p \times Z_p\right)$ and $P G\left(Z_p \times Z_p\right)$ are planar if and only if $p=2,3$. Also, we prove that the graphs $\square_2\left(\mathcal{Z}_p \times \mathbb{Z}_p\right)$ and $P G\left(\mathbb{Z}_p \times \mathbb{Z}_p\right)$ are not Eulerian for any prime $p$. Here we discuss Laplacian and algebraic connectivity of $P G\left(\mathbb{Z}_p\right), \square_2\left(\mathbb{Z}_p\right), P G\left(\mathbb{Z}_p\right), P G\left(\mathbb{Z}_p \times \mathbb{Z}_p\right), \square_2\left(\mathbb{Z}_p \times \mathbb{Z}_p\right)$ and $P G\left(\mathbb{Z}_p \times \mathbb{Z}_p\right)$, where $p$ is a prime. We also find their girth, vertex connectivity and discuss planarity and Eulerian properties.

Received: May 9, 2022
Revised: June 2, 2022
Accepted: June 30, 2022