Locating parameters of the total graph of $\Gamma(\mathbb{Z}_{2^{n}p^{m}})$

Abstract

Let $G = (V, E)$ be a connected graph and $d(x, y)$ be the distance between the vertices $x$ and $y$ in $V$. A subset of vertices $W = \{w_1, w_2,\ldots, w_k\}$ is called a locating set for $G$ if for every two distinct vertices $x, y\in V$, there is a vertex $w_i \in W$ such that $d(x, w_i) \neq d(y, w_i)$ for $i = 1, 2, \ldots, k$. A locating set containing the minimum number of vertices is called a metric basis for $G$ and the number of vertices in a metric basis is its metric dimension, denoted by ${\rm dim}(G)$. For a commutative ring $R$, let $\Gamma(R)$ denote the total graph of $R$, the undirected graph with vertex set $R$ and two distinct vertices $x$ and $y$ being adjacent if and only if $x+y\in Z(R)$, where $Z(R)$ is the set of zero-divisors of $R$. In this paper, we compute some types of locating parameters, namely metric dimension, edge metric dimension, fractional and strong metric dimension for the total graph $\Gamma(\Bbb{Z}_{2^{n}p^{m}})$ where $p$ is a prime number greater than two and $m, n$ are positive integers.