2-Vertex self switching of umbrella graph

Abstract

By a graph $G=(V, E)$ we mean a finite undirected graph without loops or multiple edges. Let $G$ be a graph and $\sigma \subseteq V$ be a non-empty subset of $V$. Then $\sigma$ is said to be a self switching of $G$ if and only if $G \cong G^\sigma$. It can also be referred to as $|\sigma|$-vertex self-switching. The set of all self switching of the graph $G$ with cardinality $k$ is represented by $S_k(G)$ and its cardinality by $s s_k(G)$. A vertex $v$ of a graph $G$ is said to be self vertex switching if $G \cong G^v$. The set of all self vertex switchings of $G$ is denoted by $\operatorname{SS}_1(G)$ and its cardinality is given by $s s_1(G)$. If $|\sigma|=2$, we call it as a 2-vertex self switching. The set of all 2-vertex switchings of $G$ is denoted by $\operatorname{SS}_2(G)$ and its cardinality is given by $s s_2(G)$. In this paper we find the number of 2-vertex self switching vertices for the umbrella graph $U_{m, n}$.