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 \(U_{m, n}\).