# Self-switching of union of two complete graphs

## Main Article Content

## Abstract

By a graph *H* = (*V*, *E*), we mean a finite undirected graph without loops and multiple edges. Let *H* be a graph and *σ *⊆ *V* be a non–empty subset of *V*. *H*^{σ} is the graph obtained from *H* by removing all edges between *σ* and its complement *V*-*σ* and adding as edges all non-edges between *σ* and *V*-*σ*. Then *σ* is said to be a self-switching of *H* if *H* ≅ *H*^{σ}. It can also be referred to as *k*-vertex self-switching where *k* = |*σ*|. The set of all self-switchings of the graph *H* with cardinality *k* is represented by *SS*_{k}(*H*) and its cardinality by *ss*_{k}(*H*). A graph on *m *vertices in which each pair of distinct vertices are neighbors is called a complete graph and is denoted by *K*_{m}. *K*_{m}*∪ **K*_{n} is the union of two complete graphs and is disconnected. In this paper, we give necessary and sufficient conditions for *σ* to be a self-switching for the graph *H*=*K*_{m} *∪* *K*_{n} and using this, we find the cardinality *ss*_{k}(*H*).

### Downloads

## Article Details

*Gulf Journal of Mathematics*,

*16*(2), 196-203. https://doi.org/10.56947/gjom.v16i2.1880