On the Powers of Signed Graphs

TL;DR

This paper introduces the concept of the nth power of signed graphs, explores their properties, and characterizes balanced power signed graphs, providing conditions for uniqueness and balance.

Contribution

It defines the nth power of signed graphs, analyzes their properties, and characterizes balanced power signed graphs, including conditions for uniqueness.

Findings

Defined the nth power of signed graphs

Provided necessary and sufficient conditions for power graph uniqueness

Characterized balanced power signed graphs

Abstract

A signed graph is an ordered pair $\Sigma=(G,\sigma),$ where $G=(V,E)$ is the underlying graph of $\Sigma$ with a signature function $\sigma:E\rightarrow \{1,-1\}$. In this article, we define $n^{th}$ power of a signed graph and discuss some properties of these powers of signed graphs. As we can define two types of signed graphs as the power of a signed graph, necessary and sufficient conditions are given for an $n^{th}$ power of a signed graph to be unique. Also, we characterize balanced power signed graphs.