Non-inverse signed graph of a group

Abstract

Let $G$ be a group with binary operation $\ast$. The non-inverse graph (in short, $i^*$-graph) of $G$, denoted by $\Gamma$, is a simple graph with vertex set consisting of elements of $G$ and two vertices $x, y \in \Gamma$ are adjacent if $x$ and $y$ are not inverses of each other. That is, $x- y$ if and only if $x\ast y \neq i_G \neq y \ast x$, where $i_G$ is the identity element of $G$. In this paper, we extend the study of $i^\ast$-graphs to signed graphs by defining $i^\ast$-signed graphs. We characterize the graphs for which the $i^\ast$-signed graphs and negated $i^\ast$-signed graphs are balanced, sign-compatible, consistent and $k$-clusterable. We also obtain the frustration index of the $i^\ast$-signed graph. Further, we characterize the homogeneous non-inverse signed graphs and study the properties like net-regularity and switching equivalence.