Edge Mostar Indices of Cacti Graph With Fixed Cycles

Abstract

Topological invariants are the significant invariants that are used to study the physicochemical and thermodynamic characteristics of chemical compounds. Recently, a new bond additive invariant named the Mostar invariant has been introduced. For any connected graph $\mathscr{H}$ , the edge Mostar invariant is described as $M{o}_e\left(\mathscr{H}\right)=\sum _{gx\in E\left(\mathscr{H}\right)}\left|m_{\mathscr{H}}\left(g\right)-m_{\mathscr{H}}\left(x\right)\right|$ , where $m_{\mathscr{H}}\left(g\right)\left(\text{or}{m}_{\mathscr{H}}\left(x\right)\right)$ is the number of edges of $\mathscr{H}$ lying closer to vertex g (or x) than to vertex x (or g). A graph having at most one common vertex between any two cycles is called a cactus graph. In this study, we compute the greatest edge Mostar invariant for cacti graphs with a fixed number of cycles and n vertices. Moreover, we calculate the sharp upper bound of the edge Mostar invariant for cacti graphs in $ℭ\left(n,s\right)$ , where s is the number of cycles.

Keywords: Mostar invariant; cacti graphs; edge Mostar invariant; graph theory; topological invariants.