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 $\mathrm{\mathscr{H}}$, the edge Mostar invariant is described as $M{o}_e\left(\mathrm{\mathscr{H}}\right)={\displaystyle \sum _{gx\in E\left(\mathrm{\mathscr{H}}\right)}\left|m_{\mathrm{\mathscr{H}}}\left(g\right)-m_{\mathrm{\mathscr{H}}}\left(x\right)\right|}$, where $m_{\mathrm{\mathscr{H}}}\left(g\right)\left(\text{or }{m}_{\mathrm{\mathscr{H}}}\left(x\right)\right)$ is the number of edges of $\mathrm{\mathscr{H}}$ 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 $\mathrm{ℭ}\left(n,s\right)$, where s is the number of cycles.