On Weighted Vertex and Edge Mostar Index for Trees and Cacti with Fixed Parameter

Abstract

 It was introduced by Do$\check{s}$li$\acute{c}$ and Ivica et al. (Journal of Mathematical chemistry, 56(10) (2018): 2995--3013), as an innovative graph-theoretic topological identifier, the Mostar index is significant in simulating compounds thermodynamic properties in simulations, which is defined as sum of absolute values of the differences among $n_{u}(\mathrm{e}|\Omega)$ and $n_{v}(\mathrm{e}|\Omega)$ over all lines $\mathrm{e}=uv\in\Omega$, where $n_{u}(\mathrm{e}|\Omega)$ (resp. $n_{v}(\mathrm{e}|\Omega)$) is the collection of vertices of $\Omega$ closer to vertex $u$ (resp. $v$) than to vertex $v$ (resp. $u$). Let $\mathbb{C}(n,k)$ be the set of all $n$-vertex cactus graphs with exactly $k$ cycles and $T(n,d)$ be the set of all $n$-vertex tree graphs with diameter $d$. It is said that a cactus is a connected graph with blocks that comprise of either cycles or edges. Beginning with the weighted Mostar index of graphs, we developed certain transformations that either increase or decrease the index. To advance this analysis, we determine the extreme graphs where the maximum and minimum values of the weighted edge Mostar index are accomplished. Moreover, we compute the maximum weighted vertex Mostar invariant for trees with order $n$ and fixed diameter $d$.