Edge-vertex degree based Zagreb index and graph operations
Amitariddhi Sinha, Somnath Paul
TL;DR
The paper develops the edge-vertex degree framework, defining $deg^{ev}_{G}(e)$ via $deg^{ev}_{G}(e)=d_G(x)+d_G(y)-\eta_G(e)$ for an edge $e=xy$, and introduces the $ev$-degree Zagreb index $M^{ev}(G)$. It then derives explicit closed-form expressions for $M^{ev}$ under a wide range of unary operations (subset of subdivision- and semitotal-related graphs) and binary operations (join, Cartesian product, composition, corona, tensor product), as well as for the $F$-sum of graphs, expressing results in terms of classical invariants such as $M_1$, $M_2$, $F$, $m$, $\eta$, and line graph terms $L(G)$. These exact formulas enable direct analysis of how $ev$-degree Zagreb indices transform under common graph constructions. The work extends Zagreb-type indices to an edge-vertex based setting and provides a toolkit for structure-property studies in chemical graph theory and related domains.
Abstract
A graph $G$ consists of two parts, the vertices and edges. The vertices constitute the vertex set $V(G)$ and the edges, the edge set. An edge \( e=xy \), \( ev \)-dominates not only the vertices incident to it but also those adjacent to either \( x \) or \( y \). The edge-vertex degree of $e,$ $deg^{ev}_{G}(e),$ is the number of vertices in the $ev$-dominating set of $e$. In this article, we compute expressions for the $ev$-degree version of the Zagreb index of several unary and binary graph operations.