Edge-vertex degree based Zagreb index and graph operations

TL;DR

This paper introduces a new edge-vertex degree based Zagreb index for graphs and derives formulas for this index under various graph operations, expanding the understanding of graph invariants.

Contribution

It defines the edge-vertex degree based Zagreb index and provides explicit formulas for this index for several unary and binary graph operations.

Findings

Formulas for the edge-vertex degree Zagreb index under various graph operations.

Extension of Zagreb index concepts to edge-vertex degree context.

New tools for analyzing graph invariants in complex graph constructions.

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.