The First Zagreb Index Conditions for Some Hamiltonian Properties of Graphs

TL;DR

This paper establishes new conditions based on the first Zagreb index that guarantee certain Hamiltonian properties in graphs, providing bounds and criteria for graph analysis.

Contribution

It introduces the first Zagreb index conditions for Hamiltonian properties and derives an upper bound for the index, advancing graph theory understanding.

Findings

Derived new Zagreb index conditions for Hamiltonian properties

Established an upper bound for the first Zagreb index

Applied Pólya-Szegő inequality in graph theory context

Abstract

Let $G = (V, E)$ be a graph. The first Zagreb index of a graph $G$ is defined as $\sum_{u \in V} d^2(u)$, where $d(u)$ is the degree of vertex $u$ in $G$. Using the P\'{o}lya-Szeg\H{o} inequality, we in this paper present the first Zagreb index conditions for some Hamiltonian properties of a graph and an upper bound for the first Zagreb index of a graph.