THE FIRST ZAGREB INDEX, THE FORGOTTEN TOPOLOGICAL INDEX, THE INVERSE DEGREE AND SOME HAMILTONIAN PROPERTIES OF GRAPHS

Year 2025, Volume: 15 Issue: 11, 2613 - 2623, 03.11.2025

Rao Li

Abstract

Let $G = (V, E)$ be a graph. The first Zagreb index and the forgotten topological index of a graph $G$ are defined respectively as $\sum_{u \in V} d^2(u)$ and $\sum_{u \in V} d^3(u)$, where $d(u)$ is the degree of vertex $u$ in $G$. If the minimum degree of $G$ is at least one, the inverse degree of $G$ is defined as $\sum_{u \in V} \frac{1}{d(u)}$. In this paper, we, for a graph with minimum degree at least one, present an upper bound for the first Zagreb index of the graph and lower bounds for the forgotten topological index and the inverse degree of the graph. We also present sufficient conditions involving the first Zagreb index, the forgotten topological index, or the inverse degree for some Hamiltonian properties of a graph.

Keywords

The first Zagreb index , the forgotten topological index , the inverse degree , Hamiltonian graph , traceable graph

Thanks

The author would like to extend his gratitude to the referee for his or her suggestions which improve the initial version of the paper.

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