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Volume 13
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Article

The First Zagreb Index, the Laplacian Spectral Radius, and Some Hamiltonian Properties of Graphs

by
Rao Li
Department of Computer Science, Engineering and Mathematics, University of South Carolina Aiken, Aiken, SC 29801, USA
Mathematics 2025, 13(17), 2897; https://doi.org/10.3390/math13172897
Submission received: 4 August 2025 / Revised: 26 August 2025 / Accepted: 4 September 2025 / Published: 8 September 2025
(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)
Versions Notes

Abstract

The first Zagreb index of a graph G is defined as the sum of the squares of the degrees of all the vertices in G. The Laplacian spectral radius of a graph G is defined as the largest eigenvalue of the Laplacian matrix of the graph G. In this paper, we first establish inequalities on the first Zagreb index and the Laplacian spectral radius of a graph. Using the ideas of proving the inequalities, we present sufficient conditions involving the first Zagreb index and the Laplacian spectral radius for some Hamiltonian properties of graphs.
Keywords: first Zagreb index; Laplacian spectral radius; Hamiltonian graph; traceable graph first Zagreb index; Laplacian spectral radius; Hamiltonian graph; traceable graph

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MDPI and ACS Style

Li, R. The First Zagreb Index, the Laplacian Spectral Radius, and Some Hamiltonian Properties of Graphs. Mathematics 2025, 13, 2897. https://doi.org/10.3390/math13172897

AMA Style

Li R. The First Zagreb Index, the Laplacian Spectral Radius, and Some Hamiltonian Properties of Graphs. Mathematics. 2025; 13(17):2897. https://doi.org/10.3390/math13172897

Chicago/Turabian Style

Li, Rao. 2025. "The First Zagreb Index, the Laplacian Spectral Radius, and Some Hamiltonian Properties of Graphs" Mathematics 13, no. 17: 2897. https://doi.org/10.3390/math13172897

APA Style

Li, R. (2025). The First Zagreb Index, the Laplacian Spectral Radius, and Some Hamiltonian Properties of Graphs. Mathematics, 13(17), 2897. https://doi.org/10.3390/math13172897

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