Abstract

In this paper, we present a comprehensive study on the spectral properties of the signless Laplacian matrix of the maximal graph. Specifically, we characterize the spectral radius of the signless Laplacian matrix of the maximal graph $M(\Gamma(\mathbb{Z}_n))$. Moreover, we study the smallest signless Laplacian eigenvalue of the maximal graph and introduce an interaction with the algebraic connectivity of $M(\Gamma(\mathbb{Z}_n))$ for some definite values of $n$. Finally, we derive an explicit formula for the Wiener index in terms of signless Laplacian eigenvalues of the graph.

Keywords and Phrases

Signless Laplacian Spectrum, Maximal Graph, Wiener Index.

A.M.S. subject classification

05C50, 05C25, 05C75.

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