Characterization of tricyclic graphs with exactly two $Q$-main eigenvalues

TL;DR

This paper characterizes all tricyclic graphs that have exactly two $Q$-main eigenvalues, extending previous work on trees, unicyclic, and bicyclic graphs to a more complex class.

Contribution

It provides a complete characterization of tricyclic graphs with exactly two $Q$-main eigenvalues, filling a gap in spectral graph theory.

Findings

Identified all tricyclic graphs with two $Q$-main eigenvalues

Extended previous classifications to more complex graph structures

Contributed to the understanding of signless Laplacian spectra in graphs.

Abstract

The signless Laplacian matrix of a graph $G$ is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called $Q$-eigenvalues of $G$. A $Q$-eigenvalue of a graph $G$ is called a $Q$-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Chen and Huang [L. Chen, Q.X. Huang, Trees, unicyclic graphs and bicyclic graphs with exactly two $Q$-main eigenvalues, submitted for publication] characterized all trees, unicylic graphs and bicyclic graphs with exactly two main $Q$-eigenvalues, respectively. As a continuance of it, in this paper, all tricyclic graphs with exactly two $Q$-main eigenvalues are characterized.