Bicyclic graphs with exactly two main signless Laplacian eigenvalues

TL;DR

This paper characterizes all connected bicyclic graphs that have exactly two main signless Laplacian eigenvalues, expanding understanding of spectral properties of such graphs.

Contribution

It provides a complete classification of connected bicyclic graphs with precisely two main signless Laplacian eigenvalues, a novel spectral characterization.

Findings

Identifies all such graphs explicitly.

Establishes conditions for having exactly two main eigenvalues.

Contributes to spectral graph theory understanding.

Abstract

A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main eigenvalues are determined.