Tricyclic graphs with exactly two main eigenvalues

TL;DR

This paper characterizes all connected tricyclic graphs that have exactly two main eigenvalues, providing a complete classification within this specific graph class.

Contribution

It offers a complete classification of connected tricyclic graphs with exactly two main eigenvalues, filling a gap in spectral graph theory.

Findings

Identified all connected tricyclic graphs with two main eigenvalues.

Provided a structural characterization of these graphs.

Contributed to the understanding of eigenvalue properties in complex graphs.

Abstract

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