On the normalized spectrum of threshold graphs

TL;DR

This paper studies the normalized eigenvalues and energy of connected threshold graphs, deriving eigenvalues from their binary representations and equitable partitions, and characterizes those with up to five distinct eigenvalues.

Contribution

It introduces methods to compute normalized eigenvalues from binary strings and equitable partitions, and characterizes threshold graphs with limited eigenvalue diversity.

Findings

Eigenvalues derived from binary representations

Eigenvalues obtained from equitable partition matrices

Characterization of graphs with up to five eigenvalues

Abstract

In this article we investigate normalized adjacency eigenvalues (simply normalized eigenvalues) and normalized adjacency energy of connected threshold graphs. A threshold graph can always be represented as a unique binary string. Certain eigenvalues are obtained directly from its binary representation and the rest of the eigenvalues are evaluated from its normalized equitable partition matrix. Finally, we characterize threshold graphs with at most five distinct eigenvalues.