No Threshold graphs are cospectral

TL;DR

This paper derives an explicit formula for the characteristic polynomial of threshold graphs based on their binary sequence, demonstrating that no two nonisomorphic threshold graphs are cospectral, and providing a formula for their adjacency matrix determinants.

Contribution

It introduces a new explicit formula for the characteristic polynomial of threshold graphs and proves their spectral uniqueness among nonisomorphic graphs.

Findings

Explicit formula for characteristic polynomial from binary sequence

Proof that no two nonisomorphic threshold graphs are cospectral

Formula for the determinant of the adjacency matrix

Abstract

A threshold graph G on n vertices is defined by binary sequence of length n. In this paper we present an explicit formula for computing the characteristic polynomial of a threshold graph from its binary sequence. Applications include obtaining a formula for the determinant of adjacency matrix of a threshold graph and showing that no two nonisomorphic threshold graphs are cospectral.