Annihilating polynomial, Jordan canonical from, and generalized spectral characterizations of Eulerian graphs

TL;DR

This paper explores spectral properties of Eulerian graphs, revealing new polynomial characterizations and demonstrating that certain Eulerian graphs are uniquely identified by their generalized spectrum.

Contribution

It introduces novel polynomial characterizations of Eulerian graphs using Jordan canonical forms and shows these graphs can be uniquely determined by their generalized spectrum.

Findings

Square-root of characteristic polynomial is an annihilating polynomial over GF(2)

Family of Eulerian graphs determined by their generalized spectrum

Simplifies previous spectral characterization results

Abstract

Let $G$ be an Eulerian graph on $n$ vertices with adjacency matrix $A$ and characteristic polynomial $\phi(x)$. We show that when $n$ is even (resp. odd), the square-root of $\phi(x)$ (resp. $x\phi(x)$) is an annihilating polynomial of $A$, over $\mathbb{F}_2$. The result was achieved by applying the Jordan canonical form of $A$ over the algebraic closure $\bar{\mathbb{F}}_2$. Based on this, we show a family of Eulerian graphs are determined by their generalized spectrum among all Eulerian graphs, which significantly simplifies and strengthens the previous result.