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

Kunyue Li, Wei Wang, Hao Zhang

Abstract

Let $G$ be an Eulerian graph on $n$ vertices with adjacency matrix $A$ and characteristic polynomial $φ(x)$. We show that when $n$ is even (resp. odd), the square-root of $φ(x)$ (resp. $xφ(x)$) 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.

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

Abstract

Let be an Eulerian graph on vertices with adjacency matrix and characteristic polynomial . We show that when is even (resp. odd), the square-root of (resp. ) is an annihilating polynomial of , over . The result was achieved by applying the Jordan canonical form of over the algebraic closure . 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.

Paper Structure

This paper contains 7 sections, 20 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and terminologies
  4. Main strategy
  5. Smith normal form and the level of $Q$
  6. Proof of Theorem \ref{['main']}
  7. Proof of Theorem \ref{['thm:mainresult']}

Key Result

Theorem 1

If ${2^{-\lfloor n/2 \rfloor}}{\det W }$ (which is always an integer) is odd and square-free, then $G$ is DGS.

Theorems & Definitions (35)

  • Theorem 1: Wang Wang2
  • Theorem 2
  • Example 1
  • Remark 1
  • Theorem 3: Johnson and Newman JN; Wang and Xu WX1
  • Definition 4
  • Theorem 5: Wang and Xu WX1
  • Definition 6
  • Lemma 1: QWWZ
  • Corollary 1: WX1
  • ...and 25 more