Let G be an Eulerian graph on n vertices with adjacency matrix A and characteristic polynomial <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\phi (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We show that when n is even (resp. odd), the square-root of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\phi (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x\phi (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) is an annihilating polynomial of A, over <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {F}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. The result was achieved by applying the Jordan canonical form of A over the algebraic closure <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\bar{\mathbb {F}}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. 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 form, and generalized spectral characterizations of Eulerian graphs

期刊论文
发表日期：

2025年12月7日

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Kunyue Li,
Wei Wang,
Hao Zhang

摘要

Let G be an Eulerian graph on n vertices with adjacency matrix A and characteristic polynomial $\phi (x)$ $ϕ (x)$ . We show that when n is even (resp. odd), the square-root of $\phi (x)$ $ϕ (x)$ (resp. $x\phi (x)$ $x ϕ (x)$ ) is an annihilating polynomial of A, over $\mathbb {F}_2$ $F_{2}$ . The result was achieved by applying the Jordan canonical form of A over the algebraic closure $\bar{\mathbb {F}}_2$ ${\bar{F}}_{2}$ . 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.