An invariant for cospectral graphs is a property shared by all cospectral graphs. Let G and H be two graphs of order n with signless Laplacian matrices Q(G) and Q(H), respectively. In this paper, we give two new invariants for Q-cospectral graphs. Specifically, we show that if G, H are Q-cospectral, then <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$e^{\text {T}}Q^{s}(G)e \equiv e^{\text {T}}Q^{s}(H)e \pmod {16}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>e</mi> <mtext>T</mtext> </msup> <msup> <mi>Q</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mi>e</mi> <mo>≡</mo> <msup> <mi>e</mi> <mtext>T</mtext> </msup> <msup> <mi>Q</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> <mi>e</mi> <mspace width="4.44443pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>16</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for any integer <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where e is the all-ones vector. As an important application, we obtain that under some conditions, every graph Q-cospectral with G is determined by its generalized Q-spectrum.

New arithmetic invariants for Q-cospectral graphs

期刊论文
发表日期：

2026年7月28日

查看全文

Lihong Qiu,
Zhihong Li,
Yizhe Ji,
Wei Wang

摘要

An invariant for cospectral graphs is a property shared by all cospectral graphs. Let G and H be two graphs of order n with signless Laplacian matrices Q(G) and Q(H), respectively. In this paper, we give two new invariants for Q-cospectral graphs. Specifically, we show that if G, H are Q-cospectral, then $e^{\text {T}}Q^{s}(G)e \equiv e^{\text {T}}Q^{s}(H)e \pmod {16}$ $e^{T} Q^{s} (G) e \equiv e^{T} Q^{s} (H) e (mod 16)$ for any integer $s\ge 1$ $s \geq 1$ , where e is the all-ones vector. As an important application, we obtain that under some conditions, every graph Q-cospectral with G is determined by its generalized Q-spectrum.