<p>The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. In this paper, we consider the family of graphs which contain no <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$K_r-E(\cup _{i=1}^{\ell } P_{k_i})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mi>r</mi> </msub> <mo>-</mo> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <msubsup> <mo>∪</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>ℓ</mi> </msubsup> <msub> <mi>P</mi> <msub> <mi>k</mi> <mi>i</mi> </msub> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-minor, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$V(P_i)\cap V(P_j)=\emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1\le i\ne j\le \ell $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>≤</mo> <mi>ℓ</mi> </mrow> </math></EquationSource> </InlineEquation>. We show that the extremal graph is obtained by joining a graph <i>L</i> on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$r-3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>-</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> vertices to the disjoint union of some copies of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$K_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and isolated vertices. Moreover, when there exists at least one <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$k_i\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, the unique extremal graph is determined.</p>

Maximum spread of $K_r-E(\cup _{i=1}^{\ell } P_{k_i})$-minor-free graphs

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发表日期：

2026年3月17日

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Junjie Wang,
Yaoping Hou,
Jiaxin Zheng

摘要

The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. In this paper, we consider the family of graphs which contain no $K_r-E(\cup _{i=1}^{\ell } P_{k_i})$ $K_{r} - E (\cup_{i = 1}^{ℓ} P_{k_{i}})$ -minor, where $V(P_i)\cap V(P_j)=\emptyset $ $V (P_{i}) \cap V (P_{j}) = \emptyset$ for $1\le i\ne j\le \ell $ $1 \leq i \neq j \leq ℓ$ . We show that the extremal graph is obtained by joining a graph L on $r-3$ $r - 3$ vertices to the disjoint union of some copies of $K_2$ $K_{2}$ and isolated vertices. Moreover, when there exists at least one $k_i\ge 3$ $k_{i} \geq 3$ , the unique extremal graph is determined.

Maximum spread of \(K_r-E(\cup _{i=1}^{\ell } P_{k_i})\)-minor-free graphs

摘要