The signless Laplacian matrix in graph spectra theory is a remarkable matrix of graphs, and it is extensively studied by researchers. In 1981, Cvetković pointed out 12 directions in further investigations of graph spectra, one of which is “classifying and ordering graphs”. Along this classic direction, we pay our attention to the order of the largest eigenvalue of the signless Laplacian matrix of graphs, which is usually called the signless Laplacian spectral radius or the Q-index of a graph. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {G}(m, g)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {G}(m, \ge g)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mo>≥</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) be the family of connected graphs on m edges with girth g (resp. no less than g), where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$g\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we firstly order the first <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(\lfloor \frac{g}{2}\rfloor +2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mrow> <mo>⌊</mo> <mfrac> <mi>g</mi> <mn>2</mn> </mfrac> <mo>⌋</mo> </mrow> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> largest Q-indices of graphs in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {G}(m, g)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$m\ge 3g\ge 12$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>3</mn> <mi>g</mi> <mo>≥</mo> <mn>12</mn> </mrow> </math></EquationSource> </InlineEquation>. Secondly, we order the first <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(\lfloor \frac{g}{2}\rfloor +3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mrow> <mo>⌊</mo> <mfrac> <mi>g</mi> <mn>2</mn> </mfrac> <mo>⌋</mo> </mrow> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> largest Q-indices of graphs in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {G}(m, \ge g)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mo>≥</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$m\ge 3g\ge 12$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>3</mn> <mi>g</mi> <mo>≥</mo> <mn>12</mn> </mrow> </math></EquationSource> </InlineEquation>. As a complement, we give the first five largest Q-indices of graphs in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {G}(m, 3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$m\ge 9$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>9</mn> </mrow> </math></EquationSource> </InlineEquation>. Finally, we give the order of the first eleven largest Q-indices of all connected graphs with size m.

Ordering signless Laplacian spectral radii of graphs with given size and girth

期刊论文
发表日期：

2025年12月15日

查看全文

Yarong Hu,
Zhenzhen Lou,
Qiongxiang Huang

摘要

The signless Laplacian matrix in graph spectra theory is a remarkable matrix of graphs, and it is extensively studied by researchers. In 1981, Cvetković pointed out 12 directions in further investigations of graph spectra, one of which is “classifying and ordering graphs”. Along this classic direction, we pay our attention to the order of the largest eigenvalue of the signless Laplacian matrix of graphs, which is usually called the signless Laplacian spectral radius or the Q-index of a graph. Let $\mathbb {G}(m, g)$ $G (m, g)$ (resp. $\mathbb {G}(m, \ge g)$ $G (m, \geq g)$ ) be the family of connected graphs on m edges with girth g (resp. no less than g), where $g\ge 3$ $g \geq 3$ . In this paper, we firstly order the first $(\lfloor \frac{g}{2}\rfloor +2)$ $(⌊ \frac{g}{2} ⌋ + 2)$ largest Q-indices of graphs in $\mathbb {G}(m, g)$ $G (m, g)$ , where $m\ge 3g\ge 12$ $m \geq 3 g \geq 12$ . Secondly, we order the first $(\lfloor \frac{g}{2}\rfloor +3)$ $(⌊ \frac{g}{2} ⌋ + 3)$ largest Q-indices of graphs in $\mathbb {G}(m, \ge g)$ $G (m, \geq g)$ , where $m\ge 3g\ge 12$ $m \geq 3 g \geq 12$ . As a complement, we give the first five largest Q-indices of graphs in $\mathbb {G}(m, 3)$ $G (m, 3)$ with $m\ge 9$ $m \geq 9$ . Finally, we give the order of the first eleven largest Q-indices of all connected graphs with size m.