The notion of Ricci curvature on graphs was introduced by Lin, Lu, and Yau (Tohoku Math. J., 2011) as a modification of Ollivier (J. Funct. Anal., 2009). Recently, a powerful limit-free formulation of Lin-Lu-Yau curvature in terms of the graph Laplacian was given in Münch-Wojciechowski (Adv. Math., 2019). Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$F_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> be the friendship graph obtained from k triangles by sharing a common vertex and let T be the graph obtained from a triangle and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$K_{1,3}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> by adding a matching between every leaf of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K_{1,3}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> and a vertex of the triangle. In this paper, we classify all the simple connected <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>-free graphs with positive Lin-Lu-Yau curvature for minimum degree at least 2: the cycles <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C_3,C_5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>C</mi> <mn>5</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, the friendship graphs <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$F_2,F_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>F</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, the line graph of the Petersen graph, and T.

Graphs with positive Lin-Lu-Yau curvature without quadrilaterals

期刊论文
发表日期：

2026年6月26日

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Huiqiu Lin,
Zhe You

摘要

The notion of Ricci curvature on graphs was introduced by Lin, Lu, and Yau (Tohoku Math. J., 2011) as a modification of Ollivier (J. Funct. Anal., 2009). Recently, a powerful limit-free formulation of Lin-Lu-Yau curvature in terms of the graph Laplacian was given in Münch-Wojciechowski (Adv. Math., 2019). Let $F_k$ $F_{k}$ be the friendship graph obtained from k triangles by sharing a common vertex and let T be the graph obtained from a triangle and $K_{1,3}$ $K_{1, 3}$ by adding a matching between every leaf of $K_{1,3}$ $K_{1, 3}$ and a vertex of the triangle. In this paper, we classify all the simple connected $C_4$ $C_{4}$ -free graphs with positive Lin-Lu-Yau curvature for minimum degree at least 2: the cycles $C_3,C_5$ $C_{3}, C_{5}$ , the friendship graphs $F_2,F_3$ $F_{2}, F_{3}$ , the line graph of the Petersen graph, and T.