In this paper, we consider Wang’s <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$CD_p(m,{\mathcal {K}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <msub> <mi>D</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi mathvariant="script">K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> condition on graphs, which depends on the p-Laplacian <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Delta _p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and is an extension of the classical Bakry-Émery <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$CD(m,{\mathcal {K}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mi>D</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi mathvariant="script">K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the p-curvature is non-negative at some vertices in the case <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, while it approaches to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$-\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> in the case of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$1<p<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. In addition, we observe that a crucial property of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Gamma _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Γ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> on Cartesian products does no longer hold for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Gamma _2^p$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Γ</mi> <mn>2</mn> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation> in the case of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$p > 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$p > 2.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>2</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

A brief note about p-curvature on graphs

期刊论文
发表日期：

2026年7月29日

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Chunyang Hu

摘要

In this paper, we consider Wang’s $CD_p(m,{\mathcal {K}})$ $C D_{p} (m, K)$ condition on graphs, which depends on the p-Laplacian $\Delta _p$ $Δ_{p}$ for $p>1$ $p > 1$ and is an extension of the classical Bakry-Émery $CD(m,{\mathcal {K}})$ $C D (m, K)$ curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the p-curvature is non-negative at some vertices in the case $p\ge 2$ $p \geq 2$ , while it approaches to $-\infty $ $- \infty$ in the case of $1<p<2$ $1 2$ $p > 2$ . As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for $p > 2.$ $p > 2 .$