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<p>Herein we study the relationship on graphs between being (metric) doubling and these two properties: being <i>p</i>-parabolic and satisfying the Cheeger isoperimetric inequality. We prove that if a uniform graph <i>G</i> satisfies the (Cheeger) isoperimetric inequality, then <i>G</i> is not (metric) doubling and see that the converse is not true. We also prove that if <i>G</i> is a doubling graph with doubling constant <i>C</i>, then it is <i>p</i>-parabolic for every <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p \ge \log _2(C)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <msub> <mo>log</mo> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and see that the converse is not true. Furthermore, we see that being doubling does not imply being <i>p</i>-parabolic for every <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(1&lt; p &lt; \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>p</mi> <mo>&lt;</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Finally, we see that a quasi-isometry between manifolds whose Ricci curvature is bounded below preserves being doubling and also, that an manifold with bounded Ricci curvature below is doubling if and only any uniform graph quasi-isometric to it is doubling.</p>

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Isoperimetric Inequality, p-parabolicity and Doubling Graphs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月28日

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  • Álvaro Martínez-Pérez,
  • José M. Rodríguez

摘要

Herein we study the relationship on graphs between being (metric) doubling and these two properties: being p-parabolic and satisfying the Cheeger isoperimetric inequality. We prove that if a uniform graph G satisfies the (Cheeger) isoperimetric inequality, then G is not (metric) doubling and see that the converse is not true. We also prove that if G is a doubling graph with doubling constant C, then it is p-parabolic for every \(p \ge \log _2(C)\) p ≥ log 2 ( C ) and see that the converse is not true. Furthermore, we see that being doubling does not imply being p-parabolic for every \(1< p < \infty \) 1 < p < ∞ . Finally, we see that a quasi-isometry between manifolds whose Ricci curvature is bounded below preserves being doubling and also, that an manifold with bounded Ricci curvature below is doubling if and only any uniform graph quasi-isometric to it is doubling.