Isoperimetric Inequality, p-parabolicity and Doubling Graphs

Abstract 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 $$p \ge \log _2(C)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:msub> <mml:mo>log</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and see that the converse is not true. Furthermore, we see that being doubling does not imply being p -parabolic for every $$1&lt; p &lt; \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . 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.