General Cheeger inequalities for -Laplacians on graphs

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• 作者：Matthias Keller^a ; ^{matthias.keller@uni-potsdam.de" class="auth_mail" title="E-mail the corresponding author} ; Delio Mugnolo^b ; ^{delio.mugnolo@fernuni-hagen.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Cheeger inequalities ; Spectral theory of graphs ; Intrinsic metrics for Dirichlet forms
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：147
• 期：Complete
• 页码：80-95
• 全文大小：689 K

文摘

We prove Cheeger inequalities for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301663&_mathId=si6.gif&_user=111111111&_pii=S0362546X16301663&_rdoc=1&_issn=0362546X&md5=d9fc72a41e5d5570c28920276648795d" title="Click to view the MathML source">pclass="mathContainer hidden">class="mathCode">$p$-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved by a novel definition of the measure of the boundary which uses the idea of intrinsic metrics. For the non-normalized case, our bounds on the spectral gap of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301663&_mathId=si6.gif&_user=111111111&_pii=S0362546X16301663&_rdoc=1&_issn=0362546X&md5=d9fc72a41e5d5570c28920276648795d" title="Click to view the MathML source">pclass="mathContainer hidden">class="mathCode">$p$-Laplacians are already significantly better for finite graphs and for infinite graphs they yield non-trivial bounds even in the case of unbounded vertex degree. We, furthermore, give upper bounds by the Cheeger constant and by the exponential volume growth of distance balls.