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Counting cycles and finite dimensional Lp norms

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We obtain sharp bounds for the number ofn-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. We prove sharp estimates on both\sum _{\scriptscriptstyle{i=\mathrm{1}}}^{\scriptscriptstyle{n}}x_{\scriptscriptstyle {i}}^{\scriptscriptstyle{k}}and\sum _{\scriptscriptstyle{i=\mathrm{1}}}^{\scriptscriptstyle{n}}\vert x_{\scriptscriptstyle{{i}}}\vert ^{\scriptscriptstyle{{k}}}, subject to the constraints that\sum _{\scriptscriptstyle{i=\mathrm{1}}}^{\scriptscriptstyle{n}}x_{\scriptscriptstyle {i}}^{\scriptscriptstyle{\mathrm{2}}}=Cand\sum _{\scriptscriptstyle{i=\mathrm{1}}}^{\scriptscriptstyle{n}}x_{\scriptscriptstyle {{i}}}=\mathrm{0}.

© 2002 Elsevier Science (USA)

Document Type: Research Article

DOI: http://dx.doi.org/10.1016/S0196-8858(02)00037-4

Publication date: November 1, 2002

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