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Ramseyramsey (talk | contribs) m →Cheeger inequality: pointing out the inequality for graphs with being regular. |
Ramseyramsey (talk | contribs) m →Cheeger inequality: pointing out the inequality for graphs with being regular. |
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For general graphs that are not necessarily regular, an alternative inequality is given by Chung <ref>{{cite book |last =Chung |first1 =Fan |author-link =Fan Chung |year =1997 |title =Spectral Graph Theory |isbn =0821803158 |mr =1421568 |url =http://www.math.ucsd.edu/~fan/research/revised.html |postscript = [first 4 chapters are available in the website] |editor =American Mathematical Society |publisher =Providence, R. I.}}</ref>
:<math> 2 {\mathbf \lambda} \geq {\mathbf h}(G) \geq \frac{\lambda^2(G)}{2}</math>
Cheeger constant <math>{\mathbf h}(G)</math>
: <math> {\mathbf h}(G) = \min_{\emptyset \not =S\subset V(G)}\frac{|\partial(S)|}{\min({\mathrm{vol}}(S), {\mathrm{vol}}(\bar{S}))}</math>
where <math>{\mathrm{vol}}(Y)</math> is the sum of degrees of vertices in <math>Y</math> and <math>\lambda</math> is the least nontrivial eigenvalue of the normalized Laplacian provided <math>G</math> is connected.
== Hoffman-Delsarte inequality ==
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