Spectral graph theory: Difference between revisions

When the graph ''G'' is ''d''-regular, there is a relationship between ''h''(''G'') and the spectral gap ''d'' − λ₂ of ''G''. An inequality due to Dodziuk<ref>J.Dodziuk, Difference Equations, Isoperimetric inequality and Transience of Certain Random Walks, Trans. Amer. Math. Soc. 284 (1984), no. 2, 787-794.</ref> and independently [[Noga Alon|Alon]] and [[Vitali Milman|Milman]]{{Sfn|Alon|Spencer|2011}} states that<ref>Theorem 2.4 in {{harvtxt|Hoory|Linial|WidgersonWigderson|2006}}</ref>

For general connected graphs that are not necessarily regular, an alternative inequality is given by Chung<ref name=chung>{{cite book |lastlast1 =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>{{rp|35}}

:<math> \frac{1}{2} {\lambda} \le {\mathbf h}(G) \le \sqrt{2 \Delta \lambda},</math>

where <math>\Delta</math> is the maximum degree, <math>\lambda</math> is the least nontrivial eigenvalue of the (non-normalized) Laplacian, and <math>{\mathbf h}(G)</math> is the (normalized) Cheeger constant

where <math>{\mathrm{vol}}(Y)</math> is the numbersum of degrees of vertices in <math>Y</math>.

This bound has been applied to establish e.g. algebraic proofs of the [[Erdős–Ko–Rado theorem]] and its analogue for intersecting families of subspaces over [[finite field]]s.<ref>{{Cite book|title=Erdős-Ko-Rado theorems : algebraic approaches|lastlast1=1949-Godsil|firstfirst1=Godsil, C. D. (Christopher David)|otherslast2=Meagher, |first2=Karen (College teacher)|isbn=9781107128446|location=Cambridge, United Kingdom|oclc=935456305|year = 2016}}</ref>

Spectral graph theory emerged in the 1950s and 1960s. Besides [[graph theory|graph theoretic]] research on the relationship between structural and spectral properties of graphs, another major source was research in [[quantum chemistry]], but the connections between these two lines of work were not discovered until much later.<ref name= cvet2>''Eigenspaces of Graphs'', by [[Dragoš Cvetković]], Peter Rowlinson, Slobodan Simić (1997) {{isbn|0-521-57352-1}}</ref> The 1980 monograph ''Spectra of Graphs''<ref>Dragoš M. Cvetković, Michael Doob, [[Horst Sachs]], ''Spectra of Graphs'' (1980)</ref> by Cvetković, Doob, and Sachs summarised nearly all research to date in the area. In 1988 it was updated by the survey ''Recent Results in the Theory of Graph Spectra''.<ref>{{cite book|first1=Dragoš M.|last1=Cvetković |first2=Michael |last2=Doob |first3=Ivan |last3=Gutman |first4=A. |last4=Torgasev |title=Recent Results in the Theory of Graph Spectra |series=Annals of Discrete mathematics |number=36 |year=1988 |isbn=0-444-70361-6 |url=http://www.sciencedirect.com/science/bookseries/01675060/36}}</ref> The 3rd edition of ''Spectra of Graphs'' (1995) contains a summary of the further recent contributions to the subject.<ref name= cvet2/> Discrete geometric analysis created and developed by [[Toshikazu Sunada]] in the 2000s deals with spectral graph theory in terms of discrete Laplacians associated with weighted graphs,<ref>{{citation | last = Sunada | first = Toshikazu | journal = Proceedings of Symposia in Pure Mathematics | pages = 51–86 | title = Discrete geometric analysis | volume = 77 | year = 2008| doi = 10.1090/pspum/077/2459864 | isbn = 9780821844717 }}.</ref> and finds application in various fields, including [[Spectral shape analysis|shape analysis]]. In most recent years, the spectral graph theory has expanded to vertex-varying graphs often encountered in many real-life applications.<ref>{{Cite journal|lastlast1=Shuman|firstfirst1=David I|last2=Ricaud|first2=Benjamin|last3=Vandergheynst|first3=Pierre|date=March 2016|title=Vertex-frequency analysis on graphs|journal=Applied and Computational Harmonic Analysis|volume=40|issue=2|pages=260–291|doi=10.1016/j.acha.2015.02.005|issn=1063-5203|arxiv=1307.5708|s2cid=16487065 }}</ref><ref>{{Cite journal|lastlast1=Stankovic|firstfirst1=Ljubisa|last2=Dakovic|first2=Milos|last3=Sejdic|first3=Ervin|date=July 2017|title=Vertex-Frequency Analysis: A Way to Localize Graph Spectral Components [Lecture Notes]|journal=IEEE Signal Processing Magazine|language=en-US|volume=34|issue=4|pages=176–182|doi=10.1109/msp.2017.2696572|issn=1053-5888|bibcode=2017ISPM...34..176S|s2cid=19969572 }}</ref><ref>{{Cite journal|lastlast1=Sakiyama|firstfirst1=Akie|last2=Watanabe|first2=Kana|last3=Tanaka|first3=Yuichi|date=September 2016|title=Spectral Graph Wavelets and Filter Banks With Low Approximation Error|journal=IEEE Transactions on Signal and Information Processing over Networks|language=en-US|volume=2|issue=3|pages=230–245|doi=10.1109/tsipn.2016.2581303|s2cid=2052898 |issn=2373-776X}}</ref><ref>{{Cite journal|lastlast1=Behjat|firstfirst1=Hamid|last2=Richter|first2=Ulrike|last3=Van De Ville|first3=Dimitri|last4=Sornmo|first4=Leif|date=2016-11-15|title=Signal-Adapted Tight Frames on Graphs|journal=IEEE Transactions on Signal Processing|language=en-US|volume=64|issue=22|pages=6017–6029|doi=10.1109/tsp.2016.2591513|issn=1053-587X|bibcode=2016ITSP...64.6017B|s2cid=12844791 |url=http://infoscience.epfl.ch/record/223159}}</ref>