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{{Short description|Linear algebra aspects of graph theory}}
In [[mathematics]], '''spectral [[graph theory]]''' is the study of the properties of a [[Graph (discrete mathematics)|graph]] in relationship to the [[characteristic polynomial]], [[eigenvalue]]s, and [[eigenvector]]s of matrices associated with the graph, such as its [[adjacency matrix]] or [[Laplacian matrix]].
The adjacency matrix of a
While the adjacency matrix depends on the vertex labeling, its [[Spectrum of a matrix|spectrum]] is a [[graph invariant]], although not a complete one.
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==Cospectral graphs==
Two graphs are called '''cospectral''' or '''
[[File:Isospectral enneahedra.svg|thumb|300px|Two cospectral [[enneahedron|enneahedra]], the smallest possible cospectral [[polyhedral graph]]s]]
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A pair of graphs are said to be cospectral mates if they have the same spectrum, but are non-isomorphic.
The smallest pair of cospectral mates is {''K''<sub>1,4</sub>, ''C''<sub>4</sub>
The smallest pair of [[polyhedral graph|polyhedral]] cospectral mates are [[enneahedron|enneahedra]] with eight vertices each.<ref>{{citation|title=Topological twin graphs. Smallest pair of isospectral polyhedral graphs with eight vertices|year=1994|last1=Hosoya|last2=Nagashima|last3=Hyugaji|first1=Haruo|first2=Umpei|first3=Sachiko|author1-link=Haruo Hosoya|journal=Journal of Chemical Information and Modeling|volume=34|issue=2|pages=428–431|doi=10.1021/ci00018a033}}.</ref>
=== Finding cospectral graphs ===
[[Almost all]] [[tree (graph theory)|tree]]s are cospectral, i.e., as the number of vertices grows, the fraction of trees for which there exists a cospectral tree goes to 1.
A pair of [[regular graph]]s are cospectral if and only if their complements are cospectral.<ref>{{Cite web|url=http://www.math.uwaterloo.ca/~cgodsil/pdfs/cospectral.pdf|title=Are Almost All Graphs Cospectral?|last=Godsil|first=Chris|date=November 7, 2007}}</ref>
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| doi = 10.2307/1971195 | jstor = 1971195 }}.</ref>
Another important source of cospectral graphs are the point-collinearity graphs and the line-intersection graphs of [[incidence geometry|point-line geometries]]. These graphs are always cospectral but are often non-isomorphic.<ref>{{
==Cheeger inequality==
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More formally, the Cheeger constant ''h''(''G'') of a graph ''G'' on ''n'' vertices is defined as
: <math>h(G) = \min_{0 < |S| \le \frac{n}{2} } \frac{|\partial(S)|}{|S|},</math>
where the minimum is over all nonempty sets ''S'' of at most ''n''/2 vertices and ∂(''S'') is the ''edge boundary'' of ''S'', i.e., the set of edges with exactly one endpoint in ''S''.<ref>Definition 2.1 in {{harvtxt|Hoory|Linial|
===Cheeger inequality===
When the graph ''G'' is ''d''-regular, there is a relationship between ''h''(''G'') and the spectral gap ''d'' − λ<sub>2</sub> 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|
: <math>\
This inequality is closely related to the [[Cheeger bound]] for [[Markov chains]] and can be seen as a discrete version of [[Cheeger constant#Cheeger.27s inequality|Cheeger's inequality]] in [[Riemannian geometry]].
For general connected graphs that are not necessarily regular, an alternative inequality is given by Chung<ref name=chung>{{cite book |last1 =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}}
== Hoffman-Delsarte inequality ==▼
:<math> \frac{1}{2} {\lambda} \le {\mathbf h}(G) \le \sqrt{2 \lambda},</math>
where <math>\lambda</math> is the least nontrivial eigenvalue of the normalized Laplacian, and <math>{\mathbf h}(G)</math> is the (normalized) Cheeger constant
: <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>.
There is an eigenvalue bound for [[Independent set (graph theory)|independent sets]] in [[regular graph]]s, originally due to [[Alan J. Hoffman]] and Philippe Delsarte.<ref>{{Cite web|url=https://www.math.uwaterloo.ca/~cgodsil/pdfs/ekrs-clg.pdf|title=Erdős-Ko-Rado Theorems|last=Godsil|first=Chris|date=May 2009}}</ref>
Suppose that <math>G</math> is a <math>k</math>-regular graph on <math>n</math> vertices with least eigenvalue <math>\lambda_{\mathrm{min}}</math>. Then:<math display="block">\alpha(G) \leq \frac{n}{1 - \frac{k}{\lambda_{\mathrm{min}}}}</math>where <math>\alpha(G)</math> denotes its [[independence number]].
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|
For general graphs which are not necessarily regular, a similar upper bound for the independence number can be derived by using the maximum eigenvalue
<math> \lambda'_{max}</math> of the normalized Laplacian<ref name=chung/> of <math>G</math>:
<math display="block">\alpha(G) \leq n (1-\frac {1}{\lambda'_{\mathrm{max}}}) \frac {\mathrm{max deg}}{\mathrm{min deg}}
</math>
where <math>{\mathrm{max deg}}</math> and <math>{\mathrm{min deg}}</math> denote the maximum and minimum degree in <math>G</math>, respectively. This a consequence of a more general inequality (pp. 109 in
<ref name=chung/>):
<math display="block">{\mathrm{vol}}(X) \leq (1-\frac {1}{\lambda'_{\mathrm{max}}}) {\mathrm{vol}}(V(G))
</math>
where <math>X</math> is an independent set of vertices and <math>{\mathrm{vol}}(Y)</math> denotes the sum of degrees of vertices in <math>Y</math> .
==Historical outline==
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|
==See also==
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==References==
{{reflist}}
* {{citation|last1=Alon|last2=Spencer|title=The probabilistic method|publisher=Wiley|year=2011}}.
* {{citation| first1=Andries |last1=Brouwer| first2 = Willem H. | last2 = Haemers| author-link = Andries Brouwer| url=http://www.win.tue.nl/~aeb/2WF02/spectra.pdf |title=Spectra of Graphs |year=2011|publisher=Springer}}
* {{citation|last1=Hoory|last2=Linial|last3=Wigderson|year=2006|title=Expander graphs and their applications|url=https://www.cs.huji.ac.il/~nati/PAPERS/expander_survey.pdf}}
* {{cite book |last1 =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.}}
* {{cite book |last =Schwenk |first =A. J. |author-link =Schwenk A. J. |chapter =Almost All Trees are Cospectral |title =New Directions in the Theory of Graphs |editor-last =Harary |editor-first =Frank |editor-link =Frank Harary |publisher =[[Academic Press]] |location =New York |year =1973 |isbn =012324255X |oclc =890297242 }}
* {{cite book |last = Bogdan | first =Nica | title="A Brief Introduction to Spectral Graph Theory" | publisher = EMS Press | location = Zurich | year = 2018 | isbn =978-3-03719-188-0}}
* [https://doi.org/10.1007/978-3-662-67872-5 Pavel Kurasov (2024), ''Spectral Geometry of Graphs'', Springer(Birkhauser), Open Access (CC4.0).]
==External links==
* {{cite web| first1=
* {{cite web| first1=Daniel |last1=Spielman |url=http://cs-www.cs.yale.edu/homes/spielman/sgta/ |title=Spectral Graph Theory and its Applications |year=2007}} [presented at FOCS 2007 Conference]
* {{cite web| first1=Daniel |last1=Spielman | url=http://www.cs.yale.edu/homes/spielman/eigs/ |title=Spectral Graph Theory and its Applications |year=2004}} [course page and lecture notes]
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