Algebraic Graph Theory
- © 2001
- Chris Godsil 0 ,
- Gordon Royle 1
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada
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Department of Computer Science, University of Western Australia, Nedlands, Australia
- Includes supplementary material: sn.pub/extras
Part of the book series: Graduate Texts in Mathematics (GTM, volume 207)
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Table of contents (17 chapters)
Front matter.
- Chris Godsil, Gordon Royle
Transitive Graphs
Arc-transitive graphs, generalized polygons and moore graphs, homomorphisms, kneser graphs.
- Matrix Theory
Interlacing
Strongly regular graphs, line graphs and eigenvalues, the laplacian of a graph, cuts and flows, the rank polynomial, knots and eulerian cycles, back matter.
- graph theory
- homomorphism
- Laplace operator
- combinatorics
Authors and Affiliations
Chris Godsil
Gordon Royle
Bibliographic Information
Book Title : Algebraic Graph Theory
Authors : Chris Godsil, Gordon Royle
Series Title : Graduate Texts in Mathematics
DOI : https://doi.org/10.1007/978-1-4613-0163-9
Publisher : Springer New York, NY
eBook Packages : Springer Book Archive
Copyright Information : Springer-Verlag New York, Inc. 2001
Hardcover ISBN : 978-0-387-95241-3 Published: 20 April 2001
Softcover ISBN : 978-0-387-95220-8 Published: 20 April 2001
eBook ISBN : 978-1-4613-0163-9 Published: 01 December 2013
Series ISSN : 0072-5285
Series E-ISSN : 2197-5612
Edition Number : 1
Number of Pages : XIX, 443
Topics : Combinatorics
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Graph Theory
An introduction to algebraic graph theory.
Cesar O. Aguilar
Department of mathematics, suny geneseo.
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Frequency Assignment Problems (FAPs) arise when transmitters need to be allocated frequencies with the aim of minimizing interference, whilst maintaining an efficient use of the radio spectrum. In this thesis FAPs are seen as generalised graph colouring problems, where transmitters are represented by vertices, and their interactions by weighted edges. Solving FAPs often relies on known structural properties to facilitate algorithms. When no structural information is available explicitly, obtaining it from numerical data is difficult. This lack of structural information is a key underlying motivation for the research work in this thesis. If there are TV transmitters to be assigned, we assume as given an N x N "influence matrix" W with entries Wij representing influence between transmitters i and j. From this matrix we derive the Laplacian matrix L = D—W, where D is a diagonal matrix whose entries da are the sum of all influences working in transmitter i. The focus of this thesis is the study of mathematical properties of the matrix L. We généralisé certain properties of the Laplacian eigenvalues and eigenvectors that hold for simple graphs. We also observe and discuss changes in the shape of the Laplacian eigenvalue spectrum due to modifications of a FAP. We include a number of computational experiments and generated simulated examples of FAPs for which we explicitly calculate eigenvalues and eigenvectors in order to test the developed theoretical results. We find that the Laplacians prove useful in identifying certain types of problems, providing structured approach to reducing the original FAP to smaller size subproblems, hence assisting existing heuristic algorithms for solving frequency assignments. In that sense we conclude that analysis of the Laplacians is a useful tool for better understanding of FAPs.
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Bessy, Stéphane. "Some problems in graph theory and graphs algorithmic theory." Habilitation à diriger des recherches, Université Montpellier II - Sciences et Techniques du Languedoc, 2012. http://tel.archives-ouvertes.fr/tel-00806716.
Myers, Joseph Samuel. "Extremal theory of graph minors and directed graphs." Thesis, University of Cambridge, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.619614.
Johnson, Chase R. "Molecular Graph Theory." Digital WPI, 2010. https://digitalcommons.wpi.edu/etd-theses/1179.
Feghali, Carl. "Topics in graph colouring and extremal graph theory." Thesis, Durham University, 2016. http://etheses.dur.ac.uk/11790/.
Nikwigize, Adolphe. "Graph theory : Route problems." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-17397.
Berg, Deborah. "Connections Between Voting Theory and Graph Theory." Scholarship @ Claremont, 2005. https://scholarship.claremont.edu/hmc_theses/178.
Hatt, Justin Dale. "Online assessment of graph theory." Thesis, Brunel University, 2016. http://bura.brunel.ac.uk/handle/2438/13389.
Keevash, Peter. "Topics in extremal graph theory." Thesis, University of Cambridge, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.619938.
Law, Ka-ho, and 羅家豪. "Some results in graph theory." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2010. http://hub.hku.hk/bib/B44899816.
Morrison, Julie Lindsay. "Computational graph theory in bioinformatics." Thesis, University of Strathclyde, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.435114.
Richer, Duncan Christopher. "Graph theory and combinatorial games." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.621916.
Eggemann, Nicole. "Some applications of graph theory." Thesis, Brunel University, 2009. http://bura.brunel.ac.uk/handle/2438/3953.
Nieh, Ari. "Fractional Analogues in Graph Theory." Scholarship @ Claremont, 2001. https://scholarship.claremont.edu/hmc_theses/131.
Reed, Bruce. "A semi-strong perfect graph theorem /." Thesis, McGill University, 1986. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=72812.
Letzter, Shoham. "Extremal graph theory with emphasis on Ramsey theory." Thesis, University of Cambridge, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.709415.
Meek, Darrin Leigh. "On graph approximation heuristics : an application to vertex cover on planar graphs." Thesis, Georgia Institute of Technology, 1991. http://hdl.handle.net/1853/24088.
Zuffi, Lorenzo. "Simplicial Complexes From Graphs Toward Graph Persistence." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/13519/.
Anderson, Jon K. "Genetic algorithms applied to graph theory." Virtual Press, 1999. http://liblink.bsu.edu/uhtbin/catkey/1136714.
Peng, Richard. "Algorithm Design Using Spectral Graph Theory." Research Showcase @ CMU, 2013. http://repository.cmu.edu/dissertations/277.
Islam, Mustafa R. "A hypertext graph theory reference system." Virtual Press, 1993. http://liblink.bsu.edu/uhtbin/catkey/879844.
Edwards, C. S. "Some extremal problems in graph theory." Thesis, University of Reading, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.373467.
Garbe, Frederik. "Extremal graph theory via structural analysis." Thesis, University of Birmingham, 2018. http://etheses.bham.ac.uk//id/eprint/8869/.
Grinshpun, Andrey Vadim. "Some problems in Graph Ramsey Theory." Thesis, Massachusetts Institute of Technology, 2015. http://hdl.handle.net/1721.1/97767.
Pappone, Francesco. "Graph neural networks: theory and applications." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23893/.
Parks, David J. "Graph theory in America, 1876-1950." Thesis, Open University, 2012. http://oro.open.ac.uk/54663/.
Schuerger, Houston S. "Contributions to Geometry and Graph Theory." Thesis, University of North Texas, 2020. https://digital.library.unt.edu/ark:/67531/metadc1707341/.
Weaver, Robert Wooddell. "Some problems in structural graph theory /." The Ohio State University, 1986. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487268021746449.
Florkowski, Stanley F. "Spectral graph theory of the Hypercube." Thesis, Monterey, Calif. : Naval Postgraduate School, 2008. http://edocs.nps.edu/npspubs/scholarly/theses/2008/Dec/08Dec%5FFlorkowski.pdf.
Han, Lin. "Graph generative models from information theory." Thesis, University of York, 2012. http://etheses.whiterose.ac.uk/3726/.
Robinson, Laura Ann. "Graph Theory for the Middle School." Digital Commons @ East Tennessee State University, 2006. https://dc.etsu.edu/etd/2226.
Loveland, Susan M. "The Reconstruction Conjecture in Graph Theory." DigitalCommons@USU, 1985. https://digitalcommons.usu.edu/etd/7022.
Yi, Peipei. "Graph query autocompletion." HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/557.
Hegde, Rajneesh. "New Tools and Results in Graph Structure Theory." Diss., Georgia Institute of Technology, 2006. http://hdl.handle.net/1853/10481.
Lopez, Christian P. "On the relationship between a graph and the cycle graph of its complement." Thesis, Edith Cowan University, Research Online, Perth, Western Australia, 1995. https://ro.ecu.edu.au/theses/1184.
Morisi, Rita. "Graph–based techniques and spectral graph theory in control and machine learning." Thesis, IMT Alti Studi Lucca, 2016. http://e-theses.imtlucca.it/188/1/Morisi_phdthesis.pdf.
Hoang, Chinh T. "Perfect graphs." Thesis, McGill University, 1985. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74011.
Hayward, Ryan B. "Two classes of perfect graphs." Thesis, McGill University, 1986. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74025.
Olariu, Stephan. "Results on perfect graphs." Thesis, McGill University, 1986. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=73997.
Waterhouse, Mary Alexandra Paula Royston Hastilow. "Coloured graph decompositions /." [St. Lucia, Qld.], 2005. http://www.library.uq.edu.au/pdfserve.php?image=thesisabs/absthe18769.pdf.
Weinstein, Lee. "Empirical study of graph properties with particular interest towards random graphs." Diss., Connect to the thesis, 2005. http://hdl.handle.net/10066/1485.
Narayanan, Bhargav. "Problems in Ramsey theory, probabilistic combinatorics and extremal graph theory." Thesis, University of Cambridge, 2015. https://www.repository.cam.ac.uk/handle/1810/252850.
Fiala, Nick C. "Some topics in combinatorial design theory and algebraic graph theory /." The Ohio State University, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=osu1486402957198077.
Burns, Jonathan. "Recursive Methods in Number Theory, Combinatorial Graph Theory, and Probability." Scholar Commons, 2014. https://scholarcommons.usf.edu/etd/5193.
Turner, Bethany. "Embeddings of Product Graphs Where One Factor is a Hypercube." VCU Scholars Compass, 2011. http://scholarscompass.vcu.edu/etd/2455.
Srikanthan, T. "Bond graph analysis." Thesis, Coventry University, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.373896.
Chen, Xujin, and 陳旭瑾. "Graph partitions and integer flows." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2004. http://hub.hku.hk/bib/B30286256.
Vella, Antoine. "A Fundamentally Topological Perspective on Graph Theory." Thesis, University of Waterloo, 2005. http://hdl.handle.net/10012/1033.
Douma, Femke. "Counting and averaging problems in graph theory." Thesis, Durham University, 2010. http://etheses.dur.ac.uk/272/.
Al-Shimary, Abbas. "Applications of graph theory to quantum computation." Thesis, University of Leeds, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.608359.
Yilma, Zelealem Belaineh. "Results in Extremal Graph and Hypergraph Theory." Research Showcase @ CMU, 2011. http://repository.cmu.edu/dissertations/49.
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relations between objects. A graph in this context is made up of vertices or nodes and lines called edges that connect them. Figure 1.1: An example of graph with 6 vertices and 7 edges. Using algebraic properties of matrices associated to graphs, we can study the combinatorial properties of graphs. For example, spectral graph theory makes use of
a visual representation of the graph G with vertex set V = {x,y,z,w} and edge set E = {{x,y},{x,z},{y,z},{z,w}} is shown in Figure 1.1. Although a visual representation of a small graph might be useful, it is important to remember that a graph is just a pair of sets V and E where E ⊂ V 2.
1 A graph of order ve. 6 2 The same graph of order ve, reprinted for convenience. 31 3 A path of length three. 31 4 A cycle of length four. 31 5 K 4 covered by length 2 paths. 33 6 K 4 = W 3, W 4, and W 5. 34 7 The same graph of order ve, reprinted again. 45 8 A subgraph. 45 9 The induced subgraph on vertices v 1, v 2, and v 4. 45
Algebraic Graph Theory "A welcome addition to the literature . . . beautifully written and wide-ranging in its coverage."—MATHEMATICAL REVIEWS "An accessible introduction to the research literature and to important open questions in modern algebraic graph theory"—L'ENSEIGNEMENT MATHEMATIQUE.
An algebraic approach to graph theory can be useful in numerous ways. There is a relatively natural intersection between the elds of algebra and graph theory, speci cally between group theory and graphs. Perhaps the most natural connection between group theory and graph theory lies in nding the automorphism group of a given graph.
Explore the latest full-text research PDFs, articles, conference papers, preprints and more on ALGEBRAIC GRAPH THEORY. Find methods information, sources, references or conduct a literature review ...
The subject of this thesis is star sets and related aspects of algebraic graph theory. There is a close link between graph theory and linear algebra and the theory behind star sets exploits this. The first chapter provides the nec essary background in linear algebra, together with some basic graph theory,
Algebra and Graph Theory by Khodakhast Bibak A thesis presented to the University of Waterloo in ful llment of the thesis requirement for the degree of ... In this thesis, we make some contributions at the interface between 'algebra' and 'graph theory'. In Chapter 1, we give an overview of the topics and also the de nitions and prelimi- ...
use of algebraic and analytic tools in a graph-theoretic setting, especially homomorphisms and measures but also convergence, moments, quantum graphs and some spectral theory. For a nite graph G, let chG(q) denote the number of proper colorings of G with q colors. Discovered by Birkho [9], this is a polynomial in q, called the chromatic polyno-
In the second half of this thesis we study a purely extremal graph theoretic problem which turned out to be connected to algebraic graph theory in many ways, even its by-product provided an elegant solution to a longstanding open problem in algebraic graph theory. The use of graph transformations in extremal graph theory has a long history. The ...
Some Applications of Graph Theory. A thesis submitted for the degree of Doctor of Philosophy by Nicole Eggemann Department of Mathematical Sciences Brunel University September 2009. Acknowledgements. This work was funded by a Marie Curie Early Stage Training Fellowship (NET-ACE-programme) under grant number MEST-CT-2004-6724.
\Algebraic Graph Theory" by Chris Godsil and Gordon Royle. Other books that I nd very helpful and that contain related material include \Modern Graph Theory" by Bela Bollobas, \Probability on Trees and Networks" by Russell Llyons and Yuval Peres, \Spectra of Graphs" by Dragos Cvetkovic, Michael Doob, and Horst Sachs, and ...
Eigenvalues of Graphs. is an eigenvalue of a graph , is an eigenvalue of the adjacency matrix , A~x =. ~x for some vector ~x. Adjacency matrix is real, symmetric ) real eigenvalues, algebraic and geometric multiplicities are equal minimal polynomial is product of linear factors for distinct eigenvalues. Eigenvalues: = 3 = 1.
1. Graph theory This section presents the basic definitions, terminology and notations of graph theory, along with some fundamental results. Further information can be found in the many standard books on the subject - for example, West [4] or (for a simpler treatment) Wilson [5]. Graphs A graph G is a pair of sets (V, E), where V is a ...
was inspired by concepts from algebraic graph theory though, and on the other hand served as a motivation to study extremal near polygons. This way the last part of the thesis should close the circle and express the appreciation I gained of both approaches. Fr ed eric Vanhove January 2011
Graph Theory. Cesar O. Aguilar. An Introduction to Algebraic Graph Theory. Cover; Preface; 1 Graphs; 2 The Adjacency Matrix; 3 Graph Colorings; 4 Laplacian Matrices; 5 Regular Graphs; 6 Quotient Graphs; Cesar O. Aguilar Department of Mathematics
by using Graph Theory. At its core, graph theory is the study of graphs as mathematical structures. In our paper, we will first cover Graph Theory as a broad topic. Then we will move on to Linear Algebra. Linear Algebra is the study of matrices. We will apply the skills discussed in these two sections to Dijkstra
In this thesis FAPs are seen as generalised graph colouring problems, where transmitters are represented by vertices, and their interactions by weighted edges. Solving FAPs often relies on known structural properties to facilitate algorithms. ... Algebraic graph theory in the analysis of frequency assignment problems Pejić, Snežana (2008) ...
A highly symmetrical graph, the Petersen graph, which is vertex-transitive, symmetric, distance-transitive, and distance-regular.It has diameter 2. Its automorphism group has 120 elements, and is in fact the symmetric group.. Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs.This is in contrast to geometric, combinatoric, or ...
Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Paci c Math Oct 19 2009 10 / 36. Eigenvalues of Graphs is an eigenvalue of a graph, is an eigenvalue of the adjacency matrix,A~x= ~xfor some vector ~x Adjacency matrix is real, symmetric )
in exploring new areas of graph theory and its applications. Ad-vanced students in graph theory may use the topics presented in this book to develop their nal-year projects, master's theses or doctoral dissertations. It is the author's hope that this publication of original re-search ideas, problems and conjectures will instigate further re-xi
CORE - Aggregating the world's open access research papers
Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. Laplace's equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. ... Schuerger, Houston S. "Contributions to Geometry and Graph Theory." Thesis, University of North Texas, 2020. https://digital.library.unt ...
A function is definable if its graph is definable. A group is ... that would matter for the wider model theory. This thesis is confirmed, for example, in the recent papers of Freitag, Jimenez, and Moosa [26, 27]. ... In Model theory with applications to algebra and analysis. Vol. 2, 59-124, London Math. Soc. Lecture Note Ser., 350 ...