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Algebraic Combinatorics

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Algebraic Combinatorics ALGEBRAIC COMBINATORICS c C D Go dsil To Gillian Preface There are p eople who feel that a combinatorial result should b e given a purely combinatorial pro of but I am not one of them For me the most interesting parts of combinatorics have always b een those overlapping other areas of mathematics This b o ok is an intro duction to some of the interac tions b etween algebra and combinatorics The rst half is devoted to the characteristic and matchings p olynomials of a graph and the second to p olynomial spaces However anyone who lo oks at the table of contents will realise that many other topics have found their way in and so I expand on this summary The characteristic p olynomial of a graph is the characteristic p olyno matrix The matchings p olynomial of a graph G with mial of its adjacency n vertices is b n2c X k n2k G k x p k =0 where pG k is the numb er of k matchings in G ie the numb er of sub graphs of G formed from k vertexdisjoint edges These denitions suggest that the characteristic p olynomial is an algebraic ob ject and the matchings p olynomial a combinatorial one Despite this these two p olynomials are closely related and therefore they have b een treated together In devel oping their theory we obtain as a bypro duct a numb er of results ab out orthogonal p olynomials The numb er of p erfect matchings in the comple ment of a graph can b e expressed as an integral involving the matchings by which we p olynomial This motivates the study of moment sequences mean sequences of combinatorial interest which can b e represented as the sequence of moments of some measure To b e brief if not cryptic a p olynomial space is obtained by asso ciat ing an inner pro duct space of p olynomials to a combinatorial structure The combinatorial structure might b e the set of all k subsets of a set of v elements the symmetric group on n letters or if the reader will b e gener n ous the unit sphere in R Given this setup it is p ossible to derive b ounds of co des and designs in the structure The derivations are on the sizes very simple and apply to a wide range of structures The resulting b ounds are often classicalthe simplest and b est known is Fishers inequality from design theory iv Preface Polynomial spaces are p erhaps imp ossibly general We distinguish one imp ortant family which corresp onds when the underlying set is nite to Q p olynomial asso ciation schemes The latter have a welldevelop ed theory thanks chiey to work of Delsarte Our approach enables us to rederive and extend much of this work In summary the theory of p olynomial spaces provides an axiomatisation of many of the applications of linear algebra to combinatorics along with a natural way of extending the theory of Q p olynomial asso ciation schemes to the case where the underlying set is innite From this discussion it is clear that to make sense of p olynomial spaces some feeling for asso ciation schemes is required Hence I have included a reasonably thorough intro duction to this topic To motivate this in turn I have also included chapters on strongly regular and distanceregular graphs Orthogonal p olynomials arise naturally in connection with p olyno mial spaces and distanceregular graphs and thus form a connecting link b etween the two parts of this b o ok My aim has b een to write a b o ok which would b e accessible to b egin ning graduate students I b elieve it could serve as a text for a numb er of t courses in combinatorics at this level and I also hop e that it will dieren prove interesting to browse in The prerequisit es for successful digestion of the material oered are Linear algebra Familiarity with the basics is taken for granted The sp ectral decomp osition of a Hermitian matrix is used more than once The theory is presented in Chapter Positive semidenite matrices app ear A brief summary of the relevant material is included in the app endix Combinatorics The basic language of graph theory is used without preamble eg spanning trees bipartite graphs and chromatic num b er Once again some of this is included in the app endix Generating functions and formal p ower series are used extensively in the rst half of the b o ok and so there is a chapter devoted to them Group theory The symmetric group creeps in o ccasionally along with automorphism groups of graphs The orthogonal group is men tioned by name at least once Ignorance By which I mean the ability to ignore the o dd paragraph devoted to unfamiliar material in the trust that it will all b e ne at the end I have not b een able to draw up a dep endence diagram for the chapters which would not b e misleading This is b ecause there are few chains of argument extending across chapter b oundaries but many cases where the Preface v material in one chapter motivates another For example it should b e p ossible to get through the chapter on asso ciation schemes without reading the preceding chapter on distanceregular graphs However these graphs provide one of the most imp ortant classes of asso ciation schemes By way of comp ensation for the lack of this traditional diagram I include some suggestions for p ossible courses The matchings p olynomial and moment sequences The characteristic p olynomial Strongly regular graphs distanceregular graphs and asso ci ation schemes Equitable partitions and co des in distanceregular graphs Polynomial spaces In making these suggestions I have made no serious attempt to consider the time it would take to cover the material indicated On the basis of my own exp erience I think it would b e p ossible to cover at most three pages p er hour of lectures On the other hand it would b e easy enough to pare down the suggestions just made For example Chapter covers formal p ower series and generating functions and dep ending on the backgrounds of ones victims this might not b e essential in and Bender Andries I have b een help ed by advice and comments from Ed Brouwer Dom de Caen Michael Do ob Mark Ellingham Tony Gardiner y Gillian Nonay Jack Ko olen Gordon Royle Bill Martin Brendan McKa J J Seidel and Akos Seress Dom in particular has made heroic eorts to protect me from my own stupidity I am very grateful for all this assistance I would also like to thank John Kimmel and Jim Geronimo of Chapman and Hall for their part in the pro duction of this b o ok Contents Preface vii Contents xi The Matchings Polynomial Recurrences Integrals Ro ok Polynomials The Hit Polynomial Stirling and Euler Numb ers Hit Polynomials and Integrals Exercises Notes and References The Characteristic Polynomial Co ecients and Recurrences Walks and the Characteristic Polynomial Eigenvectors Regular Graphs The Sp ectral Decomp osition Some Further Matrix Theory Exercises Notes and References Formal Power Series and Generating Functions Formal Power Series Limits Op erations on Power Series Exp and Log Nonlinear Equations Applications and Examples Exercises Notes and References viii Contents Walk Generating Functions Jacobis Theorem Walks and Paths A Decomp osition Formula The ChristoelDarb oux Identity ertex Reconstruction V Cosp ectral Graphs Random Walks on Graphs Exercises Notes and References of Graphs Quotients Equitable Partitions Eigenvalues and Eigenvectors WalkRegular Graphs Generalised Interlacing Covers The Sp ectral Radius of a Tree Exercises Notes and References Matchings and Walks The PathTree TreeLike Walks Consequences of Reality ChristoelDarb oux Identities Exercises Notes and References Pfaans The Pfaan of a Skew Symmetric Matrix Pfaans and Determinants Row Expansions Oriented Graphs Orientations The Diculty of Counting Perfect Matchings Exercises Notes and References Contents ix Orthogonal Polynomials The Denitions The ThreeTerm Recurrence The ChristoelDarb oux Formula Discrete Orthogonality Sturm Sequences Some Examples Exercises Notes and References Moment Sequences Moments and Walks Moment Generating Functions Hermite and Laguerre Polynomials The Chebyshev Polynomials The Charlier Polynomials Sheer Sequences of Polynomials Characterising Polynomials of Meixner Typ e The Polynomials of Meixner Typ e Exercises Notes and References Strongly Regular Graphs Basic Theory Conference Graphs Designs Orthogonal Arrays Exercises Notes and References DistanceRegula r Graphs Some Families Distance Matrices Parameters ts Quotien Imprimitive DistanceRegular Graphs Co des Completely Regular Subsets Examples Exercises Notes and References x Contents Asso ciation Schemes Generously Transitive Permutation Groups s ps and q P and QPolynomial Asso ciation Schemes Pro ducts Primitivity and Imprimitivity Co des and Antico des Equitable Partitions of Matrices Characters of Ab elian Groups Cayley Graphs Translation Schemes and Duality Exercises Notes and References Representations of DistanceRegular Graphs Representations of Graphs The Sequence of Cosines Injectivity Eigenvalue Multipliciti es Bounding the Diameter Spherical Designs Bounds for Cliques Feasible Automorphisms Exercises Notes and References Polynomial Spaces Functions The Axioms Examples The Degree of a Subset Designs The Johnson Scheme The Hamming Scheme Co ding Theory GroupInvariant Designs Weighted Designs Exercises Notes and References Contents xi QPolynomial Spaces Orthogonal Polynomials Zonal Zonal Orthogonal Polynomials Examples The Addition Rule Spherical Polynomial Spaces Harmonic Polynomials Asso ciation Schemes QPolynomial Asso ciation Schemes Incidence Matrices for Subsets v k is QPolynomial J Exercises Notes and References Tight Designs Tight Bounds Examples and NonExamples The Grassman Space Linear Programming Bigger Bounds Examples Exercises Notes and References Terminology App endix Index of Symb ols Index .
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