## Algebraic CombinatoricsThis graduate level text is distinguished both by the range of topics and the novelty of the material it treats--more than half of the material in it has previously only appeared in research papers. The first half of this book introduces the characteristic and matchings polynomials of a graph. It is instructive to consider these polynomials together because they have a number of properties in common. The matchings polynomial has links with a number of problems in combinatorial enumeration, particularly some of the current work on the combinatorics of orthogonal polynomials. This connection is discussed at some length, and is also in part the stimulus for the inclusion of chapters on orthogonal polynomials and formal power series. Many of the properties of orthogonal polynomials are derived from properties of characteristic polynomials. The second half of the book introduces the theory of polynomial spaces, which provide easy access to a number of important results in design theory, coding theory and the theory of association schemes. |

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### Contents

The Matchings Polynomial | 1 |

The Characteristic Polynomial | 19 |

Formal Power Series and Generating Functions | 37 |

Quotients of Graphs | 75 |

Matchings and Walks | 93 |

Pfaffians | 113 |

Orthogonal Polynomials | 131 |

Moment Sequences | 149 |

DistanceRegular Graphs | 195 |

Association Schemes | 221 |

Representations of DistanceRegular Graphs | 261 |

Polynomial Spaces | 285 |

Q Polynomial Spaces | 307 |

Tight Designs | 333 |

Terminology | 353 |

359 | |

### Common terms and phrases

adjacency matrix algebraic antipodal association scheme automorphism bipartite graph blocks cells Chapter characteristic polynomial classes closed walks coefficients combinatorial completely regular contains Corollary cospectral cycle deduce define Delsarte denote determine distance-regular graph edges eigenvalue eigenvalue of G eigenvector elements equal equitable partition Exercise exponential generating function finite fl,p follows formal power series graph G graph of diameter Hence i-th idempotents identity ij-entry implies imprimitive inner product isomorphic Lemma Let G linear combination mapping matchings polynomial Math multiplicity n x n non-trivial non-zero number of perfect number of vertices obtained orthogonal polynomials path perfect codes perfect matchings permutation pn(x points Pol(fi polynomial of degree Proof prove Q-polynomial radius result rows Schur Section sequence of orthogonal Show spanned spherical polynomial space strongly regular graph subset subspace Suppose t-design Theorem theory unit sphere valency vector space vertex set vertices of G walks of length weighted zero