## Graphs, Matrices, and DesignsExamines partitions and covers of graphs and digraphs, latin squares, pairwise balanced designs with prescribed block sizes, ranks and permanents, extremal graph theory, Hadamard matrices and graph factorizations. This book is designed to be of interest to applied mathematicians, computer scientists and communications researchers. |

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### Contents

Linear Operators Preserving Partition Numbers of Graphs | 19 |

Completion of the Spectrum of Orthogonal Diagonal Latin Squares | 43 |

Extremal Problems for the BondyChvátal Closure of a Graph | 73 |

Ivan Friš Department of Mathematics Statistics and Computing Science University | 92 |

Minimum Biclique Partitions of the Complete Multigraph | 93 |

Multiplicativity of Generalized Permanents over Semirings | 121 |

Pairwise Balanced Designs with Block Sizes 5t + 1 | 147 |

Pairwise Balanced Designs with Holes | 171 |

The Sum Number of Complete Bipartite Graphs | 205 |

Nora Hartsfield Department of Mathematics Western Washington University | 213 |

Maximal Partial Latin Squares | 225 |

Katherine Heinrich Department of Mathematics Simon Fraser University | 237 |

Construction of New Hadamard Matrices with Maximal Excess | 255 |

Maximum Order Digraphs for Diameter 2 or Degree 2 | 269 |

The Cycle Space of an Embedded Graph III | 295 |

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### Common terms and phrases

1-factors additional Algebra appear Applications arcs assume bicliques bipartite blocks bound called cell clique partition column Combinatorial complete components condition connected consider construction contains contradiction Corollary covering cycle define definition deleting denote designs diagonal digraph directed disjoint edges elements empty enclosing entries equal equivalent exact partition exactly example exists factors Figure four further given gives graph graph G Hadamard matrices Hence independent set integers joined least Lemma length Linear Math Mathematics matrix maximal maximum minimal norm Note obtain occupied occur one-factors orthogonal pair parallel classes path PBDH(n points possible preserves problem Proof properties prove Pullman rank Rees refer regular remaining respectively result satisfying semiring sequences Suppose Table Theorem Theory transversal treatments triangles triples unit University values vertex vertices