دانلود کتاب Formal power series and umbral chromatic polynomials of graphs

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Abstract 5 Statement of Qualifications and Research 8 Acknowledgements 9 Introduction 10 1 Power Series, Differential Operators and Umbral Calculus 20 1.1 Power Series and Differential Operators .................................. 20 1.2 Umbral Calculus .......................................................... 23 2 Posets, Incidence Algebras and Umbral Chromatic Polynomials 32 2.1 Posets and Incidence Algebras . ................... 32 2.2 Umbral Chromatic Polynomials 38 2.3 Examples .................................. 42 3 Colouring Chains and Multichains 48 3.1 Colour Partition Chains and Multichains 48 3.2 Assignment of Type Monomials ............................ 50 3.3 Examples of Colour Partition Chains and Multichains . . . . . . . . . 52 3.4 Colouring Chains and Multichains .......................... 57 4 Composition of A-Operators 58 4.1 The Umbra 0o y / .......................................................... 58 4.2 The Umbral Chromatic Polynomial ^^(G ;*) ................................ 59 4.3 The Umbra i ........................................ 67 4.4 The Umbra 0 o ^ o * * * o 0 . . . ........................................................ 68 4.5 Examples ........................................... 71 5 Compositional Inverses of A-operators 85 5.1 The Umbra 0 ..................................... 85 5.2 The Umbral Chromatic Polynomial ^(G ;x) ............................. 86 5.3 Examples ...................... 89 6 Umbral Chromatic Polynomials and p-typihcation 99 6.1 The Umbral Chromatic Polynomial Xp(G>x) • . . . . . . . . . . . . . 100 6.2 Formal Group Laws and Chromatic Polynomials . ........................... 105 6.3 The Umbral Chromatic Polynomial ^(G ;*). ........................... 108 6.4 Examples ........................................................................................... 109 6.4.1 The prime p = 2 ................. 109 6.4.2 The Prime p = 3 ................. 116 7 Products of Exponential Operators 120 7.1 The Umbra 6+ yr ................ 120 7.2 Umbra with ro * 1 . . .......................................................................... 121 7.3 The Umbral Chromatic Polynomial ^ ^ (G ; jc ) .................. 125 7.4 The Umbral Chromatic Polynomial 128 7.5 Examples .......................... 130 7.6 The Distributive Law for Umbra ................................ 134 8 Morphisms of Graphs 137 8.1 Proper Colourings and Graph Morphisms ......................................... 137 8.2 Null Graphs and Bipartite Complete Graphs....................................... 138 8.3 /w-partite Complete Graphs ................................................................. 140 Tables 143