Formal power series and umbral chromatic polynomials of graphs

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