Graphs and Matrices

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Springer Science & Business Media, Jul 23, 2010 - Mathematics - 171 pages
Graphs and Matrices provides a welcome addition to the rapidly expanding selection of literature in this field. As the title suggests, the book’s primary focus is graph theory, with an emphasis on topics relating to linear algebra and matrix theory. Information is presented at a relatively elementary level with the view of leading the student into further research. In the first part of the book matrix preliminaries are discussed and the basic properties of graph-associated matrices highlighted. Further topics include those of graph theory such as regular graphs and algebraic connectivity, Laplacian eigenvalues of threshold graphs, positive definite completion problem and graph-based matrix games. Whilst this book will be invaluable to researchers in graph theory, it may also be of benefit to a wider, cross-disciplinary readership.
 

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Contents

Preliminaries
1
12 Eigenvalues of symmetric matrices
5
13 Generalized inverses
7
14 Graphs
9
Incidence Matrix
11
21 Rank
12
22 Minors
13
23 Path matrix
15
64 Strongly regular graphs and friendship theorem
73
65 Graphs with maximum energy
76
Algebraic Connectivity
81
72 Classification of trees
83
73 Monotonicity properties of Fiedler vector
88
74 Bounds for algebraic connectivity
89
Distance Matrix of a Tree
95
81 Distance matrix of a graph
96

24 Integer generalized inverses
16
25 MoorePenrose inverse
17
26 01 Incidence matrix
19
27 Matchings in bipartite graphs
21
Adjacency Matrix
25
31 Eigenvalues of some graphs
26
32 Determinant
28
33 Bounds
31
34 Energy of a graph
36
35 Antiadjacency matrix of a directed graph
37
36 Nonsingular trees
39
Laplacian Matrix
45
41 Basic properties
46
42 Computing Laplacian eigenvalues
47
43 Matrixtree theorem
48
44 Bounds for Laplacian spectral radius
50
45 EdgeLaplacian of a tree
51
Cycles and Cuts
57
52 Fundamental matrices
59
53 Minors
60
Regular Graphs
65
62 Adjacency algebra of a regular graph
70
82 Distance matrix and Laplacian of a tree
99
83 Eigenvalues of the distance matrix of a tree
104
Resistance Distance
111
91 The triangle inequality
112
92 Network flows
113
93 A random walk on graphs
116
94 Effective resistance in electrical networks
118
95 Resistance matrix
119
Laplacian Eigenvalues of Threshold Graphs
125
102 Threshold graphs
129
103 Spectral integral variation
131
Positive Definite Completion Problem
137
112 Chordal graphs
138
113 Positive definite completion
140
Matrix Games Based on Graphs
145
122 Vertex selection games
147
123 Tournament games
148
124 Incidence matrix games
151
Hints and Solutions to Selected Exercises
159
Bibliography
165
Index
169
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