Graphs and MatricesGraphs 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. |
Contents
| 11 | |
Adjacency Matrix | 25 |
Laplacian Matrix | 45 |
Cycles and Cuts | 57 |
Regular Graphs | 65 |
Algebraic Connectivity | 81 |
Distance Matrix of a Tree | 95 |
Resistance Distance | 111 |
Laplacian Eigenvalues of Threshold Graphs | 125 |
Positive Definite Completion Problem | 137 |
Matrix Games Based on Graphs | 145 |
Hints and Solutions to Selected Exercises | 159 |
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Common terms and phrases
adjacency matrix algebraic connectivity assume bipartite called Chapter characteristic clearly column complete components conclude connected graph Consider contains coordinate Corollary corresponding cycle defined denote determinant diagonal directed distance matrix easily edges eigenvalues eigenvalues of L(G eigenvector entries equals Example exists fact Fiedler vector follows fundamental G G G g-inverse given graph G graph with V(G hence holds incidence matrix indexed induced integer inverse Laplacian least Lemma Let G Linear loss matching matrix of G n×n matrix nonsingular nonzero Note observation obtained optimal strategies orientation orthogonal matrix partitioned path Player polynomial positive definite positive semidefinite principal Proof proof is complete prove rank Recall References resistance distance respectively result satisfies Show space spanning tree square subgraph submatrix suppose symmetric matrix Theorem Theory threshold graph tree with V(T unique vector verified vertex vertices zero