ぱらぱらめくる『Graphs and Matrices』

Graphs and Matrices (Universitext)

Graphs and Matrices (Universitext)

  • 目次
  • 1 Preliminaries
  • 2 Incidence Matrix
  • 3 Adjacency Matrix
  • 4 Laplacian Matrix
  • 5 Cycles and Cuts
  • 6 Regular Graphs
  • 7 Algebraic Connectivity
  • 8 Distance Matrix of a Tree
  • 9 Resistance Distance
  • 10 Laplacian Eigenvalues of Threshold Graphs
  • 11 Positive Definite Completion Problem
  • 12 Matrix Games Based on Graphs
  • 本全体の大雑把なまとめはこちら
  • 細目次
  • 1 Preliminaries
    • 1.1 Matrices
    • 1.2 Eigenvalues of symmetric matrices
    • 1.3 Generalized inverses
    • 1.4 Graphs
  • 2 Incidence Matrix
    • 2.1 Rank
    • 2.2 Minors
    • 2.3 Path matrix
    • 2.4 Integer generalized inverses
    • 2.5 Moore-Pnrose inverse
    • 2.6 0-1 Incidence matrix
    • 2.7 Matchings in bipartite graphs
  • 3 Adjacency Matrix
    • 3.1 Eigenvalues of some graphs
    • 3.2 Determinant
    • 3.3 Bountds
    • 3.4 Energy of a graph
    • 3.5 Antiadjacency matrix of a directed graph
    • 3.6 Nonsingular trees
  • 4 Laplacian Matrix
    • 4.1 Basic properties
    • 4.2 Computing Laplacian eigenvaues
    • 4.3 Matrix-tree theorem
    • 4.4 Bounds for Laplacian spectral radius
    • 4.5 Edge-Laplacian of a tree
  • 5 Cycles and Cuts
    • 5.1 Fundamental cycles and fundamental cuts
    • 5.2 Fundamental matrices
    • 5.3 Minors
  • 6 Regular Graphs
    • 6.1 Perron-Frobenius theory
    • 6.2 Adjacency algebra of a regular graph
    • 6.3 Complement and line graph of a regular graph
    • 6.4 Strongly regular graphs and friendship theorem
    • 6.5 Grapshs with maximum energy
  • 7 Algebraic Connectivity
    • 7.1 Preliminary results
    • 7.2 Classification of trees
    • 7.3 Monotonicity properties of Fiedler vector
    • 7.4 Bounds for algebraic connectivity
  • 8 Distance Matrix of a Tree
    • 8.1 Distance matrix of a graph
    • 8.2 Distance matrix and Laplacian of a tree
    • 8.3 Eigenvalues of the distance matrix of a tree
  • 9 Resistance Distance
    • 9.1 The triangle inequality
    • 9.2 Network flows
    • 9.3 A random walk on graphs
    • 9.4 Effective resistance in electrical networks
    • 9.5 Resistance matrix
  • 10 Laplacian Eigenvalues of Thereshold Graphs
    • 10.1 Majorization
    • 10.2 Threshold graphs
    • 10.3 Spectral integral variation
  • 11 Positive Definite Completion Problem
    • 11.1 Nonsingular completion
    • 11.2 Chordal graphs
    • 11.3 Positive definite completion
  • 12 Matrix Games Based on Graphs
    • 12.1 Matrix games
    • 12.2 Vertex selection games
    • 12.3 Tournament games
    • 12.4 Incidence matrix games