1st Edition

Graph-Theoretical Matrices in Chemistry

    160 Pages
    by CRC Press

    160 Pages 52 B/W Illustrations
    by CRC Press

    Graph-Theoretical Matrices in Chemistry presents a systematic survey of graph-theoretical matrices and highlights their potential uses. This comprehensive volume is an updated, extended version of a former bestseller featuring a series of mathematical chemistry monographs. In this edition, nearly 200 graph-theoretical matrices are included.

    This second edition is organized like the previous one—after an introduction, graph-theoretical matrices are presented in five chapters: The Adjacency Matrix and Related Matrices, Incidence Matrices, The Distance Matrix and Related Matrices, Special Matrices, and Graphical Matrices. Each of these chapters is followed by a list of references.

    Among the matrices presented several are novel and some are known only to a few. The properties and potential usefulness of many of the presented graph-theoretical matrices in chemistry have yet to be investigated.

    Most of the graph-theoretical matrices presented have been used as sources of molecular descriptors usually referred to as topological indices. They are particularly concerned with a special class of graphs that represents chemical structures involving molecules. Due to its multidisciplinary scope, this book will appeal to a broad audience ranging from chemistry and mathematics to pharmacology.

    Introduction
    References

    The Adjacency Matrix and Related Matrices
    The Vertex-Adjacency Matrix of Simple Graphs
    The Linear Representation of the Vertex-Adjacency Matrix of Acyclic Structures
    Labeling Graphs Using the Randić Procedure
    The Vertex-Adjacency Matrix of Multiple Graphs
    The Atom-Connectivity Matrix
    The Bond-Electron Matrix
    The Edge-Adjacency Matrix
    The Vertex-Adjacency Matrix of Weighted Graphs
    The Vertex-Adjacency Matrix of Möbius Graphs
    The Augmented Vertex-Adjacency Matrix
    The Edge-Weighted Edge-Adjacency Matrix
    The Burden Matrix
    The Vertex-Connectivity Matrix
    The Edge-Connectivity Matrix
    The Sum-Vertex-Connectivity Matrix
    The Sum-Edge-Connectivity Matrix
    Extended Adjacency Matrices
    Zagreb Matrices
    The Hückel Matrix
    The Laplacian Matrix
    The Generalized Laplacian Matrix
    The Augmented Vertex-Degree Matrix
    Distance-Weighted Adjacency Matrix

    References

    Incidence Matrices
    The Vertex-Edge Incidence Matrix
    The Edge-Vertex Incidence Matrix
    The Edge-Cycle Incidence Matrix
    The Cycle-Edge Incidence Matrix
    The Vertex-Path Incidence Matrix
    The Weighted-Hexagon-Kekulé-Structure Incidence Matrix
    References

    The Distance Matrix and Related Matrices
    The Standard Distance Matrix or the Vertex-Distance Matrix
    Generalized Vertex-Distance Matrix
    The Vertex-Galvez Matrix
    Combinatorial Matrices
    Reciprocal Combinatorial Matrices
    The Edge-Distance Matrix
    The Vertex-Distance-Complement Matrix
    The Augmented Vertex-Distance Matrix
    The Edge-Weighted Vertex-Distance Matrix
    The Barysz Vertex-Distance Matrix
    The Complement of the Barysz Vertex-Distance Matrix
    The Reciprocal Barysz Vertex-Distance Matrix
    The Reciprocal of the Complement of the Barysz Vertex-Distance Matrix
    The Complementary Vertex-Distance Matrix
    The Reciprocal of the Complementary Vertex-Distance Matrix
    Matrix of Dominant Distances in a Graph
    The Detour Matrix
    The Detour-Path Matrix
    The Detour-Delta Matrix
    The Edge-Weighted Detour Matrix
    The Maximum-Minimum Path Matrix
    The Detour-Complement Matrix
    The Vertex-Distance Matrix and the Detour Matrix of Complete Graphs and Complete Bipartite Graphs
    The Vertex-Harary Matrix
    The Edge-Harary Matrix
    The Edge-Weighted-Harary Matrix
    The Modified Edge-Weighted-Harary Matrix
    Distance-Degree Matrices
    The Resistance-Distance Matrix
    Distance/Distance Matrices
    The Common Vertex Matrix
    References

    Special Matrices
    Adjacency-Plus-Distance Matrices
    The Distance-Sum-Connectivity Matrix
    Wiener Matrices
    The Modified Wiener Matrices
    The Reverse-Wiener Matrix
    The Reverse-Detour Matrix
    Szeged Matrices
    Reciprocal Szeged Matrices
    The Unsymmetric Szeged Matrix
    Cluj Matrices
    Reciprocal Cluj Matrices
    The Hosoya Matrix
    The Path Matrix
    The All-Path Matrix
    The Expanded Vertex-Distance Matrices
    The Quotient Matrices
    The Random-Walk Markov Matrix
    Restricted Random-Walk Matrix
    The Transfer Matrix
    References

    Graphical Matrices
    Construction of Graphical Matrices
    Numerical Realization of Graphical Matrices
    A Generalized Procedure for Constructing Graphical Matrices and for Obtaining Their Numerical Representations
    References

    Concluding Remarks
    References
    Index

    Biography

    Dušanka Janežič, PhD, University of Primorska, Faculty of Mathematics, Natural Sciences and Information Technologies, Koper, Slovenia

    Ante Miličević, PhD, The Institute for Medical Research and Occupational Health, Zagreb, Croatia

    Sonja Nikolić, PhD, The Rugjer Bošković Institute, Zagreb, Croatia

    Nenad Trinajstić, PhD, fellow of the Croatian Academy of Sciences and Arts, The Rugjer Bošković Institute, Zagreb, Croatia