1st Edition

# Graph-Theoretical Matrices in Chemistry

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**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 Vertex-Adjacency Matrix of Simple Graphs**

The Adjacency Matrix and Related Matrices

The Adjacency Matrix and Related Matrices

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

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 Standard Distance Matrix or the Vertex-Distance Matrix**

The Distance Matrix and Related Matrices

The Distance Matrix and Related Matrices

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

**Adjacency-Plus-Distance Matrices**

Special Matrices

Special 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

**Construction of Graphical Matrices**

Graphical Matrices

Graphical Matrices

Numerical Realization of Graphical Matrices

A Generalized Procedure for Constructing Graphical Matrices and for Obtaining Their Numerical Representations

References

**References**

Concluding Remarks

Concluding Remarks

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