The Distance Matrix and Related Matrices

Distance matrices are much richer algebraic structures than the adjacency matrices (Harary, 1971 Buckley and Harary, 1990). They are square symmetric V x y matrices whose entries are graph-theoretical distances between the vertices. Augmented distance matrices have nonzero values on the main diagonal. A number of distance matrices and molecular descriptors derived from them were recently reviewed by Zhou and Trinajstid (2010). 

1 THE STANDARD DISTANCE MATRIX OR THE VERTEX-DISTANCE MATRIX 

The standard distance matrix or the vertex-distance matrix (or the minimum path matrix) of a vertex-labeled connected graph G (Harary, 1971 Gutman and Polansky, 1986 Buckley and Harary, 1990 Trinajstid, 1992 Mihalic et al., 1992 Todeschini and Consonni, 2000,2009 Consonni and Todeschini, 2012), denoted by D, is a real symmetric VxV matrix whose elements are defined as 

Their vertex-distance matrices are different and are given as follows  

These kind of graphs are called twin graphs (Hosoya et al., 1994, 2001), because they possess, besides identical distance-spectra and consequently identical distance-polynomials, identical characteristic polynomials and their spectra (3.8801, 1.3557, 0.7732, 0.4773, -0.7376, -1.2464, -2.0953, -2.4069), identical matching polynomials and their spectra, and many identical graph-theoretical invariants. 

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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. 

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