Generalized Vertex-Distance Matrix

The generalized vertex-distance matrix, denoted as D, is derived from the distance matrix D by raising its elements to an exponent h 

The squared vertex-distance matrix and reciprocal squared vertex-distance matrix D have been discussed by Consonni and Todeschini (Todeschini and Consonni, 2000, 2009 Consonni and Todeschini, 2012). 

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Diudea and Gutman (1998) generalized the concept of the expanded vertex-distance matrix in order to define novel matrices using the Hadamard matrix product between the vertex-distance matrix D and a general square VxVmatrix V as... 

The classical definition for the graph centre is not helpful for cyclic graphs where often a large number of vertices (most of them topologically non-equivalent) appear as central. Recently, Bonchev, Balaban and Mekenyan 67) proposed a generalized concept for the graph centre, and several centric indices were derived on this basis. The new definition consists of four hierarchically ordered criteria based on the distance matrix 1) the smallest maximum distance in the row or column of the vertex ... 

Note. The vertex-distance-vertex-degree matrices with P = 0 were called v d matrices by Perdih [Perdih and Perdih, 2002a] and the general distance-degree matrix was denoted by G a,b,c) [Perdih and Perdih, 2004], whose elements are v vj dl [ Perdih and Perdih, 2003c], where v denotes the vertex degree 6 and a, b, and c are the parameters corresponding to p, y, and a, respectively. 

Examples of molecular descriptors derived from generalized matrices are distance distribution moments and W% indices from the distance matrix and molecular profiles from the geometry matrix. Moreover, vertex Zagreb matrix (X = 2) and modified vertex Zagreb matrix (X = —2) are a generalization of the vertex degree matrix, and the generalized... 

For vertex- and edge-weighted molecular graphs, the weighted distance matrix is generally defined as a square symmetric A x A matrix, A being the number of graph vertices. 

Derived from the Barysz distance matrix, a general weighting scheme was proposed by Ivanciuc [Ivanciuc, Ivanciuc et al, 1999a Ivanciuc, 2000a, 2000i] in terms of the conventional bond order n and any atomic property Pj. The vertex weight Wi associated with the vertex V was defined as... 

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