Reciprocal Cluj Matrices

The reciprocal Cluj matrices are matrices whose entries are the reciprocal of the corresponding entries of the symmetric Cluj matrices (Diudea and Gutman, 1998 Diudea et al., 1998). Here we present the reciprocal edge-Cluj matrix, denoted 

However, the reciprocal edge-Cluj matrix is equal to the reciprocal edge-Szeged matrix  

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Derived from Cluj matrices, the reciprocal Cluj matrices CJ are the matrices whose elements are the reciprocal of the corresponding Cluj matrix elements [Diudea, 1997c Diudea, Katona etal., 1998]. Harary indices and hyper-Harary indices are defined For these matrices... 

These are molecular indices derived from —> reciprocal Cluj matrices. The Harary-type index is calculated from the reciprocal edge-Cluj-distance matrix CJD as... 

The most popular reciprocal matrices are obtained for 2, = 1, such as the Harary matrix, reciprocal geometry matrix, reciprocal detour matrix, reciprocal Szeged matrix, reciprocal Cluj matrix. The reciprocal square distance matrix is derived from the distance matrix by setting X = 2. 

The Harary index and hyper-Harary index are obtained from, respectively, the 1 -order sparse -> reciprocal Cluj-distance matrix ... 

The most popular expanded matrices are —> expanded distance matrices, D M, derived as the Hadamard product between the —> distance matrix D and some different graph-theoretical matrix M, such as the —> Wiener matrix, —> Cluj matrices, Szeged matrix, and walk matrices. Moreover, —> expanded reciprocal distance matrices, D M, were defined by analogy with the expanded distance matrices by using the —> reciprocal distance matrix instead of the... 

Selecting different combinations of Mi and M2 matrices leads to the derivation of several Schultz-type indices. The original Schultz molecular topological index MTI is obtained for Ml = A and M2 = D, where D is the topological —> distance matrix. Typical Schultz indices are derived from (D, A, D), A, D ), (W, A, D), (W A, D ), (W, A, W), (UCJ, A, UCJ), (USZ, A, USZ), where is the reciprocal distance matrix, W is one among —> walk matrices, is the reciprocal walk matrix, UCJ and USZ the unsymmetrical Cluj and Szeged matrices, respectively. 

As an example we give the reciprocal path-Cluj matrix of Gj (see structure A in Figure 2.1) ... 

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