The Unsymmetric Szeged Matrix

The unsymmetric Szeged matrix, denoted SZ, is an unsymmetric V x V matrix defined as 

The use of unsymmetric Szeged matrices is discussed by Todeschini and Consonni (2009). 

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The square symmetric Szeged matrix SZ, of dimension xy4, is obtained from the unsymmetric Szeged matrix SZ by the relation... 

The half sum of entries in the SZe/p matrix gives the Szeged index SZ, and the hyper-Szeged index SZp, respectively. The difference between SZp and SZe gives the SZ matrix SZa = SZp — SZe- -The unsymmetric Cluj matrix, CJu, was defined on the basis of the numbers N, and Ny. 

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. 

Unsymmetrical path-Szeged matrix and symmetrical edge- and path-Szeged matrices for 2,3-dimethylhexane. VSj and CSj indicate the matrix row and column sums, respectively. 

Most of the graph-theoretical matrices are symmetrical, whereas some of them are un-symmetrical. Examples of unsymmetrical matrices are —> Szeged matrices, —> Cluj matrices, random walk Markov matrix, —> combined matrices such as the topological distance-detour distance combined matrix, and some weighted adjacency and distance matrices. 

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