Szeged matrices

Several Szeged matrices have been proposed in the literature (Diudea et al., 1997). Here, we will consider the edge-Szeged matrix, the path-Szeged matrix, and the Szeged difference matrix. 

The edge-Szeged matrix, denoted by SZ, was introduced by Gutman (1994a). It is formally defined as the Wiener matrix  

The summation of elements in the upper or lower matrix-triangle produces the hyper-Szeged index (Todeschini and Consonni, 2000,2009). 

The Szeged difference matrix, denoted by S, is defined as the difference of the edge- and path-Szeged matrices  

Research on the Szeged index and its variants was nicely summarized by Todeschini and Consonni (2000, 2009) in their handbooks and in the review article by Khadikar and his coworkers (Khadikar et al., 2005), 

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Characteristic polynomials and Hosoya-type indices were also derived from distance-valency matrices, —> distance-path matrix, reciprocal distance-path matrix, distance-delta matrix, —> Szeged matrices [Ivanciuc and Ivanciuc, 1999], —> layer matrices, and edge adjacency matrix. 

These are molecular indices derived from —> reciprocal Szeged matrices. The Harary-type index is obtained from the reciprocal edge-Szeged matrix SZ as ]Diudea, 1997c]... 

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. 

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. 

There are two types of symmetric Szeged matrices. One is the path-Szeged matrix, denoted by S2p, obtained when all the off-diagonal entries are calculated as the product of the numbers Nfp and Njp of the vertices lying closer to the vertices V and Vj, respectively, for all the pairs (i, j) of vertices. The path-Szeged matrix SZ is formally defined as... 

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. 

Reciprocal Szeged matrices, denoted as SZ, are matrices whose off-diagonal elements are the reciprocal of the corresponding elements of the Szeged matrices [Diudea, 1997a Diudea, Parv et al, 1997b] ... 

All elements equal to zero are left unchanged in the reciprocal Szeged matrices. 

Harary Szeged indices Hsz, and Hsz are calculated from reciprocal symmetric edge- and path-Szeged matrices, respectively, by applying the Wiener operator. 

Other important weighted vertex adjacency matrices are the extended adjacency matrices, the edge-Wiener matrix, —> edge-Cluj matrices, —> edge-Szeged matrices, and the random walk Markov matrix. 

Diudea, M.V., Minailiuc, O.M., Katona, G. and Gutman, I. (1997b) Szeged matrices and related... 

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

M.V. Diudea, O.M. Minailiuc, G. Katona, and I. Gutman, Szeged matrices and related numbers, MATCH Commun. Math. Comput. Chem. 35 (1997) 129-143. 

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