A path-Szeged matrix is a graph matrix defined for a connected graph using the Szeged closeness counts for all unordered pairs of distinct vertices, not only for adjacent vertices (Diudea et al. 1997; Devillers and Balaban 1999, pp. 256-257).
If 
and 
are distinct graph vertices, the 
entry is the product of the number of vertices closer
to 
than to 
and the number of vertices closer to 
than to 
. Diagonal entries are taken to be 0.
The ordinary Szeged matrix, also known as the edge-Szeged matrix, is obtained from the path-Szeged matrix by retaining only the entries corresponding to adjacent vertices. Equivalently, it is the Hadamard product of the path-Szeged matrix and the adjacency matrix.
