The Expanded Vertex-Distance Matrices

The expanded vertex-distance matrix, denoted by has been introduced by Tratch 

For acyclic graphs, the expanded vertex-distance matrix is given by 

Or in other words, the elements of the matrix A for acyclic graphs can be simply obtained from the elements of D and W by the Hadamard matrix product  

If the path-Wiener matrix, W, is substituted by the unsymmetric Cluj matrix, then the unsymmetric expanded vertex-distance matrix is obtained. This has been used for a new definition of the hyper-Wiener index (Diudea and Gutman, 1998 Ivanciuc et al, 1998 Gutman, 2004). This kind of expanded vertex-distance matrix of T2 is shown below  

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 

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