The Vertex-Galvez Matrix

The vertex-Galvez matrix of a graph G, denoted by GM, is a square asymmetric matrix that can be obtained by multiplying the vertex-adjacency matrix A and the reciprocal square vertex-distance matrix (Gdlvez et al., 1994,1995 Todeschini and Consonni, 2000, 2009)  

The vertex-Galvez matrix GM can be, for example, used to generate the charge-transfer matrix. This matrix, denoted as CTM, is defined as 

The diagonal entries of the charge-transfer matrix (CTM) represent the topological valence of the atoms in a molecule, while the off-diagonal elements (CTM)y represent a measure of the net charge transfer from the atom j to the atom i, or if the value of (CTM),y is negative, the atom / will transfer net charge to atom j. 

---

As an example, we give the vertex-Galvez matrix for T2 (see Figure 2.19). Below we give the vertex-adjacency matrix of T2, while the reciprocal square vertex-distance matrix D is given above. 

---