The Edge-Harary Matrix

The edge-Harary matrix of a graph G, denoted by IK, is the vertex-Harary matrix of the corresponding line graph UG)  

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We show below the edge-Harary matrix of Gj (see structure B in Figure 2.1), that is, the vertex-Harary matrix of L(G ) (see Figure 2.12). 

The reciprocal edge distance matrix (or edge Harary matrix), denoted as is a square... 

The edge-Harary index is defined as the half sum of the elements of the edge-weighted Harary matrix ... 

By generalization, Harary indices and hyper-Harary indices (or hyper-Harary numbers) are all molecular descriptors derived from the application of the Wiener operator to reciprocal topological matrices Harary indices are obtained from the 1 -order - sparse matrix, i.e. considering only the graph edges, while the hyper-Harary indices are from the whole matrices, i.e. considering the paths [Diudea, 1997c]. 

The total adjacency index Ay is a measure of the graph connectedness and is calculated as the sum of all the entries of the adjacency matrix of a molecular graph, vhich is tv ice the number B of graph edges [Harary, 1969a Rouvray, 1983] ... 

The Harary index and hyper-Harary index, defined only for acyclic graphs, are obtained from reciprocal Wiener matrix [Diudea, 1997c Diudea and Gutman, 1998], The two following indices are derived, respectively, from the reciprocal edge-Wiener matrix and reciprocal path-... 

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]... 

The edge-weighted Harary matrix, denoted as is a symmetric weighted adjacency matrix defined by weighting each edge between adjacent vertices V and + as the following [Lucic, Milicevic et al., 2002 Janezic, Milicevic et al., 2007] ... 

All elements equal to zero are left unchanged in the reciprocal matrix. Note that the reciprocal edge-Wiener matrix W was later called by other authors modified edge-Wiener matrix and denoted as W [Nikolic, Trinajstic et al., 2001b Janezic, Milicevic et al., 2007]. Harary Wiener indices and are derived by applying the Wiener operator to the reciprocal edge-... 

The vertex-edge incidence matrix of a graph G, denoted by VE, is an unsymmetri-cal (and, in general, not a square) Vx E matrix, which is determined by the incidences of vertices and edges in G (Harary, 1971 Johnson and Johnson, 1972 Bondy and Murty, 1976 Rouvray, 1976 Chartrand, 1977 Trinajstid, 1992 Todeschini and Consonni, 2000, 2009) ... 

The edge-weighted-Harary matrix of T2 is given as follows ... 

The modified edge-weighted-Harary matrix, denoted by is obtained from the edge-weighted-Harary matrix by... 

The modified edge-weighted-Harary matrix serves for computing the modified edge-weighted-Harary index. This index has been successfully tested in the structure-property modeling of alkanes (Lucid et al., 2002), but its use with polycyclic structures has not yet been investigated. 

In the same simple distance- (path) or degree-based category we may recall the Harary index (Hal) Harary (1969), defined as half-sum of the off-diagonal elements of the reciprocal molecular distance matrix of the molecule, as well as the Platt index (Pal) computed as the sum of the edge degrees of a molecular graph (Platt, 1952). 

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