The Edge-Distance Matrix

The edge-distance matrix of a graph G, denoted by D, is the vertex-distance matrix of the corresponding line graph L(G)  

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All the criteria defined above can also be applied to search for central edges in the graphs using the information provided from the - edge distance matrix. For example, the centre distance-based criteria ID - 4D are defined as ... 

In the initial step, the distance-based criteria ID - 4D are applied to graph vertices to order them into equivalence classes identified by ranks 1,2,3. Rank 1 is assigned to the polycentre and the maximum rank to the most external vertices. Then the same procedure is applied to the edges on the basis of the edge distance matrix. 

Bonchev centric information indices are centric indices derived from the vertex - distance matrix D and the -> edge distance matrix "D, based on the concept of - graph centre and calculated as mean information content [Bonchev et al, 1980a Bonchev, 1983 Bonchev, 1989],... 

From the edge distance matrix several - topological information indices are calculated. Moreover, the atomic and molecular descriptors already defined for the vertex distance matrix are analogously defined for the edge distance matrix. 

A Wiener-type index - edge Wiener index - can be obtained from the edge distance matrix as ... 

More interesting is the -> edge-type Schultz index derived from both the edge distance matrix and the edge adjacency matrix. 

Bond multiplicity is taken into account by augmenting the edge distance matrix with a supplementary column and row where the elements are conventional bond orders, therefore obtaining an edge distance matrix for multigraphs [Bonchev, 1983]. All the local vertex invariants and molecular descriptors defined above can also be calculated on this matrix. 

An edge-type Schultz index has been derived from the - edge adjacency matrix E and the - edge distance matrix [Estrada and Gutman, 1996 Estrada and Rodriguez, 1997] ... 

Information indices on the edge distance matrix are listed below. 

The most important edge matrices are the edge adjacency matrix and the edge distance matrix D. Edge matrices of a molecular graph (j are usually calculated from the line... 

This is because the edge-distances in a graph are equal to the distances between vertices in the corresponding line graph. We give below the edge-distance matrix of G] (see structure B in Figure 2.1) as the vertex-distance matrix of L(G,). 

There are two kinds of the distance-degree matrices one is based on the vertex-distance matrix and the vertex-degrees, and the other is based on the edge-distance matrix and edge-degrees. 

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