The Edge-Adjacency Matrix

The edge-adjacency matrix, denoted by A, of an edge-labeled connected graph G is a square ExE matrix, which is determined by the adjacencies of edges (Rouvray, 1976 Trinajstid, 1983, 1992)  

It should be noted that the vertex-adjacency matrix uniquely determines a graph, but the edge-adjacency matrix does not that is, there are known nonisomorphic graphs with identical edge-adjacency matrices. For example, a pair of nonisomorphic graphs—the three-point star and the cycle on three vertices C3— 

FIGURE 2.11 A pair of nonisomorphic graphs consisting of the 3-star and the 3-cycle C3 

FIGURE 2.12 Construction of the line graph L G ) from the graph G.  

Below we give the edge-adjacency matrix that represents both and C  

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Estrada [13] proposes the use of the edge-adjacency matrix, E, to develop new molecular descriptors. For example, G-I has five edges, which can be labeled in the following way ... 

Then the edge-adjacency matrix E of G-I is derived as the normal adjacency matrix of E-I ... 

Another application of basic subgraphs arises from the possibility of relating the invariants of molecular graphs to the occurrence numbers of some basic subgraphs. Estrada has developed this methodology for spectral moments of the edge-adjacency matrix of molecular graphs - defined as the traces of the different powers of such matrix ... 

The edge degree e, provides the simplest information related to the considered bond and is calculated from the edge adjacency matrix as follows ... 

The total edge adjacency index Ae (also known as Platt number, F [Platt, 1947 Platt, 1952]) is the sum over all entries of the edge adjacency matrix ... 

Ngs) [Gordon and Scantlebury, 1964] or Bertz branching index Bl), is the simplest graph-invariant obtained from the edge adjacency matrix which considers both vertices and edges and is calculated as ... 

From the edge adjacency matrix, a graph-theoretical invariant analogous to the -> Randic connectivity index was derived by Estrada [Estrada, 1995a] it is called edge connectivity index, denoted by e, and defined as ... 

The spectral moments of the edge adjacency matrix E were defined [Estrada, 1996] as ... 

A bond distance-weighted edge adjacency matrix is derived from the edge adjacency matrix using - bond distances calculated by computational chemistry methods as diagonal entries on selected molecular geometries [Estrada, 1997] ... 

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

Obviously, spectral moments p of the P order line graph Lj are the spectral moments of the edge adjacency matrix, and 0- order spectral moments po of any line graph L coincide with the number of vertices in the considered line graph. 

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 adjacency matrix E are listed below. 

Derived from the - edge adjacency matrix E, this is defined as ... 

This is calculated by dividing the total information content of the edge adjacency matrix elements equality by the total number of edge adjacency matrix elements Bh E-jE 2N2, 2N2 (. 2N2, (. 2N2 ... 

Estrada, E. (1996). Spectral Moments of the Edge Adjacency Matrix of Molecular Graphs. 1. Definition and Applications to the Prediction of Physical Properties of Alkanes. J.Chem.Inf... 

Estrada, E. (1997) Spectral moments of the edge-adjacency matrix of molecular graphs. 2. Molecules containing heteroatoms and QSAR applications. Journal of Chemical Information and Computer Sciences, 37, 320-328. 

Derived from the —> molecular graph Q, the edge adjacency matrix, denoted by E, or more formally as A, also called bond matrix, encodes information about the coimectivity between... 

It is a square symmetric matrix of dimension B x B, where B is the number of bonds, and is usually derived from a H-depleted molecular graph [Bonchev, 1983]. It is to be noted that the edge adjacency matrix of a graph Q is equal to the adjacency matrix of the line graph of Q [Gutman and Estrada, 1996]. 

Edge invariants can also be directly obtained by physico-chemical properties of the bonds used as the —> weighting scheme for graph edges, such as bond dipole moments, bond order indices, and so on, as well as from the —> edge adjacency matrix and edge distance matrix by applying specific matrix operators such as the row sum operator. 

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... 

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