Edge-adjacency matrix

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 ... 

Bertz branching index connection number - edge adjacency 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 ... 

Therefore, the ith edge degree calculated on the weighted edge adjacency matrix is the sum of the bond orders associated with all e, bonds adjacent to the / th edge ... 

In analogy with the bond order-weighted edge adjacency matrix, a resonance-weighted edge adjacency matrix E was also proposed, replacing the bond orders with parameters kc-x used in the Hiickel matrix and related to the resonance integral Pc-x of the bond between the heteroatom X and the carbon atom by the relationship ... 

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. 

E bond order-weighted edge adjacency matrix B B Y... 

The simplest form to represent the chemical information contained in a molecular graph is by a -> matrix representation of a molecular structure. Examples are -+ adjacency matrix A, - edge adjacency matrix E, vertex - distance matrix D, -> edge distance matrix D, - incidence matrix I, - Wiener matrix W, -> Hosoya Z-matrix Z, - Cluj matrices CJ, - detour matrix A, - Szeged matrix SZ, -> distance/distance matrix DD, and - detour/distance matrix A/D. 

Platt number total edge adjacency index -> edge adjacency matrix PLS-based variable selection - variable selection P matrix -> bond order indices (O graphical bond order)... 

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] ... 

Collected below are the information indices calculated on the most important topological matrices such as - adjacency matrix A, distance matrix D, - edge distance matrix "D, - edge adjacency matrix E, - vertex cycle matrix Cy, and -+ edge cycle matrix Ce [Bonchev et al, 1981b Bonchev, 1983]. 

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