Molecular Graph Representation of Chemical Structures

Graphs may be represented in algebraic form as matrices [3-5]. This numerical description of the structure of chemical compounds is essential for the computer manipulation of molecules and for the calculation of various topological indices and graph descriptors [6]. The computation of the E-state indices is based on the adjacency and distance matrices. 

The adjacency matrix A(G) of a molecular graph G with N vertices is the square NxN symmetric matrix in which [A],j=l if vertex Vj is adjacent to vertex Vj and [A],j=0 otherwise. The adjacency matrix is symmetric, with all elements on the main diagonal equal to zero. The sum of entries over row i or column i in A(G) is the degree of vertex Vj, 5j. Usually, the adjacency matrix is based on weighted molecular graphs in which heteroatoms are represented as vertex parameters and 

In a simple (nonweighted) connected graph, the graph distance dy between a pair of vertices V and Vj is equal to the length of the shortest path cormecting the two vertices, i.e. the number of edges on the shortest path. The distance between two adjacent vertices is 1. The distance matrix D(G) of a simple graph G with N vertices is the square NxN symmetric matrix in which [D],j=cl,j [9, 10]. 

The 5 connectivity index (atom degree), that has a central role in computing the E-state, was used in the definihon of the Zagreb topological indices [11]. Randle modified the Zagreb index M2 to obtain the connectivity index % [12]. 

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