Total Adjacency Index

This index is redundant due to the symmetry of the adjacency matrix. It can be reduced to the sum A of the upper triangular adjacency submatrix, which is called 11 the total adjacency of a molecule  

Owing to the fact that au = 1 when there is a bond between atoms i and j, the total adjacency of a molecule equals the number of bonds in it. Burton 2) applied this quantity to cluster studies though not referring to its graph-theoretical origin. 

For the two examples above, i.e., G, and G2, one obtains A = 7 and 10, respectively. The total adjacency index, however, seems to be of little use for molecular studies since it can only distinguish between molecules having different numbers of cycles (i.e., cyclomatic numbers) p  

(3) is the known Euler equation connecting the number of vertices, edges and cycles in a planar graph. 

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The total adjacency index Ay is the sum of all the entries of the adjacency matrix of a molecular graph, and is twice the - bond number B [Harary, 1969a] ... 

This is the simplest graph invariant obtained from the -> adjacency matrix A, defined as the number of bonds in the -> molecular graph (7 where multiple bonds are considered as single edges. Bond number is calculated as half the - total adjacency index Ay ... 

The most well-known complexity indices are listed below. Other molecular descriptors which give information about molecular complexity are the indices of neighbourhood symmetry and the - total adjacency index. The latter was proposed by Bonchev and Polansky [Bonchev and Polansky, 1987] as a simple measure of topological complexity, being a measure of the degree of connectedness of molecular graph. 

This is derived from the partition of the adjacency matrix A elements according to their magnitude and is a trivial quantity, as zero entries do not contribute to the - total adjacency index Av, i.e. the sum of all vertex degrees, corresponding to twice the bond number 5 ... 

This is calculated by dividing the total information content on the adjacency matrix elements magnitude by the total adjacency index Av ... 

As the diagonal entries are just the vertex degrees 6, the first term on the right is the - total adjacency index Ay The topological state index is one of the -> molecular ID numbers. 

In analogy with the total adjacency index Av, which is the sum of all the vertex degrees in the molecular graph, three other simple molecular descriptors are defined [Pogliani, 1992a] ... 

For example, the total adjacency index of 2-methylpentane isAy = l + 3 + 2 + 2 + l + l = 10, which is twice the number of edges equal to five in the H-depleted molecular graph of this molecule. Therefore, the number of entries equal to 1 in the adjacency matrix is 2B, while the number of entries equal to zero is — 2 B in particular, for acyclic graphs the total number of... 

The global connectivity of a graph can also be characterized by the average of the total adjacency index as [Bonchev and Buck, 2007]... 

The doubly normalized total adjacency index is called density index and is defined as... 

Those derived from the adjacency matrix the total adjacency index, the Zagreb group indices, the Randic connectivity index, the Platt index, the compatibility code, and the largest eigenvalue index ... 

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