List of walk and path counts calculated by DRAGON

 

Molecular walk counts are descriptors obtained from a H-depleted molecular graph [G. Rücker, C. Rücker, J.Chem.Inf.Comput.Sci. 1993, 33, 683-695]. The molecular walk count  MWCk of order k is the total number of walks of kth length in the molecular graph. A walk in a molecular graph is a sequence of pairwise adjacent edges leading from one vertex to another one; any edge can be traversed several times. The walk length is the number of edges traversed by the walk.

The molecular walk count of zero order is simply equal to the number of graph vertices (descriptor nSK) and the molecular walk count of order 1 to the number of graph edges (descriptor nBO).

DRAGON calculates molecular walk counts from order 1 up to order 10.

The molecular walk count is related to molecular branching and size, and in general to the molecular complexity of the graph.

Note that since for molecules with a large number of atoms counts of molecular walks can be very big numbers, the logarithmic transformation log (1 + MWCk) has been adopted.

 

The total walk count TWC is the total number of walks of any length (from 0 to 10 in DRAGON) in the graph.

 

Self-returning walk counts are molecular descriptors obtained from a H-depleted molecular graph, based on graph walks starting and ending on the same vertex, i.e. self-returning walks [F. Harary, Graph Theory, Addison-Wesley, Reading (MA), 1969]. The self-returning walk count SRWk of kth order is the total number of self-returning walks of length k in the graph.

It has to be noted that the self-returning walk counts of 1st and 2nd order coincide with the number of graph vertices (descriptor nSK) and twice the number of graph edges (descriptor nBO), respectively. DRAGON calculates self-returning walk counts from order 1 to order 10.

Self-returning walk counts are also known as the spectral moments of the vertex adjacency matrix because they can be calculated by the trace of the kth power of the adjacency matrix.

 

Molecular path counts are descriptors obtained from a H-depleted molecular graph, based on the graph path, which is a walk without any repeated vertices or edges [M. Randic, G.M. Brissey, R.B. Spencer, C.L. Wilkins, Computers Chem. 1979, 3, 5-13; M. Randic, MATCH (Comm.Math.Comp.Chem.) 1979, 7, 5-64]. The molecular path count MPCk of order k is the total number of paths of length k in the graph.

Path counts of order 0 and 1 coincide with the number of graph vertices (descriptor nSK) and the number of graph edges (descriptor nBO), respectively. Moreover, the path count of 2nd order is commonly known as Gordon-Scantlebury index(descriptor NGS).

DRAGON calculates molecular path counts from order 1 up to 10.

 

Molecular multiple path counts (piPCk) are counts of graph weighted paths of a given length k, where each path is weighted by the product of the conventional bond order of the involved edges. For saturated molecules, molecular multiple path counts coincide with the molecular path counts.

Molecular multiple path count of order 1 coincides with the sum of all the conventional bond orders in the graph (descriptor SCBO).

DRAGON calculates molecular multiple path counts from order 1 up to 10. Since for molecules with a large number of atoms counts of molecular walks can be very big numbers, the logarithmic transformation log (1 + piPCk) has been adopted.

 

The total path count (TPC) is the total number of paths of any length (from 0 to the maximum path length) in the graph.

 

The conventional bond-order ID number (piID) is the total number of weighted paths obtained by summing the weights of all paths of any length (from 0 to the maximum path length) in the graph. The weight of each path is calculated by multiplying the conventional bond order of all the edges of the path [M. Randic, P.C. Jurs, Quant.Struct.-Act.Relat. 1989, 8, 39-48]. This ID number accounts for multiple bonds in the molecule; for saturated molecules each edge weight is equal to one, therefore the conventional bond-order ID number piID coincides with the total path count TPC.

DRAGON applies the logarithmic transformation log (1 + x) to the total path count TPC and the conventional bond-order ID number piID.

 

The ratio of multiple path count over path count (PCR) is defined as the conventional bond-order ID number piID divided by the total path count TPC.

 

The difference between multiple path count and path count (PCD) is defined as the difference between the conventional bond-order ID number piID and the total path count TPC.

 

The Randic ID number (CID) is the molecular identification number proposed first [M. Randic, J.Chem.Inf.Comput.Sci. 1984, 24, 164-175] and is defined as a total weighted path count obtained by summing the weights of all paths of any length (from 0 to the maximum path length) in the graph. The weight of each path is calculated by multiplying the reciprocal square root of the edge connectivity (see Table below) of all edges of the path.

 

(d i , d j )	edge connectivity
(1, 2)	0.7071
(1, 3)	0.5774
(1, 4)	0.5000
(2, 2)	0.5000
(2, 3)	0.4082
(2, 4)	0.3536
(3, 3)	0.3333
(3, 4)	0.2887
(4, 4)	0.2500

 

Edge connectivities. d refers to the vertex degree of the two vertices incident to the considered edge.

 

The Balaban ID number (BID) is a total weighted path count obtained by summing the weights of all paths of any length (from 0 to the maximum path length) in the graph. The weight of each path is calculated by multiplying the weights of all edges of the path, the edge weight being the reciprocal square root of the product of the vertex distance degrees of the two atoms incident to the edge [A.T. Balaban, Numerical Modelling of Chemical Structures: Local Graph Invariants and Topological Indices in Graph Theory and Topology in Chemistry, R.B. King, D.H. Rouvray (Eds.), Elsevier, Amsterdam (The Netherlands), pp. 159-176, 1987].