Graph, edges

II. By virture of the fundamental relationship, the elements of a graph, edges and vertices, form a connected system. In other words, any two vertices can be joined by a path consisting of a sequence of edges and vertices. 

The information analogue of the Randic index, %R, was defined 83) by partitioning the graph edges into classes, depending on the equality of their partial connectivity... 

Our graph has a cycle that is a finite sequence of graph edges, whose beginning and end coincide. This cycle corresponds to the cyclic conversion of the intermediates. In our case the cycle is unique. 

Edge weights are obtained if the reaction (both direct and reverse) rates corresponding to the graph edges are divided by the concentrations of the reacting intermediates... 

In ref. 1, a complex mechanism was represented as an open graphical sequence every graph edge (step) jointed nodes (substances). It is likely that it was Temkin who was the first to suggest the representation of catalytic conversions by a cycle on the graph. 

Graph edges oriented in a definite direction which indicates the order of interconnection between the nodes are called directed arcs. Their orientation is indicated by the arrows placed either on the arcs or near them. A graph, a cycle, and a tree containing directed arcs are called directed. A directed cycle is also called a contour. 

In the second stage of molecule reconstruction, the chemical graph is converted to a chemical molecular structure. The vertices in the graph become the atoms in the chemical molecular structure, and the graph edges become the bonds, and their chemical valences, charges, and properties are validated by a set of chemical knowledge rules (Karl 2007). 

By generalization, Harary indices and hyper-Harary indices (or hyper-Harary numbers) are all molecular descriptors derived from the application of the Wiener operator to reciprocal topological matrices Harary indices are obtained from the 1 -order - sparse matrix, i.e. considering only the graph edges, while the hyper-Harary indices are from the whole matrices, i.e. considering the paths [Diudea, 1997c]. 

Hosoya graph decomposition is a graph edge decomposition defined as ... 

A molecular weighted path number obtained by weighting graph edges with -> conventional bond order [Randic and Jurs, 1989] ... 

The sum of the elements of each i th row is a constant equal to 3B, B being the number of graph edges. 

Referring to the subgraph order, rth order indices can be defined as -+ count descriptors, i.e. the number of rth order subgraphs in the graph The zero order index is simply the - atom number A, i.e. the number of graph vertices 1 order index is the -> bond number B, i.e. the number of graph edges 2" order index is the... 

The resistance distance is based on electrical network theory and is defined as the effective electrical resistance between two vertices (nodes) when a battery is connected across them and each graph edge is considered as a resistor taking a value of 1 ohm. 

The total adjacency index Ay is a measure of the graph connectedness and is calculated as the sum of all the entries of the adjacency matrix of a molecular graph, vhich is tv ice the number B of graph edges [Harary, 1969a Rouvray, 1983] ... 

---