Branching in graphs and molecules
Branching at a point is defined in terms of the degrees of the central points in the star graphs, \(K_{1,n}\), \(n=0,1,2,3,... \). As atoms are added to a structure (i.e., as points are added to a graph), the new structure in which the atom has been affixed to the precursor atom with the higher degree is the more branched. A branching index consistent with this axiom is \(B_ M=B_ R+B_ S+r+s\), where \(B_ M\), \(B_ R\), and \(B_ S\) are the branching indices of the new structure and two precursor structures, respectively, and r and s are the degrees of the precursor atoms before they are connected by a bond (line). If \(B_ M=0\) for methane, then for other molecules it is simply equal to the number of pairs of adjacent bonds, or the number of lines in the line graph of the molecular graph. The iterated line graphs are then used to obtain a more complete ordering. Our results are compared to those of other approaches and help clarify branching and its relation to chemical properties such as boiling point.
scientific article; zbMATH DE number 4029633