WEIGHTED GRAPHS

Every molecule is represented by a graph. The simpler molecules, i.e. diatomic, triatomic, etc. have degeneracies. To lift the degeneracies one needs to add the weighted average atomic number to the graph's description. Thus "r" is designated as the weighted average atomic number of a graph...and it forms an ordered triplet with the polygonality, "n", and the connectivity "p" of the respective graph. The triplet is thus designated in the form of a Cartesian triplet (n, p, r) in n-p-r space. Thus "n" is defined as 2E/F and "p" is defined as 2E/V from the Euler relation, or its analog for graphs known as the Schlaefli relation...V - E + F = 2 or 1, respectively. Little is gained from this analysis and plotting data for small molecules...but for medium and large sized molecules the triplet (n, p, r) has empirical importance and the n-p-r coordinates thus may be useful as descriptors in a structure-property analysis. It is also to be noted that n-p-r ratios and products...i.e. n/p and p/r etc. and np and npr etc. could produce other, alternative molecular descriptors, in this connection, for further structure-property analysis. The sheer plotting of n-p-r data in 3-space for enough molecules may lead to new insights between otherwise disparate molecules, that could lead to new directions in the research of their chemistries thus...MJB