A Geometric Topological Index via Threshold Partitions

Source abstract

We introduce a new topological index δ(G) for a molecular graph G on n vertices, defined as the Euclidean distance from the degree partition d(G), viewed as a point in the degree-partition polytope DP(n), to the nearest threshold partition in TP(n), the set of 2n−1 extreme points of DP(n). The index has an intrinsic geometric interpretation: it measures how far the degree sequence of a molecule lies from the class of threshold graphs. We establish three results. First, for the cycle Cn (n ≥ 4), δ(Cn) = √ 4n − 14. Second, every connected organic molecule on n ≥ 6 vertices with maximum degree at most 4 satisfies δ(G) > 0. Third, for any tree T on n vertices with ℓ pendent vertices, δ(T) 2 ≥ n − ℓ − 3; for alkane carbon skeletons this gives a lower bound controlled by the number of methyl groups. A computational study across alkane isomers C1–C15 and a library of cyclic molecules confirms that δ increases monotonically with cyclomatic number and achieves Spearman ρ ≈ 0.922 with normal boiling point (largely via molecular size) and ρ ≈ 0.891 with experimental log P; partial correlation controlling for carbon count remains significant for log P (r = 0.555, p = 0.003) but not for boiling point (p = 0.74).