We introduce a new topological index \(\delta(G)\) 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 \(\mathrm(n)\), to the nearest threshold partition in \(\mathrm(n)\), the set of \(2^{n-1}\) extreme points of \(\mathrm(n)\). 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 \(C_n\) (\(n\geq 4\)), \(\delta(C_n)=\sqrt{4n-14}\). Second, every connected organic molecule on \(n\geq 6\) vertices with maximum degree at most~\(4\) satisfies \(\delta(G)>0\). Third, for any tree \(T\) on \(n\) vertices with \(\ell\) pendent vertices, \(\delta(T)^{2}\geq n-\ell-3\); for alkane carbon skeletons this gives a lower bound controlled by the number of methyl groups. A computational study across alkane isomers \(C_{1}\)-\(C_{15}\) and a library of cyclic molecules confirms that \(\delta\) increases monotonically with cyclomatic number and achieves Spearman \(\rho\approx 0.922\) with normal boiling point (largely via molecular size) and \(\rho\approx 0.891\) 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 (\(p=0.74\)).