A Natural Identity Connecting the Szeged and Quadratic Mostar Index: Theory, Extremal Results, and Applications

Abstract

The Szeged index and the Mostar index are widely used graph-theoretic invariants that capture, respectively, routing diversity and structural imbalance in a connected graph. Despite sharing the same local quantities $n_u(e)$ and $n_v(e)$ at each edge, these two indices have generally been studied in isolation. We establish a clean identity for every tree $T$ on $n$ vertices: $4\cdot\mathrm{Sz}(T) + \mathrm{Mo}_2(T) = n^2(n-1),$ where $\mathrm{Mo}_2(T) = \sum_{e}(n_u(e) - n_v(e))^2$ is the quadratic Mostar index. The proof is one line: every edge of a tree is a bridge, so $n_u(e) + n_v(e) = n$, and the elementary identity $(a+b)^2 = 4ab + (a-b)^2$ does the rest. The consequences are surprisingly far-reaching---extremal behaviour, a Cauchy--Schwarz bound linking all three classical indices, and a power-mean chain for the generalised Mostar family all fall out cleanly. Two applications illustrate the identity's interpretive value: in chemical graph theory it makes precise why the Szeged and Mostar indices rank alkane isomers in opposite order, and in network modelling it quantifies an exact trade-off between routing diversity and link-load imbalance. Computational verification across all non-isomorphic trees on up to ten vertices confirms the identity without exception.