An Alternative Proof of the Sombor Index Minimizing Property of Greedy Trees

Ivan Damnjanović, Dragan Stevanović

Recently, Gutman defined a new graph invariant which is named the Sombor index $\operatorname{SO}(G)$ of a graph~$G$ and is computed via the expression \[ peratorname{SO}(G)=um_{uim v}qrt{ẹg(u)^2+ẹg(v)^2}, \] where $\deg(u)$ represents the degree of the vertex $u$ in $G$ and the summing is performed across all the unordered pairs of adjacent vertices $u$ and $v$. Damnjanović et al.~have implemented an earlier result obtain by Wang in order to show that, among all the trees $\mathcal{T}_D$ that have a specified degree sequence~$D$, the greedy tree must attain the minimum Sombor index. Here we provide an alternative proof of this same result by constructing an auxiliary graph invariant named the pseudo-Sombor index and without relying on any other earlier results.