The interplay between vertex degrees and edge-based topological invariants has recently gained significant traction in chemical graph theory. Among these, the Euler-Sombor index, defined by \( \sum \sqrt{d_u^2+d_v^2+d_u d_v} \), has emerged as a promising descriptor. In this paper we study its one-parameter generalization \( SO(G) \), where the edge-weight function is \( \varphi_\lambda(x,y)=\sqrt{x^2+y^2+\lambda xy} \) for \( \lambda\in[-2,2) \).
For \( \lambda\in[0,2) \), we prove that every connected graph which attains the minimum of \( SO \) among all connected graphs with fixed numbers of vertices and edges must be almost regular. This yields complete characterisations of the extremal graphs among trees, unicyclic graphs, bicyclic graphs, and more generally \( k \)-cyclic graphs for \( k \ge 3 \).
The case \( \lambda\in[-2,0) \) is more delicate: the kernel \( \varphi_\lambda \) induces a more subtle ordering on edge contributions. We give a complete classification of this ordering and use it to prove that the path \( P_n \) is the unique graph that attains the minimum of \( SO \) among all trees of order \( n \), and the cycle \( C_n \) is the unique graph for which the minimum is attained among all unicyclic graphs of order \( n \), for every \( \lambda\in[-2,0) \).
Our results unify and extend several known extremal results, including those for the ordinary Euler-Sombor index (\( \lambda=1 \)) and the classical Sombor index (\( \lambda=0 \)).