Resolving Open Problems on the Euler Sombor Index

TL;DR

This paper solves key open problems about the Euler Sombor index, determining extremal graphs with minimum and maximum index values for various classes of graphs, advancing the understanding of this new topological measure.

Contribution

The paper completely resolves two challenging open problems on the Euler Sombor index, identifying extremal graphs for fixed order, girth, and number of leaves.

Findings

Minimum EUS among unicyclic graphs with fixed order and girth identified.

Extremal graphs for maximum EUS with fixed order and number of leaves characterized.

Extremal structures for EUS in broader graph classes established.

Abstract

Recently, the Euler Sombor index $(EUS)$ was introduced as a novel degree-based topological index. For a graph $G$, the Euler Sombor index is defined as $$EUS(G) = \sum_{v_i v_j \in E(G)} \sqrt{d_i^2 + d_j^2 + d_i d_j},$$ where $d_i$ and $d_j$ denote the degrees of the vertices $v_i$ and $v_j$, respectively. Very recently, Khanra and Das \textbf{\bf [Euler Sombor index of trees, unicyclic and chemical graphs, \emph{MATCH Commun. Math. Comput. Chem.} \textbf{94} (2025) 525--548]} proposed several open problems concerning the Euler Sombor index. This paper completely resolves two of the most challenging problems posed therein. First, we determine the minimum value of the $EUS$ index among all unicyclic graphs of a fixed order and prescribed girth, and we characterize the extremal graphs that attain this minimum. Building on this result, we further establish the minimum $EUS$ index within the broader class of connected graphs of the same order and girth, and identify the corresponding extremal structures. In addition, we classify all connected graphs that attain the maximum Euler Sombor index $(EUS)$ when both the order and the number of leaves are fixed.