Abstract
Molecular descriptors play a significant role in the quantitative studies on structure-property and structure-activity relationships. One of the popular degree-based topological index, symmetric division deg \( (SDD) \) index is a chemically useful descriptor.
The \( SDD \) index of a graph G is defined as \[ SDD(G)=\sum\limits_{v_iv_j\in E(G)}\,\left(\frac{d_i}{d_j}+\frac{d_j}{d_i}\right), \] where \( d_i \) is the degree of the vertex \( v_i\in V(G) \). Very recently, Ali et al. [Symmetric
division deg index: Extremal results and bounds, MATCH Commun. Math. Comput. Chem.90 (2023) 263-299] mentioned several open problems on symmetric division deg index of graphs. One of them is as follows:
Characterize graphs attaining the minimum \( SDD \) index over the class of all those \( n \)-order connected graphs of minimum degree \( \delta \) that are not \( \delta \)-regular.
In this paper we completely solved the above problem.
