Proof of a conjecture on symmetric division deg index of graphs. (English) Zbl 1542.92218
Summary: 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
\[
SDD(G) = \sum_{v_iv_j\in E(G)}\left(\frac{d_i}{d_j} + \frac{d_j}{d_i}\right),
\]
where \(d_i\) is the degree of the vertex \(v_i\in V(G) \). Very recently, A. Ali et al. [MATCH Commun. Math. Comput. Chem. 90, No. 2, 263–299 (2023; Zbl 1519.92349)] 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.
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.
MSC:
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
| 05C92 | Chemical graph theory |
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
Citations:
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