University of Isfahan Transactions on Combinatorics 2251-8657 14 3 2025 09 01 Minimal graphs with respect to the multiplicative version of some vertex-degree-based topological indices 157 172 28369 10.22108/toc.2024.139624.2119 EN Mehdi Eliasi Department of Mathematics , Khansar Faculty, University of Isfahan, Isfahan, Iran Journal Article 2023 10 28 As a real-valued function, a graphical parameter is defined on the class of finite simple graphs, and remains invariant under graph isomorphism. In mathematical chemistry, vertex-degree-based topological indices are the graph parameters of the general form of $p_{\phi}(G)=\sum_{uv\in E(G)}\phi(d(u),d(v))$, where $\phi$ represents a real-valued symmetric function, and $d(u)$ shows the degree of $u\in V(G)$. In this paper, it is proved that if $\phi$ has certain conditions, then the graph among those with $n$ vertices and $m$ edges, whose difference between the maximum and minimum degrees is at most $1$, has the minimal value of $p_{\phi}$. Moreover, it is demonstrated that some well-known topological indices are able to satisfy these certain conditions, and the given indices can be treated in a unified manner. https://toc.ui.ac.ir/article_28369_69de69f3082f6df35d742356561470b1.pdf