minimal graphs with respect to the multiplicative version of some vertex-degree-based topological indices

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 $p_{phi}(g)=sum_{uvin e(g)}phi(d(u),d(v))$, where $phi$ represents a real-valued symmetric function, and $d(u)$ shows the degree of $uin v(g)$. 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.