Relations between ordinary and multiplicative degree-based topological indices

Abstract

Let $G$ be a simple connected graph with $n$ vertices and $m$
edges, and sequence of vertex degrees $d_1\geq d_2\geq\cdots\geq d_n >0$. If vertices $i$ and $j$ are adjacent, we write $i\sim j$.
Denote by $\Pi_1$, $\Pi_1^*$, $Q_{\alpha}$ and $H_{\alpha}$ the multiplicative Zagreb index, multiplicative sum Zagreb index, general first Zagreb index, and general sum-connectivity index, respectively. These indices are defined as $\Pi_1= \prod_{i=1}^n d_i^2$, $\Pi_1^*=\prod_{i\sim j}(d_i+d_j)$, $Q_{\alpha}=\sum_{i=1}^n d_i^{\alpha}$ and $H_{\alpha}=\sum_{i\sim j}(d_i+d_j)^{\alpha}$. We establish upper and lower bounds for the differences $H_{\alpha} - m \left(\Pi_1^*\right)^{\frac{\alpha}{m}}$ and $ Q_{\alpha}-n\left(\Pi_1\right)^{\frac{\alpha}{2n}}$. In this way we generalize a number of results that were earlier reported in the literature.