### The relations between the Sombor index and Merrifield-Simmons index

#### Abstract

The \emph{Sombor index} $SO(G)$ of a graph $G$ is defined as $SO(G)=\sum\limits_{uv\in E(G)}(d_{G}(u)^{2}+d_{G}(v)^{2})^{\frac{1}{2}}$, while the \emph{Merrifield-Simmons index} $i(G)$ of a graph $G$ is defined as $i(G)=\sum\limits_{k\geq 0}i(G;k)$, where $d_{G}(x)$ is the degree of any one given vertex $x$ in $G$ and $i(G;k)$ denotes the number of $k$-membered independent sets of $G$. In this paper, we investigate the relations between the Sombor index and Merrifield-Simmons index. First, we compare the Sombor index with Merrifield-Simmons index for some special graph families, including chemical graphs, bipartite graphs, graphs with restricted number of edges or cut vertices and power graphs, and so on. Second, we determine sharp bounds on the difference between Sombor index and Merrifield-Simmons index for general graphs, connected graphs and some special connected graphs, including self-centered graphs and graphs with given independence number.

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