On the maximal Sombor index of quasi-tree graphs

The Sombor index $SO(G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d_G^2(u)+d_G^2(v)}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G=(V,E)$ is called a quasi-tree, if there exists $u\in V(G)$ such that $G-u$ is a tree. Denote $\mathcal{Q}(n,k)$={$G$: $G$ is a quasi-tree graph of order $n$ with $G-u$ being a tree and $d_G(u)=k$}. In this paper, we determined the maximum, the second maximum and the third maximum Sombor indices of all quasi-tree graphs in $\mathcal{Q}(n,k)$, respectively. Moreover, we characterized their corresponding extremal graphs, respectively.