For a graph $G$, the Sombor index $SO(G)$ of $G$ is defined as

$$SO(G)=\sum _{uv\in E(G)}\sqrt{d_G(u{)}^2+d_G(v{)}^2},$$

where $d_G(u)$ is the degree of the vertex $u$ in $G$. A cactus is a connected graph in which each block is either an edge or a cycle. Let $\mathcal{G}(n,k)$ be the set of cacti of order $n$ and with $k$ cycles. Obviously, $\mathcal{G}(n,0)$ is the set of all trees and $\mathcal{G}(n,1)$ is the set of all unicyclic graphs, then the cacti of order $n$ and with $k(k\ge 2)$ cycles is a generalization of cycle number $k$. In this paper, we establish a sharp upper bound for the Sombor index of a cactus in $\mathcal{G}(n,k)$ and characterize the corresponding extremal graphs. In addition, for the case when $n\ge 6k-3$, we give a sharp lower bound for the Sombor index of a cactus in $\mathcal{G}(n,k)$ and characterize the corresponding extremal graphs as well. We also propose a conjecture about the minimum value of sombor index among $\mathcal{G}(n,k)$ when $n\ge 3k$.

A graph is outerplanar if it has a crossing-free embedding in the plane such that all vertices are on the boundary of its outer face. An outerplanar graph is maximal outerplanar if no edge can be added without losing outerplanarity. The Wiener index of a graph G is the sum of distances between all pairs of vertices of G. We show that for a maximal outerplanar graph G on n vertices, W(K₁∨P_n−1) ≤W(G)≤W( $P_n^2$), where K₁∨P_n−1 is the graph obtained from joining a vertex to each vertex of P_n−1 and $P_n^2$ is the square of P_n.