On graphs with integer sombor indices

‎Sombor index of a graph $G$ is defined by $SO(G) = \sum_{uv \in E(G)} \sqrt{d^2_G(u)+d^2_G(v)}$, where $d_G(v)$ is the degree of the vertex $v$ of $G$. An $r$-degree graph is a graph whose degree sequence includes exactly $r$ distinctive numbers. In this article, we study $r$-degree connected graphs with integer Sombor index for $r \in \{5, 6, 7\}$. We show that: if $G$ is a 5-degree connected graph of order $n$ with integer Sombor index then $n \geq 50$ and the equality occurs if only if $G$ is a bipartite graph of size 420 with $SO(G) = 14830$; if $G$ is a 6-degree connected graph of order $n$ with integer Sombor index then $n \geq 75$ and the equality is established only for the bipartite graph of size $1080$; and if $G$ is a 7-degree connected graph of order $n$ with integer Sombor index then $n \geq 101$ and the equality is established only for the bipartite graph of size $1680$.