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 $1680$.