On graphs with integer Sombor index

The Sombor index of a simple graph G is defined as SO(G) = Sigma(uv is an element of E(G)) root d(u)(2)+d(v)(2), where d(u) is the degree of the vertex u. A r-degree graph is a graph whose degree sequence has exactly r distinct numbers. In this paper we study the r-degree graphs whose Sombor index is integer. We characterize all 1-degree and 2-degree graphs with integer Sombor index. We show that if G is a 3-degree connected graph of order n, with integer Sombor index, then n >= 25 and the equality happens if and only if G is a graph that is constructed from a 5-regular bipartite graph of order 18, say K, by joining 7 isolated vertices to every vertex of one of the parts of K. In addition, we find that if G is a 4-degree connected graph of order n and with integer Sombor index, then n >= 30 and the equality holds just for five bipartite graphs. Our results are determined the smallest connected r-degree graph with integer Sombor index, for r = 3 and r = 4.