The Sombor index

Question

1: If $e$ is an edge in $G$ connecting $u$ and $v$, then the degree of e in the line graph $L(G)$ is equal to $d‎_{G}(u)+d‎_{G}(v)-2$ so ‎$‎‎d‎_{L(G)}=d‎_{G}(u)+d‎_{G}(v)-2‎‎‎$.Then ‎ $\sum‎_{‎‎e\in V(L(G))}=‎\sum‎_{uv\in E(V(G))}(d‎_{G}(u)+d‎_{G}(v) -2)=‎[‎\sum‎_{uv\in E(V(G))} d‎_{G}(u)‎^{2}‎] -2m(G)$. ($m$ is the number of edges)

2:‎Let $G=(V,E)$ be a finite simple graph‎. ‎The Sombor index $SO(G)$ of $G$ is defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^{2}+d_v^{2}}$‎, ‎where $d_u$ is the degree of vertex $u$ in $G$‎.

‎> I got the sombor index value for many graphs, but none of them are integer number. For which graph is the Sombor index value an integer?

I would like to find graphs where the value of the Sambor index ‎$‎L(G)‎$ is ‎an integer. Please help me.

Answer 1

You have a sum of terms that look like $\sqrt{d_u^2+d_v^2}$. Now, $d_u^2+d_v^2$ is an integer, and the square root of an integer is either an integer or an irrational. Now we can ask whether some square roots could have some linear dependencies, and the answer is essentially no (see for instance here). This means that $SO(G)$ is an integer if and only if each summand is integer. This implies that each summand is of the form $\sqrt{a^2+b^2}$, where $a^2+b^2 = c^2$ for an integer $c$, i.e., $(a, b, c)$ is a Pythagorean triple. An example of such a graph would be $K_{3,4}$, since $(3,4,5)$ is a Pythagorean triple.