‎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 $1680$.

Based on the geometric background of Sombor index and motivating by the higher order connectivity index and the Sombor index, we introduce the pathcoordinate of a path in a graph and a degree-point in a higher dimensional coordinate system, and define the higher order Sombor index of a graph. We first consider mathematical properties of the higher order Sombor index, show that the higher order connectivity index of a starlike tree is completely determined by its branches and that starlike trees with a given maximum degree which have the same higher order Sombor indices are isomorphic. Then, we determine the extremal values of the second order Sombor index for all trees with n vertices and characterize the corresponding extremal trees. Finally, the chemical importance of the second order Sombor index is investigated and it is shown that the new index is useful in predicting physicochemical properties with high accuracy compared to some well-established.

The Sombor index of the graph G is a degree based topological index, deffined as SO ∑,uvϵ,E(G) ,d2u + d2v, where du is the degree of the vertex u, and E(G) is the edge set of G. Bounds on SO are established in terms of graph energy, size of minimum vertex cover, matching number, and induced matching number.

Let $G=(V,E)$ be a finite simple graph. The Sombor index $SO(G)$ of $G$ is defined as $\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of vertex $u$ in $G$. In this paper, we study this index for certain graphs and we examine the effects on $SO(G)$ when $G$ is modified by operations on vertex and edge of $G$. Also we present bounds for the Sombor index of join and corona product of two graphs.

Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. The Sombor index of the graph $G$ is a degree-based topological index, defined as$$SO(G)=\sum_{uv \in E}\sqrt{d(u)^2+d(v)^2},$$in which $d(x)$ is the degree of the vertex $x \in V$ for $x=u, v$. \\In this paper, we introduce a new topological index called the entire Sombor index of a graph which is defined as the sum of the terms $\sqrt{d(x)^2+d(y)^2}$ where $x$ is either adjacent or incident to $y$ and $x, y \in V \cup E$. We obtain exact values of this new topological index in some graphs families. Some important properties of this index are obtained.

‎The Sombor index $(SO)$ is a recently invented vertex-degree-based molecular structure-descriptor‎. ‎Let $M_1$ be the first Zagreb index‎. ‎The fact that $SO$ is bounded from below by $M_{1}/\sqrt{2}$ and from above by $M_{1}$ is well-known and easy to prove‎. ‎In this paper‎, ‎we improve these bounds‎.

‎Let G=(V‎, ‎E) be a graph with vertex set V(G) and edge set E(G)‎. ‎The Sombor index of a graph G‎, ‎SO(G)‎, ‎is defined as ∑uv∈ E(G) √(d2u+d2v), ‎where du is the degree of vertex u in V(G)‎. ‎In the present paper‎, ‎we determine the lower bound for the Sombor index of edge corona‎, ‎R-edge and R-vertex corona products of two graphs‎. ‎We also compute the exact value for the Sombor index of the line graphs of subdivision of tadpol‎, ‎ladder and wheel graphs‎.

Let G be a simple graph. The elliptic Sombor index of G is defined as $$    ESO(G) = \sum_{uv} \left(d_{u}+ d_{v} \right)\sqrt{d^{2}_{u}+d^{2}_{v}}, $$  where d_{u} denotes the degree of the vertex u, and the sum runs over the set of edges of G. In this paper we solve the extremal value problem of ESO over the set of (connected) chemical graphs and over the set of chemical trees, with equal number of vertices.

Let $ G $ be a finite simple graph. The Sombor index of $ G $ isdefined as $ \sum\nolimits_{uv\in E(G)} \sqrt{d_{u}^{2}+d_{v}^{2}} $where $d_{u}$ and $d_{v}$ represent the degrees of vertices $ u$ and$v$ in $ G $, respectively. The sum of the absolute values of theadjacency eigenvalues defines the energy of a graph. This paper aimsto enhance the current connections between the Sombor index and theenergy of graphs. Additionally, we provide some upper bounds for theSombor index of triangle-free, square-free, $K_r$-free andtripartite graphs in terms of order, size and minimum degree.

The Sombor index of the graph G is a degree based topological index, de_ned as SO= ∑,uv, E(G) √, dG(u)2 + dG(v)2, where dG(u) is the degree of the vertex u of G and E(G) is the edge set of G. In this paper we calculate SO of some graph transformations.