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Gutman proposed a topological index called the Sombor index, which was defined as

Operations in graph theory have a significant influence in the theoretical and application aspect of the domain. Topological indices serve as a crucial component in chemical graph theory linked with some molecular structure. Recently, Gutman initiated the study on the Sombor index. In this paper, the computation of some bounds for Sombor index of graph operation notably join, cartesian product, corona product, lexicographic product, tensor product and strong product is carried out. The computation has been utilized to determine the upper bounds of the index for the specified graph operations for some standard graphs like the path and cycle graphs.

The biological, chemical, and structural properties of molecular graphs can be predicted by using topological indices. The lower and upper bounds of fundamental topological indices are well-studied. In this paper, by using comparison theorem for integrals, we obtain new bounds for some topological indices namely, Randić index, Sombor index, harmonic index and forgotten index.

For a positive real number $k$, the $k$-Sombor index of a graph $G$, introduced by Réti et al, is defined as

In this paper, for $k\ge 1$ we present the extremal values of the $k$-Sombor index of trees with some given parameters, such as matching number, the number of pendant vertices, diameter. This generalizes the relevant results on Sombor index due to Chen, Li and Wang ((2002). Extremal values on the Sombor index of trees, MATCH Communications in Mathematical and in Computer Chemistry, 87, 23–49).

Handling $S{O}_k(G)$ appears to be different for $k<1$ in contrast to the case when $k\ge 1$. To demonstrate this, we also characterize the extremal trees with respect to the ${\mathrm{\text{SO}}}_{\frac{1}{2}}$ with matching number, the number of pendant vertices and diameter. In addition, three relevant conjectures are proposed.

The comaximal graph $\mathrm{\Gamma }(R)$ of a commutative ring $R$ is a simple graph with vertex set $R$ and two distinct vertices $u$ and $v$ of $\mathrm{\Gamma }(R)$ are adjacent if and only if $uR+vR=R$. In this paper, we find the sharp bounds for the Sombor index for comaximal graphs of integer modulo ring ${\mathbb{Z}}_n$ and give the corresponding extremal graphs. Also, we find the Sombor eigenvalues and the bounds for the Sombor energy of comaximal graphs of ${\mathbb{Z}}_n$.

Buckyballs are closed spherical carbon cages which are beneficial in several industries. Their chemical graphs provide structural invariants called topological invariants. This study is devoted to investigating the relation between degree-based topological index called Sombor index and physical properties of buckyballs. The sample contains buckyballs consisting of 54, 58, 60, 70, 74, 76, 78, 80, 82, 84, 86 and 90 carbon atoms.

The topological indices with degree bases can be used to display the physical characteristics of buckyballs. The numerical values of topological indices describe the topological structure of a molecule. The physical properties include binding energy, Ramsauer–Townsend effect (RT-1, RT-2), shape resonances (SR-1, SR-2), and heat of formation of buckyballs.

The Sombor index $\text{SO}$ of a graph $G$ is defined as

In this paper, we extend the recently introduced vertex-degree-based topological index, the Sombor index, and we call it general Sombor index. The general Sombor index generalizes both the forgotten index and the Sombor index. We present the bounds in terms of other important graph parameters for general Sombor index. We also explore the Nordhaus–Gaddum-type result for the general Sombor index. We present further the relations between general Sombor index and other generalized indices: general Randić index and general sum-connectivity index.