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Let G = (V (G) , E (G)) be a simple connected graph where V (G) is a vertex set and E (G) is an edge set. The Sombor index of graph G is a vertex-degree-based topological index, which is introduced by Gutman et al. and defined as S O (G) = ∑ u v ∈ E (G) d 2 (u) + d 2 (v) where d (u) denotes the degree of vertex u. In this paper, we give the extremal value of the Sombor index among all maximal outerplanar graphs with given order n and characterize those maximal outerplanar graphs with the extremal value. [ABSTRACT FROM AUTHOR]

Based on the geometric background of Sombor index and motivating by the higher order connectivity index and the Sombor index, we introduce the path-coordinate 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 indices. [ABSTRACT FROM AUTHOR]

I. Gutman (2022) constructed six new graph invariants based on geometric parameters, and named them Sombor-index-like graph invariants, denoted by S O 1 , S O 2 , ... , S O 6 . Z. Tang, H. Deng (2022) and Z. Tang, Q. Li, H. Deng (2023) investigated the chemical applicability and extremal values of these Sombor-index-like graph invariants, and raised some open problems, see Z. Tang, Q. Li, H. Deng (2023). We consider the first open problem formulated at the end of Z. Tang, Q. Li, H. Deng (2023). We obtain the extremal values of the graph invariants S O 5 and S O 6 among all trees and molecular trees of order n, and characterize the trees and molecular trees that achieve the extremal values, respectively. Thus, the problem is completely solved. [ABSTRACT FROM AUTHOR]

A novel geometric method is proposed for constructing vertex‐degree‐based molecular structure descriptors, which is the perimeter of an ellipse whose focal points represent the degree‐point and its dual‐point of a pair of adjacent vertices in a graph G=(V,E)$$ G=\left(V,E\right) $$. We first analyze the chemical applicability of this elliptic Sombor index, and then establish its mathematical properties and determine the extremal values for this elliptic Sombor index among all (molecular) trees and characterize the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

Unicyclic graph is one kind of typical graph for indices basing degree of vertex. Various chemical structures can be exhibited in unicyclic graph as well. A graph is said to be unicyclic if the graph is connected and | V (G) | = | E (G) |. Recently, the Sombor index, which is defined by S O = S O (G) = ∑ u v ∈ E (G) d G 2 (u) + d G 2 (v) , was proposed by Gutman. This paper establishes distinct bounds for this index of unicyclic graph with girth l , as well as specific bounds of chemical unicyclic graph with girth l. • Some bounds and extremal graphs of Sombor index over unicyclic graph are given. • The minimum value and graph of Sombor index over chemical unicyclic graph are gotten. • With a order condition, the maximum value and graphs of Sombor index over chemical unicyclic graph are determined. [ABSTRACT FROM AUTHOR]