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Let G be a graph with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as SO(G)=∑uv∈E(G)dG(u)2+dG(v)2 and SOred(G)=∑uv∈E(G)dG(u)−12+dG(v)−12, respectively, where dG(v) is the degree of the vertex v in G. We denote by Hn,ν the graph constructed from the star Sn by adding ν edge(s), 0≤ν≤n−2, between a fixed pendent vertex and ν other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph Hn,ν has the maximum Sombor index among all connected ν-cyclic graphs of order n, where 0≤ν≤n−2. In some earlier works, the validity of this conjecture was proved for ν≤5. In this paper, we confirm that this conjecture is true, when ν=6. The Sombor index in the case that the number of pendent vertices is less than or equal to n−ν−2 is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained.

In this paper, a 2-gyrogroup G(n) of order 2n, n≥3, is constructed in which every proper subgyrogroup is either a cyclic or a dihedral group. It is proved that the subgyrogroup lattice and normal subgyrogroup lattice of G(n) are isomorphic to the subgroup lattice and normal subgroup lattice of the dihedral group of order 2n, which causes us to use the name dihedral gyrogroup for this class of gyrogroups of order 2n. Moreover, all proper subgyrogroups of G(n) are subgroups.

A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. The aim of this paper is to compute the automorphism group of the Euclidean graph of a carbon nanotorus. We prove that this group is a semidirect product of a dihedral group by a group of order 2.