On a Conjecture about the Sombor Index of Graphs

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