On a Conjecture about the Sombor Index of Graphs

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)= n-ary sumation uv & ISIN;E(G)dG(u)2+dG(v)2 and SOred(G)= n-ary sumation uv & ISIN;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,nu the graph constructed from the star Sn by adding nu edge(s), 0 & LE;nu & LE;n-2, between a fixed pendent vertex and nu other pendent vertices. Reti et al. [T. Reti, T Doslic and A. Ali, On the Sombor index of graphs, Contrib. Math. 3 (2021) 11-18] proposed a conjecture that the graph Hn,nu has the maximum Sombor index among all connected nu-cyclic graphs of order n, where 0 & LE;nu & LE;n-2. In some earlier works, the validity of this conjecture was proved for nu & LE;5. In this paper, we confirm that this conjecture is true, when nu=6. The Sombor index in the case that the number of pendent vertices is less than or equal to n-nu-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.</p>