Open problems on Sombor index of unicyclic and bicyclic graphs.

Let Γ = (V , E) be a graph of order p. Recently, the Sombor index is introduced, defined as S O (Γ) = ∑ v i v j ∈ E (Γ) d Γ (v i) 2 + d Γ (v j) 2 , where d Γ (v i) is the degree of the vertex v i in Γ. Cruz and Rada [4] obtained an upper bound on the Sombor index of unicyclic and bicyclic graphs of order p , but did not characterize the extremal graphs. In the same paper, they mentioned that the maximal graphs over the set of unicyclic and bicyclic graphs with respect to Sombor index, is an interesting problem that remains open. In this paper we completely solve these problems. • Sombor index of a graph Γ is defined as S O (Γ) = ∑ v i v j ∈ E (Γ) d Γ (v i) 2 + d Γ (v j) 2 . • To determine maximal graph over the set of unicyclic graphs with respect to Sombor index. • To characterize the maximal bicyclic graph with respect to Sombor index. [ABSTRACT FROM AUTHOR]