Unicyclic graphs with the first three smallest and largest first general Zagreb index

The first general Zagreb index of a graph G is defined as M ₁^α(G) = Σ_uεV(G) d(U) ^α, where d(u) denotes the degree of the vertex u in G and α is an arbitrary real number except 0 and 1. A graph is called unicyclic if it is connected and contains a unique cycle. In this paper, we characterize all unicyclic graphs with the smallest, the second and third smallest values of the first general Zagreb index. The same is done for unicyclic graphs with the largest, the second and third largest values of this index.