Bicyclic graphs with the first three smallest and largest values of the first general Zagreb index

Let G be a simple connected graph with vertex set V(G) and α a real number other than 0 and 1. The first general Zagreb index of G is defined as M₁^α(G) = Σ_νεV(G) d(v) ^α, where d(v) is the degree of v. If G has n vertices and n + 1 edges, then it is called a bicyclic graph. In this paper, for arbitrary n ≥ 5, we characterize all bicyclic graphs on n vertices with the first three smallest and largest values of the first general Zagreb index when α > 1, with the largest and the first three smallest values of the first general Zagreb index when α < 0, and with the smallest and the first three largest values of the first general Zagreb index when 0 < α < 1; for every sufficiently large n, we characterize all bicyclic graphs on n vertices with the second and third smallest values of the first general Zagreb index when 0 < α < 1, and with the second and third largest values of the first general Zagreb index when α < 0.