Complete solution for unicyclic graphs with minimum general Randić index

The general Randić index of a (molecular) graph G is defined by R_α(G) = Σ_uv(d(u)d(v))^α, where d(u) denotes the degree of a vertex u in G, uv runs over the edge set of G and α is an arbitrary real number. Wu and Zhang studied unicyclic graphs with minimum general Randić index. For α ≥ -1 they got the unique minimum unicyclic graph by distinguishing or into intervals α ≥ 0 and -1 ≤ α <0. But, unfortunately, for α <-1 they could not completely solve this minimum problem. At the end they figured out two classes G and H of possible minimum unicyclic graphs. In this paper, we completely solve this problem by showing that for α <-1 and n ≥ 5, the unique unicyclic graph with minimum general Randić index is either S _n ⁺ or t_[n-3/2],[n-3/2]^k, where S_n ⁺ denotes the unicyclic graph obtained from the star S_n on n vertices by joining its two vertices of degree 1, whereas t_[n-3/2],[n-3/2]^* denotes the unicyclic graph that has a triangle as its unique cycle and the vertices not on the cycle are leaves that are adjacent to two vertices of the triangle such that the numbers of leaf vertices on the two branches are almost equal. Furthermore, we observe that as α approaches to -1, if t_[n-3/2],[n-3/2]^k has the minimum value then n must be considerably large, that is to say, when α is near -1, it is almost sure that S_n ⁺ is the unique minimum unicyclic graph, whereas when α is at a distance from -1, it is almost sure that t_[n-3/2],[n-3/2]^k is the unique minimum unicyclic graph. In particular, for α ≤ -2, S_n ⁺ for 5 ≤ n ≤ 41 and t_[n-3/2],[n-3/2]^k for n ≥ 42, respectively, is the unique unicyclic graph with minimum general Randić index.