A unified approach to extremal trees with respect to geometric-arithmetic, szeged and edge szeged indices

The second and third geometric-arithmetic indices GA₂(G) and GA₃(G) of a graph G are defined, respectively, as Σ_uvεE(G) √n_u(e,G) n_v(e,G) / 1/2[n_u(e,G) + n_v(e,G)] and Σ_uvεE(G) √m_u(e, G) m_v(e,G) / 1/2[m_u(e,G) + m _v(e, G)] , where e = uv is one edge in G, n_u(e, G) denotes the number of vertices in G lying closer to u than to v and m_u(e, G) denotes the number of edges in G lying closer to u than to v. The Szeged and edge Szeged indices are defined, respectively, as Sz(G) = Σ _uvεE(G) n_u (e, G) · n_v(e, G) and Sz_e(G) = Σ_uvεE(G) m_u (e, G) · m_v(e, G). In this paper, we provide a unified approach to characterize the tree with the minimum and maximum GA₂, GA ₃, Sz and Sz_e indices among the set of trees with given order and pendent vertices, respectively. As applications, we deduce a result of [2] concerning tree with the maximum GA₂ index and a result of [3] concerning tree with the maximum GA₃ index.