The second-minimum and second-maximum values of exponential Zagreb indices among trees

Exponential of the first Zagreb index and the second Zagreb index of a graph G are defined as e(M1) (G) = Sigma(uv is an element of E(G)) e(d(u)+d(v)), and e(M2) (G) = Sigma(uv is an element of E(G)) e(d(u)d(v)), where E(G) denotes the set of edges and d(u) denotes the degree of vertex u. The extremal trees of these two indices among trees have been determined in previous studies. In this paper, we give the second-minimum and the second-maximum values of e(M1) and e(M2) among trees.