EXTREMAL FIRST AND SECOND ZAGREB INDICES OF APEX TREES

Let G be a simple connected graph with edge set E(G) and vertex set V (G). The first and the second Zagreb indices of the graph G are defined as M-1(G) = [GRAPHICS] (d(v))(2) and M-2(G) = [GRAPHICS] d(u)d (v), respectively, where d(v) is the degree of the vertex v. A graph G is called an apex tree [8] if it contains a vertex x such that G-x is a tree. For any integer k >= 1 the graph G is called k-apex tree if there exists a subset X of V (G) of cardinality k such that G-X is a tree and for any Y subset of V (G) and vertical bar Y vertical bar< k, G-Y is not a tree. In this work we have determined upper and lower bounds of M-1(G) and an upper bound of M-2(G) in k apex trees. The corresponding extremal k-apex trees are also characterized in each case.