Leaves and inverse degree of a graph

There are many long-standing conjectures on trees, and many invariants of graphs arose from Graffiti conjectures. Let diam(G) and id(G) = Σ_u∈V(G) 1/deg_G(u) be the diameter and the inverse degree of a connected graph G, respectively. The notation n_d(G) indicates the number of vertices of degree d in G. We show that (i) diam(G) + id(G) ≤ 3/2 |G|. (ii) id(G) ≤ 1/2 |G|+1 if G is hamiltonian. (iii) id(G) ≤ _n1(T)+|M|-1/2 for a certain spanning tree T and a largest matching M of G. (iv) For a tree T on n vertices, 2 ≤ 2 · id(T ) - n ≤ n₁(T), diam(T) + n₁(T) + 1 ≤ 2 · id(T ), diam(T) + id(T ) ≤ 2n-n₁(T)-1/2n₂(T)+1, and 2·id(T ) ≤ 2(n₁(T )+|M|)-1 for a largest matching M of T.