An inequality between the diameter and the inverse dual degree of a tree

Let T be a nontrivial tree with diameter D(T) and radius R(T). Let I(T) be the inverse dual degree of T which is defined to be ∑u∈V(T)1/d̄(u), where d̄(u)=(∑v∈N(u)d(v))/d(u) for u∈V(T). For any longest path P of T, denote by a(P) the number of vertices outside P with degree at least 2, b(P) the number of vertices on P with degree at least 3 and distance at least 2 to each of the end-vertices of P, and c(P) the number of vertices adjacent to one of the end-vertices of P and with degree at least 3. In this note we prove that I(T)⩾D(T)/2+a(P)/3+b(P)/10+c(P)/12+56. As a corollary we then getI(T)⩾R(T)+1/3if D(T) is odd,R(T)+5/6if D(T) is even,with equality if and only if T is a path of at least four vertices. The latter inequality strengthens a conjecture made by the program Graffiti.