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

Abstract

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) I/d(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 + 5/6. As a corollary we then get { R(T) + 1/3 if D(T) is odd, I(T) ≥ { R(T) + 5/6 if D(T) is even, with equality if and on..