On Harmonic Index and Diameter of Unicyclic Graphs

The Harmonic index H(G) of a graph G is defined as the sum of the weights dfrac{2}{d(u)+d(v)} of all edges uv of G, where d(u) denotes the degree of the vertex u in G. In this work, we prove the conjecture dfrac{H(G)}{D(G)} geq dfrac{1}{2}+dfrac{1}{3(n1)} given by Jianxi Liu in 2013 when G is a unicyclic graph and give a better bound dfrac{H(G)}{D(G)}geq dfrac{1}{2}+dfrac{2}{3(n2)}, where n is the order and D(G) is the diameter of the graph G.