NOTES ON THE HARMONIC INDEX OF GRAPHS

For a connected graph G of order n, the harmonic index of G is the sum of weights 2 d(u)+d(v) over all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. In this note, we prove that H(G) ≥ χDP (G)− 2 2 + 2(χDP (G)− 1) n + χDP (G)− 2 + 2(n− χDP (G)) n , and this bound is sharp for all n and 2 ≤ χDP (G) ≤ n, where χDP (G) is the DP -chromatic number of G. This generalizes the previous lower bounds on H(G). Moreover, we also determine the tree with minimum harmonic index among trees in Tn,l, where Tn,l is the set of trees of order n with a given segment sequence l. AMS classification: 05C09; 05C15.