Extremal values on the harmonic number of trees

Abstract

Let G=(V(G), E(G)) be a simple connected graph. The harmonic number of G, denoted by H(G), is defined as the sum of the weights 2/(d(u)+d(v)) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, some extremal problems on the harmonic number of trees are studied. The extremal values on the harmonic number of trees with given graphic parameters, such as pendant number, matching number, domination number and diameter, are determined. The corresponding extremal graphs are characterized, respectively.

Keywords:

• trees
• harmonic number
• matching number
• domination number
• diameter

2010 AMS classification::

• 05C05

Acknowledgements

The authors would like to express their sincere gratitude to the referees for a very careful reading of the paper and for all their insightful comments and valuable suggestions, which led to a number of improvements in this paper.

Financially supported by the National Natural Science Foundation of China (grant nos. 11271149 and 11371062), the Programme for New Century Excellent Talents in University (grant no. NCET-13-0817) and the Special Fund for Basic Scientific Research of Central Colleges (grant no. CCNU13F020)).