Some Extremal Problems On Harmonic Index Of A Graph

In order to study the branching degree of carbon atom skeleton for saturated ydrocarbons, M. Randic proposed an important topological index (named Randic ndex)in 1975. Another variant of the Randic index, named the harmonic index H(G), where H(G)=∑uv∈E(G) 2/(d(u)+d(v)), d(u) is the degree of vertex u and the summatio goes over all edges uv of G. It closely relates to the physical chemical properties of nolecule. In this paper, we study the harmonic index of some graphs. This paper is divided nto the following: In the first part, we introduce the basic terminology and notations, some mportant topological index, current status of research on harmonic index and the nain result of this paper; In the second part, we determine the graph with minimum harmonic indeximong two classes of trees(trees with a given matching number and trees of order n with domination number γ); In the third part, we determine the graph with minimum harmonic index amongill unicyclic graphs with a given matching number and give the corresponding xtremal graphs;In the fourth part, we determine the graph with maximum harmonic index among;raph with a given degree sequence and give the corresponding extremal graphs.; In the finally part, we also proposed some problems for further study on...