Maximum General Randi(?) Index Of Quasi-tree Graphs

In studying the extent of branching of the carbon-atom skeleton of saturatedhydrocarbons, Randi(?) in 1975 proposed an important topological index- branching index: R =∑_uv∈E(d(u)d(v))^?, where d(u) denotes the degree of a vertex u. Branching index is also called the connectivity index or Randi(?) index. The Randi(?) index is closely correlated with a variety of physico-chemicalproperties of molecular. Later in 1998, Bollob(?)s. Erd(?)s and Ami(?) et al. independently generalized this index to be the general Randi(?) index: R_α(G)=∑_uv∈E(d(u)d(v))^α, whereαis an arbitrary real number. The general Randi(?) index has attracted the wide attention of both theoretical chemists and mathematicians. Up to now, research interest on general Randi(?) index focus mainly on the problem of determining the extremal graphs in a given class of graphs. A graph G is called a quasi-tree graph, if there exists u∈V(G) such that G- u is a tree. In this paper, forα≥1 andα≤-2.1, the maximum value of the general Randi(?) index and the corresponding extremal quasi-tree praphs(other than trees) are determined. For 0 <α< 1 and -2.1 <α< 0, some properties for such quasi-tree graphs are also established.