Bicyclic Graphs With The Minimum General Randic Index

In 1975, Randic [1] proposed an important topological index of a (molecular)graph in his research on molecular structures, which is closely related with manychemical properties, and it is called Randic index or is known as the connectivityindex [2, 3]. Later, Bollobas and Erd¨os [4] generalized this index, which wascalled the general Randic index. The general Randic index Rα(G) of a graph Gis defined by Rα(G) = uv∈E(G)(d(u)d(v))α,where d(u) denotes the degree ofa vertex u in G andα(= 0) is an arbitrary real number. Bollobas and Erd¨os [4]gave a sharp lower bound of Rαfor ?1≤α< 0 in restricted to the graphs (mayhave isolated vertices) of given size. Clark and Moon [5] gave several extremaland probabilistic results of Rαfor certain families of trees. Hu, Li and Yuan [6]gave the minimum general Randic index of trees and its extremal graph, andbasically Wu and Zhang [7] gave the minimum general Randic index of unicyclicgraphs and its extremal graph.Let G be a simple connected graph. We call it bicyclic graph if there existtwo edges e1,e2∈E(G) such that G ? {e1,e2} is a spanning tree of G. |P|denotes the length of the path P. For an integer k≥1, the k- -graph denotesa bicyclic graph obtained from two disjoint cycles by joining them with a pathof length k. The∞-graph denotes a bicyclic graph consisting of two cycles withexactly one common vertex. Theθ-graph denotes a bicyclic graph consisting oftwo vertices of degree three joined by three independent paths Q1,Q2 and Q3(where |Q3|≥|Q2|≥|Q1|).In this thesis, we study the minimum general Randic index of the bicyclicgraphs of order n. The minimum general Randic index of the bicyclic graphs oforder n and its extremal graphs are determined forα≥1. Let G be a bicyclicgraph of order n. The followings are our main results:(1) Rα(G)≥6·6α+(n-5)·4αforα> 0 and the equality holds if and only ifG is one of followings: (i) a k- -graph with k > 1 for n≥7; (ii) aθ-graphwith |Q1| > 1 for n≥5.(2) Rα(G)≥(n?5)(n-2)α+2·(2n-4)α+(3n-6)α+2·6αfor -1≤α< 0and n≥5 and the equality holds if and only if G～= Sn+ +, where Sn++denotes the bicyclic graph of order n obtained from the star Sn with nvertices {v1,v2,v3,···,vn} by adding two edges e1 = v1v2 and e2 = v2v3to it.