Remarks on a conjecture about Randic index and graph radius

LetG be a nontrivial connected graph. The radiusr.G/ ofG is the minimum eccent- ricity among eccentricities of all vertices in G. The Randiindex of G is defined as R.G/D P uv2E.G/ 1 p d G.u/dG.v/ , and the Harmonic index is defined asH.G/D P uv2E.G/ 2 dG.u/CdG.v/ , where dG.x/ is the degree of the vertex x in G. In 1988, Fajtlowicz conjectured that for any connected graphG,R.G/ r.G/ 1. This conjecture remains still open so far. More recently, Deng et al. proved that this conjecture is true for connected graphs with cyclomatic number no more than 4 by means of Harmonic index. In this short paper, we use a class of composite gra- phs to construct infinite classes of connected graphs, with cyclomatic number greater than 4, for which the above conjecture holds. In particular, for any given positive odd number k 7, we construct a connected graph with cyclomatic numberk such that the above conjecture holds for this graph. 2010 Mathematics Subject Classification: 05C07; 05C12; 05C76