Note: A proof of a conjecture on the Randić index of graphs with given girth

Authors:
Xueliang Li;Jianxi Liu
Affiliations:
Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, PR China;Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, PR China
Venue:
Discrete Applied Mathematics
Year:
2009

Citing 3
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On the Randić index and girth of graphs

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Abstract

The Randic index R(G) of a graph G is defined by R(G)=@?"u"v1d(u)d(v), where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche, Hansen and Zheng proposed the following conjecture: For any connected graph on n=3 vertices with Randic index R and girth g, R+g=n-3+2n-1+72andR@?g=3n-9+32n-1+32 with equalities if and only if G=S"n^+. This paper is devoted to giving a confirmative proof to this conjecture.