ar X iv : 0 90 6 . 52 30 v 1 [ m at h . C O ] 2 9 Ju n 20 09 Randić index , diameter and the average distance ∗

نویسندگان

  • Xueliang Li
  • Yongtang Shi
چکیده

The Randić index of a graph G, denoted by R(G), is defined as the sum of 1/ √ d(u)d(v) over all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we partially solve two conjectures on the Randić index R(G) with relations to the diameter D(G) and the average distance μ(G) of a graph G. We prove that for any connected graph G of order n with minimum degree δ(G), if δ(G) ≥ 5, then R(G) −D(G) ≥ √ 2 − n+1 2 ; if δ(G) ≥ n/5 and n ≥ 15, R(G) D(G) ≥ n−3+2 √ 2 2n−2 and R(G) ≥ μ(G). Furthermore, for any arbitrary real number ε (0 < ε < 1), if δ(G) ≥ εn, then R(G) D(G) ≥ n−3+2 √ 2 2n−2 and R(G) ≥ μ(G) hold for sufficiently large n.

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تاریخ انتشار 2009