Sharp bounds for the Randić index of graphs with given minimum and maximum degree
نویسندگان
چکیده
The Randić index of a graph G, written R(G), is the sum of 1 √ d(u)d(v) over all edges uv in E(G). Let d and D be positive integers d < D. In this paper, we prove that if G is a graph with minimum degree d and maximum degree D, then R(G) ≥ √ dD d+Dn; equality holds only when G is an n-vertex (d,D)-biregular. Furthermore, we show that if G is an n-vertex connected graph with minimum degree d and maximum degree D, then R(G) ≤ n2 − D−1 i=d 1 2 ( 1 √ i − 1 √ i+1 2 ; it is sharp for infinitely many n, and we characterize when equality holds in the bound.
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