For a connected undirected graph G = (V;E) with vertex set {1; 2;: :: ; n} and degrees di, for 1≤ i≤ n, we show that ABC(G)≤ → (n-1)(|E|-R-1(G)); where R-1(G) = ∑ (i;j)ϵE 1/didj is the Randić index. This bound allows us to obtain some maximal results for the ABC index with elementary proofs and to improve all the upper bounds in [20], as well as some in [17], using lower bounds for R-1(G) found in the literature and some new ones found through the application of majorization.
Cornaro, A., Bianchi, M., Torriero, A., Palacios, J. (2016). New Upper Bounds for the ABC Index. MATCH, 76(1), 117-130.
New Upper Bounds for the ABC Index
Cornaro, Alessandra;Bianchi, Monica;Torriero, Anna;
2016
Abstract
For a connected undirected graph G = (V;E) with vertex set {1; 2;: :: ; n} and degrees di, for 1≤ i≤ n, we show that ABC(G)≤ → (n-1)(|E|-R-1(G)); where R-1(G) = ∑ (i;j)ϵE 1/didj is the Randić index. This bound allows us to obtain some maximal results for the ABC index with elementary proofs and to improve all the upper bounds in [20], as well as some in [17], using lower bounds for R-1(G) found in the literature and some new ones found through the application of majorization.
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