For a connected undirected graph $G=(V,E)$ with vertex set $\{1, 2, \ldots, n\}$ and degrees $d_i$, for $1\le i \le n$, we show that $$ABC(G) \le \sqrt{(n-1)(|E|-R_{-1}(G))},$$ where $\displaystyle R_{-1}(G)=\sum_{(i,j)\in E}\frac{1}{d_id_j}$ is the Randi\'c 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. L., New Upper Bounds for the ABC Index, <<MATCH>>, 2016; 2016 (1): 117-130 [http://hdl.handle.net/10807/72308]

### New Upper Bounds for the ABC Index

#### Abstract

For a connected undirected graph $G=(V,E)$ with vertex set $\{1, 2, \ldots, n\}$ and degrees $d_i$, for $1\le i \le n$, we show that $$ABC(G) \le \sqrt{(n-1)(|E|-R_{-1}(G))},$$ where $\displaystyle R_{-1}(G)=\sum_{(i,j)\in E}\frac{1}{d_id_j}$ is the Randi\'c 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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Cornaro, A., Bianchi, M., Torriero, A., Palacios, J. L., New Upper Bounds for the ABC Index, <>, 2016; 2016 (1): 117-130 [http://hdl.handle.net/10807/72308]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10807/72308
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