# New Upper Bounds for the ABC Index

Monica Bianchi, Alessandra Cornaro, Anna Torriero, José Luis Palacios

Risultato della ricerca: Contributo in rivistaArticolo in rivista

14 Citazioni (Scopus)

## 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.
Lingua originale English 117-130 14 Match 2016 Pubblicato - 2016

## Keywords

• Atom-bond connectivity index
• Majorization
• Randic index
• Schur-convex functions

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