New Upper Bounds for the ABC Index

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

Research output: Contribution to journalArticle

14 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)117-130
Number of pages14
Publication statusPublished - 2016


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


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