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

Cornaro, Alessandra;Bianchi, Monica;Torriero, Anna;
2016

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.
Inglese
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]
File in questo prodotto:
File Dimensione Formato  
match76n1_117-130.pdf

non disponibili

Tipologia file ?: Versione Editoriale (PDF)
Licenza: Non specificato
Dimensione 301.89 kB
Formato Unknown
301.89 kB Unknown   Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10807/72308
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 17
  • ???jsp.display-item.citation.isi??? 14
social impact