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\r\n$$ABC(G) \le \sqrt{(n-1)(|E|-R_{-1}(G))},$$\r\nwhere $\displaystyle R_{-1}(G)=\sum_{(i,j)\in E}\frac{1}{d_id_j}$ is the Randi\'c index.\r\nThis 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 |
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pagine (da-a) | 117-130 |
Numero di pagine | 14 |
Rivista | Match |
Volume | 2016 |
Numero di pubblicazione | 1 |
Stato di pubblicazione | Pubblicato - 2016 |
All Science Journal Classification (ASJC) codes
- ???subjectarea.asjc.1600.1600???
- ???subjectarea.asjc.1700.1706???
- ???subjectarea.asjc.1700.1703???
- ???subjectarea.asjc.2600.2604???
Keywords
- Atom-bond connectivity index
- Majorization
- Randic index
- Schur-convex functions