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Abstract: O and Shi proved that the Randić index of any graph G with minimum degree at least δ and maximum degree at most Δ is at least sqrt(δΔ)/(δ+Δ) |G|, with equality if and only if the graph is (δ, Δ)-biregular. In this note we give a short proof via a more general statement. As an application of our more general result, we classify for any given degree range which graphs minimise (or maximise) the generalised Randić index for any exponent, and describe the transitions between different types of behaviour precisely.
Keywords: Randić index, bounded-degree graph, extremal problem
Published in RUP: 03.11.2025; Views: 344; Downloads: 2
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Abstract: In the paper we study maximal values of Balaban and sum-Balaban index, and correct some results appearing in the literature which are only partially correct. Henceforth, we were able to solve a conjecture of M. Aouchiche, G. Caporossi and P. Hansen regarding the comparison of Balaban and Randić index. In addition, we showed that for every k and large enough n, the first k graphs of order n with the largest value of Balaban index are trees. We conclude the paper with a result about the accumulation points of sum-Balaban index.
Keywords: topological index, Balaban index, sum-Balaban index, Randić index
Published in RUP: 03.01.2022; Views: 2514; Downloads: 25
Full text (310,27 KB)