Lower bounds for the Randi\'c index in terms of matching number

Lower bounds for the Randi\'c index in terms of matching number

We investigate how small the Randi\'c index of a graph can be in terms of its matching number, and prove several results. We give best-possible linear bounds for graphs of small excess and for subcubic graphs; in the former case the size of excess we permit is qualitatively the best possible. W...

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Hauptverfasser: Akbari, Saieed, Nezhad, Sina Ghasemi, Ghazizadeh, Reyhane, Haslegrave, John, Tohidi, Elahe
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creator Akbari, Saieed
Nezhad, Sina Ghasemi
Ghazizadeh, Reyhane
Haslegrave, John
Tohidi, Elahe
description We investigate how small the Randi\'c index of a graph can be in terms of its matching number, and prove several results. We give best-possible linear bounds for graphs of small excess and for subcubic graphs; in the former case the size of excess we permit is qualitatively the best possible. We show that a linear bound holds for any sparse hereditary graph class (such as planar graphs). In general, however, we show that it can be much smaller than linear. We determine the asymptotic growth rate of the minimum Randi\'c index for graphs with a near perfect matching, and conjecture that the same bounds hold for all graphs.
doi_str_mv 10.48550/arxiv.2402.12884
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title Lower bounds for the Randi\'c index in terms of matching number
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