The extremal generalised Randić index for a given degree range
arxiv(2024)
摘要
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
√(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. We apply the latter to classify the graphs in any
given degree range which minimise (or maximise) the generalised Randić index
for any exponent, and describe the transitions between different types of
behaviour precisely.
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