A (forgotten) upper bound for the spectral radius of a graph

The best degree-based upper bound for the spectral radius is due to Liu and Weng. This paper begins by demonstrating that a (forgotten) upper bound for the spectral radius dating from 1983 is equivalent to their much more recent bound. This bound is then used to compare lower bounds for the clique number. A series of sharp upper bounds for the signless Laplacian spectral radius is then proposed as another application. Finally a new lower bound for generalised r-partite graphs is proved, by extending a result due to Erdos.

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