A (FORGOTTEN) UPPER BOUND FOR THE SPECTRAL RADIUS OF A GRAPH

Abstract

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 line graph degree-based upper bounds for the Q-index is then proposedand compared experimentally with a graph based bound. Finally a new lower bound for generalised $r$-partite graphs is proved, by extending a result due to Erdős.