Abstract
It was conjectured by Li and Feng in 1979 that the unicyclic graph formed by a cycle of order g linking to an endvertex of a path of length k minimizes the spectral radius of all unicyclic graphs of order g+k and girth g. In 1987, Cao proved that this conjecture is true for k ≥ g(g − 2)/8 and false for k = 2 and sufficiently large g. In this note, we show that g > 12 suffices for the counterexample and give more counterexamples with large girth for any integer k > 1.
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