Unicyclic Graphs of Minimal Spectral Radius

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.