Unicyclic Graphs of Minimal Spectral Radius

<正>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.