Faber-Krahn type inequality for unicyclic graphs
  The Faber-Krahn inequality states that the ball has minimal first Dirichlet eigenvalue among all bounded domains with the fixed volume in $\mathbb{R}^n$. In this paper, we investigate the similar inequality for unicyclic graphs. The results show that the Faber-Krahn type inequality also holds for unicyclic graphs with a given graphic unicyclic degree sequence with minor conditions. 
Guang-Jun Zhang, Jie Zhang and Xiao-Dong Zhang
math.CO