Minimizing the least eigenvalue of unicyclic graphs with fixed diameter

Let U ( n , d ) be the set of unicyclic graphs on n vertices with diameter d . In this article, we determine the unique graph with minimal least eigenvalue among all graphs in U ( n , d ) . It is found that the extremal graph is different from that for the corresponding problem on maximal eigenvalue as done by Liu et al. [H.Q. Liu, M. Lu, F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007) 449–457].