Minimizing the least eigenvalue of unicyclic graphs with fixed diameter

Let U(n,d)U(n,d) be the set of unicyclic graphs on nn vertices with diameter dd. In this article, we determine the unique graph with minimal least eigenvalue among all graphs in U(n,d)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].