The least eigenvalue of unicyclic graphs with [formula omitted] vertices and [formula omitted] pendant vertices

Abstract

Let U ( n , k ) be the set of unicyclic graphs with n vertices and k pendant vertices. In this paper, we determine the unique graph with the minimal least eigenvalue among all graphs in U ( n , k ) . The work is related with that of Guo [S.G. Guo, The spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices, Linear Algebra Appl. 408 (2005) 78–85], which determined the unicyclic graph with the maximal spectral radius in U ( n , k ) . We can observe that the extremal graph on the least eigenvalue is different from that on the spectral radius.