Ordering non-bipartite unicyclic graphs with pendant vertices by the least Q-eigenvalue

A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, 2013 ) and Liu et al. (Electron. J. Linear Algebra 26:333-344, 2013 ) determined, independently, the unique unicyclic graph whose least Q -eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices. In this paper, we extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q -eigenvalue.