The smallest values of algebraic connectivity for unicyclic graphs

The algebraic connectivity of GG is the second smallest eigenvalue of its Laplacian matrix. Let UnUn be the set of all unicyclic graphs of order nn. In this paper, we will provide the ordering of unicyclic graphs in UnUn up to the last seven graphs according to their algebraic connectivities when n≥13n≥13. This extends the results of Liu and Liu [Y. Liu, Y. Liu, The ordering of unicyclic graphs with the smallest algebraic connectivity, Discrete Math. 309 (2009) 4315–4325] and Guo [J.-M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702–5711].