The smallest values of algebraic connectivity for unicyclic graphs Original Research Article

The algebraic connectivity of image is the second smallest eigenvalue of its Laplacian matrix. Let image be the set of all unicyclic graphs of order image. In this paper, we will provide the ordering of unicyclic graphs in image up to the last seven graphs according to their algebraic connectivities when image. 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].