The ordering of unicyclic graphs with the smallest algebraic connectivity

Fielder [M. Fielder, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298–305] has turned out that GG is connected if and only if its algebraic connectivity a(G)>0a(G)>0. In 1998, Fallat and Kirkland [S.M. Fallat, S. Kirkland, Extremizing algebraic connectivity subject to graph theoretic constraints, Electron. J. Linear Algebra 3 (1998) 48–74] posed a conjecture: if GG is a connected graph on nn vertices with girth g≥3g≥3, then a(G)≥a(Cn,g)a(G)≥a(Cn,g) and that equality holds if and only if GG is isomorphic to Cn,gCn,g. In 2007, Guo [J.M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702–5711] gave an affirmatively answer for the conjecture. In this paper, we determine the second and the third smallest algebraic connectivity among all unicyclic graphs with n(n≥12) vertices.