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 G is connected if and only if its algebraic connectivity a ( 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 G is a connected graph on n vertices with girth g ≥ 3 , then a ( G ) ≥ a ( C n , g ) and that equality holds if and only if G is isomorphic to C n , 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.