  We investigate the bounds on algebraic connectivity of graphs subject to
constraints on the number of edges, vertices, and topology. We show that the
algebraic connectivity for any tree on $n$ vertices and with maximum degree $d$
is bounded above by $2(d-2) \frac{1}{n}+O(\frac{\ln n}{n^{2}}) .$ We then
investigate upper bounds on algebraic connectivity for cubic graphs. We show
that algebraic connectivity of a cubic graph of girth $g$ is bounded above by
$3-2^{3/2}\cos(\pi/\lfloor g/2\rfloor) ,$ which is an improvement over the
bound found by Nilli [A. Nilli, Electron. J. Combin., 11(9), 2004]. Finally, we
propose several conjectures and open questions.
