ELA, Volume 8, pp. 1-13, January 2001, abstract.
The algebraic connectivity of two trees connected
by an edge of infinite weight
Jason J. Molitierno and Michael Neumann
Let T_1 and T_2 be two weighted trees with algebraic
connectivities mu(T_1) and mu(T_2), respectively.
A vertex on one of the trees is connected to a vertex
on the other by an edge of weight w to obtain a new tree
hat{T}_w. By interlacing properties of eigenvalues of
symmetric matrices it is known that
mu(hat{T}_w) <= min{mu(T_1), mu(T_2)} =: m.
It is determined precisely when mu(hat{T}_w) tends to m as
w tends to infinity. Finally, a possible interpretation is
given of this result to the theory of electrical circuits
and Kirchoff's laws.