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.