The algebraic connectivity of two trees connected by an edge of infinite weight

Abstract

Let T1 and T2 be two weighted trees with algebraic connectivities μ(T1) and μ(T2), 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 T̂w. By interlacing properties of eigenvalues of symmetric matrices it is known that μ(T̂w) ≤ min{μ(T1), μ(T2)} =: m. It is determined precisely when μ(T̂w) → m as w → ∞. Finally, a possible interpretation is given of this result to the theory of electrical circuits and Kirchoff's laws.