The Algebraic Connectivity of Two Trees Connected by an Edge of Infinite Weight

Document Type


Publication Date



Let T 1 and T 2 be two weighted trees with algebraic connectivities μ(T 1 ) and μ(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 T ^ w . By interlacing properties of eigenvalues of symmetric matrices it is known that μ(T ^ w )≤min{μ(T 1 ),μ(T 2 )}=:m. It is determined precisely when μ(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.


At the time of publication Jason Molitierno was affiliated with University of Connecticut.