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

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Electronic Journal of Linear Algebra, Vol: 8, Issue: 1

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Molitierno, Jason J.; Neumann, Michael
University of Wyoming Libraries
article description
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.