Algebraic Connectivity Of Weighted Trees

In this thesis, some needed notions and results on graph theory and Laplacian matrix arefirstly introduced. We have investigated how the algebraic connectivity of a weighted tree behaveswhen the tree is perturbed by removing one of its branches and replacing it with another. Utilizequadratic form and Perron value of bottleneck matrices, we study how the algebraic connectivityof a weighted tree changes under shifting its branches and changing its edges weights and obtainmore entire information about algebraic connectivity of weighted trees than before.(1)When branches not containing any characteristic vertex are shifted from vertices with lessnonnegative valuation or larger nonpositive valuation to other vertices with larger nonnegativevaluation or less nonpositive valuation while their edges weights might change less at same time,or exchange larger weight of edges close to characteristic vertices with smaller weight of edgesoutside along pathes which each path contains a characteristic vertex, or reduce weights of someedges, its algebraic connectivity is non-increasing.(2) When branches not containing any characteristic vertex are shifted from vertices withlarger nonnegative valuation or less nonpositive valuation to other vertices with less nonnegativevaluation or larger nonpositive valuation along pathes which each path contains a characteristicvertex while their edges weights might change larger at same time, or exchange smaller weight ofedges close to characteristic vertices with larger weight of edges outside along pathes which eachpath contains a characteristic vertex, or add weights of some edges, its algebraic connectivity isnon-decreasing.