ALGEBRAIC CONNECTIVITY OF WEIGHED GRAPHS UNDER SHIFTING COMPONENTS

Abstract

Integrating quadratic form of algebraic connectivity and Perron value of bottleneck matrices, we investigate how the algebraic connectivity of a connected weighted graph behaves under shifting components. Generally speaking, when shift components not containing characteristic vertex from less positive (larger negative) valuation vertices to larger positive (less negative) valuation vertices, or reduce weights of some edges, or add some new blocks, its algebraic connectivity is nonincreasing; when shift components along paths from blocks to other block closer to characteristic block (vertex), or increase weights of some edges, or delete some blocks, its algebraic connectivity is non-decreasing. Therefore, algebraic connectivity could be regarded as a measure of central tendency about blocks of a connected weighted graph with characteristic block (vertex) as its center.