Effects on the algebraic connectivity of weighted graphs under edge rotations

Abstract

For a weighted graph G, the rotation of an edge $u{v}_1$ from $v_1$ to a vertex $v_2$ is defined as follows: delete the edge $u{v}_1$, set $w\left(u{v}_2\right)$ as $w\left(u{v}_1\right)+w\left(u{v}_2\right)$ if $u{v}_2$ is an edge of G; otherwise, add a new edge $u{v}_2$ and set $w\left(u{v}_2\right)=w\left(u{v}_1\right)$, where $w\left(u{v}_1\right)$ and $w\left(u{v}_2\right)$ are the weights of the edges $u{v}_1$ and $u{v}_2$, respectively. In this paper, effects on the algebraic connectivity of weighted graphs under edge rotations are studied. For a weighted graph, a sufficient condition for an edge rotation to reduce its algebraic connectivity and a necessary condition for an edge rotation to improve its algebraic connectivity are proposed based on Fiedler vectors of the graph. As applications, we show that, by using a series of edge rotations, a pair of pendent paths (a pendent tree) of a weighted graph can be transformed into one pendent path (pendent edges attached at a common vertex) of the graph with the algebraic connectivity decreasing (increasing) monotonically. These results extend previous findings of reducing the algebraic connectivity of unweighted graphs by using edge rotations.