Effects on the algebraic connectivity of weighted graphs under edge rotations

For a weighted graph G, the rotation of an edge uv₁ from v₁ to a vertex v₂ is defined as follows: delete the edge uv₁, set w(uv₂) as w(uv₁)+w(uv₂) if uv₂ is an edge of G; otherwise, add a new edge uv₂ and set w(uv₂)=w(uv₁), where w(uv₁) and w(uv₂) are the weights of the edges uv₁ and uv₂, 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.