05C50 Online Seminar: Sukanta Pati

Let G be a connected simple graph and L be its Laplacian matrix. Let v be a cut vertex and B be a branch at v. Assume that v_1 is the only vertex in B adjacent to v. Let P be a path that starts at v_1 and stays inside B. It is shown that the algebraic connectivity decreases if we do an appropriate sliding operation along the path. Similar results for trees were known decades ago. The techniques involve the notions of Perron branches and bottleneck matrices developed by Kirkland, Neumann, Shader and others.