Increase or Decrease of the Second Eigenvalue of Laplacian Matrices Caused by Adding an Edge to Directed Graphs

Abstract

Abstract. Increase or decrease of “the second eigenvalue (the second smallest real part of all eigenvalues)” of the graph Laplacian caused by the addition of an edge to directed graphs is considered. It is proved that the addition of an upstrem edge to a directed path or a directed tree graph decreases the second eigenvalue, while the addition of a downstream edge does not. The extension of those results to the edge-addtion in a directed acyclic graph with a spanning tree is also discussed.