The Infimum on Laplacian Eigenvalues of a Connected Extended Graph: An Edge-Grafting Perspective

Consider a connected extended graph G of size N+1 generated by a connected graph G of size N together with one additional node and some directed edges, in which the extra node is named the virtual leader and the directed edges are originated from the virtual leader to some nodes in G. Define an N × N matrix L + H as the extended Laplacian matrix of G, in which L is the Laplacian matrix of G and H is an nonnegative diagonal matrix with at least one diagonal element being 1. In this brief, we focus on the eigenvalues of the extended Laplacian matrix L + H (i.e., the Laplacian eigenvalues) associated with a connected G and acquire the infimum for them by using the edge-grafting approach. It is found that, the infimum on the Laplacian eigenvalues of a connected G obtained here corresponds to a chain together with a virtual leader and a directed edge, in which either one of the two ending nodes with degree 1 in the chain is informed by the virtual leader. Also with the help of the edge-grafting approach, the infimum on the algebraic connectivity associated with a connected G is gotten.