Consider a connected k-regular undirected network (i.e., a network in which every node has degree k and there is only one component).
a) Show that the uniform vector 1 = (1,1,1,...) is an eigenvector of the adjacency matrix with eigenvalue k. In a connected network there is only one eigenvector with all elements positive and hence the eigenvector 1 gives, by definition, the eigenvector centrality of our k-regular network and the centralities are the same for every vertex. c) You should find that, like the eigenvector centralities, the Katz centralities of all nodes are the same. Name a centrality measure that could give different centrality values for different nodes in a regular network.