Leading Eigenvector of a Regular Graph

Let \(G\) be a \(k\)-regular graph (that is, all nodes have degree \(k\).)

Prove that the vector \(\mathbf{1}\) of all \(1\)s is an eigenvector of the adjacency matrix \(\mathbf{A}\) of \(G\), and determine its eigenvalue.

Note: We’ll soon discuss in class that finding an eigenvector of \(\mathbf{A}\) is one way to mathematically express the idea of centrality or importance in a network. This result says that, when using the so-called eigenvector centrality, all nodes in a \(k\)-regular graph are equally important.