The eigenvector centrality of a vertex in a graph, is a self-referential centrality, which basically says that a vertex with a high value of eigenvector centrality is one that is adjacent to highly eigenvector-central vertices.
From this, how does one infer that it is a more generalized version of degree centrality, where the most central vertices are the ones adjacent to vertices with a high degree? The definition of the eigenvector centrality tells nothing about how degree plays a role.