I'm trying to understand when the most central node of a graph is the same when measured according to the degree- and eigenvalue-centrality measures. That is, if $\mathbf{x}$ is the principal eigenvalue of $A(G)$ and $\mathbf{x}_v$ is its largest component under what circumstances will $d_G(v)$ be largest?
The graph $G_1$ comprising $K_3$ joined to $K_{1,3}$ by a single edge has as largest component of its principal eigenvector the vertex of $K_3$ of degree 3, which is not the largest degree vertex; the degree-4 vertex at the root of the star is. OTOH if I interconnect the three degree-1 star vertices with 2 edges and add two more "bridge" edges each between a $K_3$ vertex and a star vertex (so that all vertices are now degree-3 except for one degree-4) then the largest component of $G_2$'s principal e'vector is the degree-4 root of the star.
Can anybody shed any light on this or refer me to where it might be discussed in the literature?
Many thanks.