The inverse eigenvector centrality problem for connected undirected graphs is completely characterized by conditions on stable sets and their neighborhoods.
[10]Michini, C.The stable set problem: some structural properties and relaxations.4OR 11 (2013), 199–200
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A Complete Characterization of the Inverse Eigenvector Centrality Problem for Undirected Graphs
The inverse eigenvector centrality problem for connected undirected graphs is completely characterized by conditions on stable sets and their neighborhoods.