A graph H is a star complement for an eigenvalue \(\mu \) in its extension G if \(\mu \) is not in the spectrum of H, but appears as an eigenvalue of G with multiplicity \(|V(G)|-|V(H)|\) . The eigenvalues 0 and \(-\,1\) pose certain obstruction to the general theory, as for every H there is a finite number of graphs containing H as a star complement for \(\mu \) if and only if \(\mu \notin \{0,-\,1\}\) . We establish some structural properties of regular graphs having a path as a star complement for any of these eigenvalues. It occurs that, for a fixed path, either there are no regular extensions or there is an infinite family of such extensions. We also discuss corona products \(T\circ K_1\) as star complements for 0 in regular graphs, where T is either a star or a path. Regular extensions are characterized, and in some particular cases they are completely determined. In contrast to the previous case, here we encounter the existence of a unique regular extension among regular graphs with a given vertex degree greater than 2.
