Star complements for ±2 in signed graphs

Abstract

In this article, we investigate connected signed graphs which have a connected star complement for both -2-2 and 2 (i.e. simultaneously for the two eigenvalues), where -2-2 (resp. 2) is the least (largest) eigenvalue of the adjacency matrix of a signed graph under consideration. We determine all such star complements and their maximal extensions (again, relative to both eigenvalues). As an application, we provide a new proof of the result which identifies all signed graphs that have no eigenvalues other than -2-2 and 2.