A Database of Star Complements of Graphs

We take G to be an undirected graph without loops or multiple edges, with vertex set V (G) = f1; : : : ; ng, and with (0; 1)-adjacency matrix A. Let P denote the orthogonal projection of IR onto the eigenspace E( ) of A, and let fe1; : : : ; eng be the standard orthonormal basis of IR. Since E( ) is spanned by the vectors Pej (j = 1; : : : ; n) there exists X V (G) such that the vectors Pej (j 2 X) form a basis for E( ). Such a subset X of V (G) is called a star set for in G. (The terminology re ects the fact that the vectors Pe1; : : : ; Pen form a eutactic star in the sense of Seidel [23]. In the context of star partitions [10, Section 7.1], star sets are called star cells.) An equivalent de nition which is useful in a computational context is the following: if has multiplicity k then a star set for in G is a set X of k vertices of G such that is not an eigenvalue of G X [10, Theorem 7.2.9]. Here G X is the subgraph of G induced by X , the complement of X in V (G). Accordingly, the graph G X is called the star complement for corresponding to X. (Such graphs are called -basic subgraphs in [15].) If a single vertex is deleted from a graph then by the Interlacing Theorem [5, Theorem 0.19] the multiplicity of an eigenvalue changes by 1 at most. Accordingly, the deletion of any r vertices from X (0 < r < k) results in a graph for which is an eigenvalue of multiplicity k r. We can also make the following observations: (i) if K is an induced subgraph of G not having as an eigenvalue then G has a star complement for containing K [21, Proposition 1.1], (ii) a connected graph has a connected star complement for each eigenvalue [21, Theorem 2.4]. (The second result is ascribed to S. Penrice in [15].)