On the Displacement of Eigenvalues when Removing a Twin Vertex

Twin vertices of a graph have the same open neighbourhood. If they are not adjacent, then they are called duplicates and contribute the eigenvalue zero to the adjacency matrix. Otherwise they are termed co-duplicates, when they contribute $-1$ as an eigenvalue of the adjacency matrix. On removing a twin vertex from a graph, the spectrum of the adjacency matrix does not only lose the eigenvalue $0$ or $-1$. The perturbation sends a rippling effect to the spectrum. The simple eigenvalues are displaced. We obtain a closed formula for the characteristic polynomial of a graph with twin vertices in terms of two polynomials associated with the perturbed graph. These are used to obtain estimates of the displacements in the spectrum caused by the perturbation

Poncelet's Theorem, Paraorthogonal Polynomials and the Numerical Range of Compressed Multiplication Operators

There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [11]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the theory of OPUC.Comment: 46 pages, 4 figures; minor revisions from v1; accepted for publication in Adv. Mat