On the Displacement of Eigenvalues when Removing a Twin Vertex
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Twin vertices of a graph have the same common neighbours. If they are 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 closed formulae 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.
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