4 research outputs found
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 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 or . 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
Strong Cospectrality and Twin Vertices in Weighted Graphs
We explore algebraic and spectral properties of weighted graphs containing
twin vertices that are useful in quantum state transfer. We extend the notion
of adjacency strong cospectrality to arbitrary Hermitian matrices, with focus
on the generalized adjacency matrix and the generalized normalized adjacency
matrix. We then determine necessary and sufficient conditions such that a pair
of twin vertices in a weighted graph exhibits strong cospectrality with respect
to the above-mentioned matrices. We also generalize known results about
equitable and almost equitable partitions, and use these to determine which
joins of the form , where is either the complete or empty graph,
exhibit strong cospectrality.Comment: 25 pages, 6 figure
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