Abstract
Given an Hermitian matrix A whose graph G is a simple undirected graph and its eigenvalues , we suppose the status of each vertex in the graph is known for each eigenvalue of A. We investigate the change of the multiplicity of each eigenvalue , when we add a pendent vertex with given value to a particular vertex in the graph via an edge with given weight. It is shown how each multiplicity changes based on this information. The results are applied to show that more than one eigenvalue may increase in multiplicity with the addition of just one vertex. The intended focus is trees, but the analysis is given for general graphs .
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Toyonaga, K., Johnson, C.R., Uhrig, R. (2017). Multiplicities: Adding a Vertex to a Graph. In: Bebiano, N. (eds) Applied and Computational Matrix Analysis. MAT-TRIAD 2015. Springer Proceedings in Mathematics & Statistics, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-49984-0_8
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DOI: https://doi.org/10.1007/978-3-319-49984-0_8
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