Mathematics Colloquia and Seminars

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We review the properties of eigenvectors for the graph Laplacian matrix, aiming at predicting a specific eigenvalue/vector from the geometry of the graph. We focus on eigenvectors that have zero components and extend the pioneering results of Merris (1998) on graph transformations
that preserve a given eigenvalue λ or shift it in a simple way. These transformations enable us to obtain eigenvalues/vectors combinatorially instead of numerically; in particular we show that graphs having eigenvalues λ = 1, 2, . . . , 6 up to six vertices can be obtained from a short list
of graphs. For the converse problem of a λ subgraph G of a λ graph G”, we prove results and conjecture that G and G” are connected by two of the simple transformations described above.