For every rational number x ∈ (0, 1), we construct a pair of graphs, one regular and one nonregular with adjacency matrices A1 and A2, having the property that A1 - xJ and A2 - xJ have the same spectrum (J is the all-ones matrix). This solves a problem of Van Dam and the second author. For some values of x, we have generated the smallest examples (with respect to the number of vertices) by computer. © 2006 Elsevier Inc. All rights reserved.
Chesnokov, A. A., & Haemers, W. H. (2006). Regularity and the generalized adjacency spectra of graphs. Linear Algebra and Its Applications, 416(2–3), 1033–1037. https://doi.org/10.1016/j.laa.2006.01.026