Geodesic
Cospectral Pairs of Regular Graphs with Different Connectivity
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 577-584

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For vertex- and edge-connectivity we construct infinitely many pairs of regular graphs with the same spectrum, but with different connectivity.
Keywords: graph spectrum, vertex-connectivity, edge-connectivity, spectral characterization 
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Haemers, Willem H. Cospectral Pairs of Regular Graphs with Different Connectivity. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 577-584. http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a12/