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Graphs can be associated with a matrix according to some rule and we can findthe spectrum of a graph with respect to that matrix. Two graphs are cospectralif they have the same spectrum. Constructions of cospectral graphs help usestablish patterns about structural information not preserved by the spectrum.We generalize a construction for cospectral graphs previously given for thedistance Laplacian matrix to a larger family of graphs. In addition, we showthat with appropriate assumptions this generalized construction extends to theadjacency matrix, combinatorial Laplacian matrix, signless Laplacian matrix,normalized Laplacian matrix, and distance matrix.

Cospectral constructions for several graph matrices using cousin vertices