A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the new distance matrix, with natural extensions to the distance Laplacian and distance signless Laplacian, in addition to the new adjacency matrix, with natural extensions to the Laplacian and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues.
Available at: http://works.bepress.com/leslie-hogben/136/
This article is published as Catral, Minerva, Lorenzo Ciardo, Leslie Hogben, and Carolyn Reinhart. "Spectra of products of digraphs." The Electronic Journal of Linear Algebra 36, no. 36 (2020): 744-763. DOI: 10.13001/ela.2020.5243. Posted with permission.