On the Spectra of Symmetric Cylindrical Constructs

In this article, following [A.~Daneshgar, M.~Hejrati, M.~Madani, {\it On cylindrical graph construction and its applications}, EJC, 23(1) p1.29, 45, 2016] we study the spectra of symmetric cylindrical constructs, generalizing some well-known results on the spectra of a variety of graph products, graph subdivisions by V.~B.~Mnuhin (1980) and the spectra of GI-graphs (see [M.~Conder, T.~Pisanski, and A.~{\v{Z}}itnik, {\it GI-graphs: a new class of graphs with many symmetries}, 40, 209--231 (2014)] and references therein). In particular, we show that for bsymmetric cylinders with no internal vertex the spectra is actually equal to the eigenvalues of a perturbation of the base, and using this, we study the spectra of sparsifications of complete graphs by tree-cylinders. We also, show that a specific version of this construction gives rise to a class of highly symmetric graphs as a generalization of Petersen and Coxeter graphs.