  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.
