On the Spectra of Symmetric Cylindrical Constructs

TL;DR

This paper investigates the spectra of symmetric cylindrical graph constructs, generalizing known results, and introduces new classes of highly symmetric graphs, including perturbations of base graphs and sparsifications of complete graphs.

Contribution

It generalizes spectral results for cylindrical graph constructs and introduces new highly symmetric graph classes, extending previous work on graph spectra and symmetries.

Findings

Spectra of bsymmetric cylinders without internal vertices equal eigenvalues of a base perturbation.

Spectra of sparsified complete graphs by tree-cylinders are characterized.

A new class of highly symmetric graphs generalizing Petersen and Coxeter graphs is introduced.

Abstract

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.