On the eigenvalues of Cayley graphs on generalized dihedral groups

Abstract

‎Let $Gamma$ be a graph with adjacency eigenvalues $lambda_1leqlambda_2leqldotsleqlambda_n$‎. ‎Then the energy of‎ ‎$Gamma$‎, ‎a concept defined in 1978 by Gutman‎, ‎is defined as $mathcal{E}(G)=sum_{i=1}^n|lambda_i|$‎. ‎Also‎ ‎the Estrada index of $Gamma$‎, ‎which is defined in 2000 by Ernesto Estrada‎, ‎is defined as $EE(Gamma)=sum_{i=1}^ne^{lambda_i}$‎. ‎In this paper‎, ‎we compute the eigenvalues‎, ‎energy and Estrada index of Cayley graphs on generalized dihedral groups‎. ‎As an application‎, ‎we‎ ‎compute these items for honeycomb toroidal graphs and Cayley graphs on dihedral groups‎.