On di-Cayley graphs and their spectrum

Abstract

Given a group $G$ and three subsets $S_{\ell },S_r,S_m\subset G$, we consider di-Cayley graphs $DX(G;S_{\ell },S_r,S_m)$ and di-Cayley sum graphs $D{X}^{+}(G;S_{\ell },S_r,S_m)$, directed generalizations of the bi-Cayley (sum) graphs $BX(G;S_{\ell },S_r,S_m)$ and $B{X}^{+}(G;S_{\ell },S_r,S_m)$. We refer to these four kinds of graphs collectively as $X^{\ast }(G;S_{\ell },S_r,S_m)$. First, we give the basic properties of these graphs and compute their adjacency matrices. Then, we obtain the eigenvalues of $X^{\ast }(G;S_{\ell },S_r,S_m)$ in terms of the spectra of the associated Cayley graphs $X(G,S)$ with $S\in \{S_{\ell },S_r,S_m\}$ in two ways, using adjacency matrices and using irreducible characters of $G$.

35 pages, 6 tables, 6 figures