2015/04/11 by Xiaogang Liu, Sanming Zhou
Mathematics · #Advanced Topics in Algebra #Cayley graph #Cayley transform #Combinatorics #Commutative property #Commutative ring #Discrete mathematics #Finite Group Theory Research #Generating set of a group #Geometry #Graph #Graph theory and applications #Mathematics #Quotient ring #Ring (chemistry) #Unitary state #Vertex (graph theory) #Vertex-transitive graph #Voltage graph #math.CO #msc:05C25 #msc:05C50
paper · pdf · doi:10.1016/j.laa.2015.03.037
published as Linear Algebra and its Applications 479 (2015) 73--90
openalex publication_date 2015/04/11 · arxiv created 2015/04/12 · arxiv updated 2015/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The purpose of this paper is to study spectral properties of a family of Cayley graphs on finite commutative rings. Let R be such a ring and R^× its set of units. Let QR=\u2: u∈ R^×\ and TR=QR∪(-QR). We define the quadratic unitary Cayley graph of R, denoted by GR, to be the Cayley graph on the additive group of R with respect to TR; that is, GR has vertex set R such that x, y ∈ R are adjacent if and only if x-y∈ TR. It is well known that any finite commutative ring R can be decomposed as R=R1× R2×⋯× Rs, where each Ri is a local ring with maximal ideal Mi. Let R0 be a local ring with maximal ideal M0 such that |R0|/|M0| ≡ 3 (\mod 4). We determine the spectra of GR and GR0× R under the condition that |Ri|/|Mi|≡ 1 (\mod 4) for 1 ≤ i ≤ s. We compute the energies and spectral moments of such quadratic unitary Cayley graphs, and determine when such a graph is hyperenergetic or Ramanujan.