Quadratic unitary Cayley graphs of finite commutative rings

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Abstract

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× be its set of units. Let QR={u2:uR×} 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,yR are adjacent if and only if xyTR. 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(mod4). We determine the spectra of GR and GR0×R under the condition that |Ri|/|Mi|1(mod4) for 1is. We compute the energies and spectral moments of such quadratic unitary Cayley graphs, and determine when such a graph is hyperenergetic or Ramanujan.

MSC

05C50
05C25

Keywords

Spectrum
Quadratic unitary Cayley graph
Ramanujan graph
Energy of a graph
Spectral moment

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