On the Nonorientable Genus of the Generalized Unit and Unitary Cayley Graphs of a Commutative Ring

Let [Formula: see text] be a commutative ring and [Formula: see text] the multiplicative group of unit elements of [Formula: see text]. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, [Formula: see text], corresponding to a multiplicative subgroup [Formula: see text] of [Formula: see text] and a nonempty subset [Formula: see text] of [Formula: see text] with [Formula: see text], as the graph with vertex set [Formula: see text]and two distinct vertices [Formula: see text] and [Formula: see text] being adjacent if and only if there exists [Formula: see text] such that [Formula: see text]. In this paper, we characterize all Artinian rings [Formula: see text] for which [Formula: see text] is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring [Formula: see text] for which [Formula: see text] has finite nonorientable genus, [Formula: see text] must be a finite ring. Finally, it is proved that for a given positive integer [Formula: see text], the number of finite rings [Formula: see text] for which [Formula: see text] has nonorientable genus [Formula: see text] is finite.

UNIT AND UNITARY CAYLEY GRAPHS FOR THE RING OF EISENSTEIN INTEGERS MODULO \(n\)

Let \({E}_{n}\) be the ring of Eisenstein integers modulo \(n\). We denote by \(G({E}_{n})\) and \(G_{{E}_{n}}\), the unit graph and the unitary Cayley graph of \({E}_{n}\), respectively. In this paper, we obtain the value of the diameter, the girth, the clique number and the chromatic number of these graphs. We also prove that for each \(n>1\), the graphs \(G(E_{n})\) and \(G_{E_{n}}\) are Hamiltonian.

A characterization of rings whose unitary Cayley graphs are planar

Let [Formula: see text] be a ring with identity. The unitary Cayley graph of [Formula: see text] is the simple graph with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are linked by an edge if and only if [Formula: see text] is a unit of [Formula: see text]. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar.

Gutman Index and Harary Index of Unitary Cayley Graphs

In this paper, we determine the Gutman Index and Harary Index of Unitary Cayley Graphs. The Unitary Cayley Graph Xn is the graph with vertex set V(Xn) ={u|u∈ Zn} and edge set {uv|gcd(u−v, n) = 1 and u, v ∈ Zn }, where Zn ={0,1,...,n−1}.

Hamiltonian properties on a class of circulant interconnection networks

Classes of circulant graphs play an important role in modeling interconnection networks in parallel and distributed computing. They also find applications in modeling quantum spin networks supporting the perfect state transfer. It has been noticed that unitary Cayley graphs as a class of circulant graphs possess many good properties such as small diameter, mirror symmetry, recursive structure, regularity, etc. and therefore can serve as a model for efficient interconnection networks. In this paper we go a step further and analyze some other characteristics of unitary Cayley graphs important for the modeling of a good interconnection network. We show that all unitary Cayley graphs are hamiltonian. More precisely, every unitary Cayley graph is hamiltonian-laceable (up to one exception for X6) if it is bipartite, and hamiltonianconnected if it is not. We prove this by presenting an explicit construction of hamiltonian paths on Xnm using the hamiltonian paths on Xn and Xm for gcd(n,m) = 1. Moreover, we also prove that every unitary Cayley graph is bipancyclic and every nonbipartite unitary Cayley graph is pancyclic.

On the Diameter of Unitary Cayley Graphs of Rings

The unitary Cayley graph of a ringR, denoted Γ(R), is the simple graph defined on all elements ofR, and where two verticesxandyare adjacent if and only ifx−yis a unit inR. The largest distance between all pairs of vertices of a graphGis called the diameter ofGand is denoted by diam(G). It is proved that for each integern≥ 1, there exists a ringRsuch that diam(Γ(R)) =n. We also show that diam(Γ(R)) ∊ {1, 2, 3,∞} for a ringRwithR/J(R) self-injective and classify all those rings with diam(Γ(R)) = 1, 2, 3, and ∞, respectively.

Polynomials of unitary Cayley graphs

The unitary Cayley graph Xn has the vertex set Zn = {0,1,2,..., n-1} and vertices a and b are adjacent, if and only if gcd(a-b,n) = 1. In this paper, we present some properties of the clique, independence and distance polynomials of the unitary Cayley graphs and generalize some of the results from [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. In addition, using some properties of Laplacian polynomial we determine the number of minimal spanning trees of any unitary Cayley graph.

BALANCED UNITARY CAYLEY SIGRAPHS OVER FINITE COMMUTATIVE RINGS

Let R be a finite commutative ring with identity 1. The unitary Cayley graph of R, denoted by GR, is the graph whose vertex set is R and the edge set {{a, b} : a, b ∈ R and a - b ∈ R×}, where R× is the group of units of R. We define the unitary Cayley signed graph (or unitary Cayley sigraph in short) to be an ordered pair 𝒮R = (GR, σ), where GR is the unitary Cayley graph over R with signature σ : E(GR) → {1, -1} given by [Formula: see text] In this paper, we give a criterion on R for SR to be balanced (every cycle in 𝒮R is positive) and a criterion for its line graph L(𝒮R) to be balanced. We characterize all finite commutative rings with the property that the marked sigraph 𝒮R,μ is canonically consistent. Moreover, we give a characterization of all finite commutative rings where 𝒮R, η(𝒮R) and L(𝒮R) are hyperenergetic balanced.