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.

Integral Mixed Cayley Graphs over Abelian Groups

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The mixed Cayley graph $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\left\{ (a,b): b-a\in S \right\}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over an abelian group $\Gamma$ in terms of its connection set $S$.