On the genus of generalized unit and unitary Cayley graphs of a commutative ring

Genus of total graphs from rings: A survey

“…More specifically, we present results on characterization of planar, toroidal and projective total graphs and these are found in [3][4][5]. In Section 3, we present results on the genus of the complement of the total graph of commutative rings and generalized total graphs [8,[10][11][12][13][14].…”

The classi cation of rings with its genus of class of graphs

“…The total graph of a commutative ring R , denoted by T (Γ(R)), is an undirected graph with vertex set as R and the distinct vertices x and y are adjacent if and only if x + y ∈ Z(R) where Z(R) is the set of all zero divisors of R . The total graph (as in [4]) has been investigated in [3,7,15], and several generalizations of the total graph have been studied in [1,5,[8][9][10]16].…”

Classification of rings with toroidal Jacobson graph

“…In [4], Anderson and Livingston modified and studied the zero-divisor graph Γ(R) as the graph with the nonzero zero-divisors Z(R) * of R as the vertex set. While they focus just on the zero-divisors of the rings (see [1], [2], [3], [4], [10]), there are many other kinds of graphs associated to rings, some of which have been extensively studied, see for example [5], [6], [13], [17], [18].…”

“…Genus two zero-divisor graphs of local rings are studied by Bloomfield and Wickham [12]. Also various research articles have been published on the genera of the graphs constructed out of the rings [6], [12], [19]. In [8], the authors classified the finite commutative rings R for which J R is planar (see Theorem 4.3).…”