On the Genus of the Total Graph of a Commutative Ring

DOMINATION IN THE TOTAL GRAPH ON ℤn

“…Suppose there exist i = j such that x j = x i + lp 1 where l ∈ Z + . From this there exists some k (0 ≤ k ≤ p 1 − 1) such that x m ≡ k (mod p 1 ) for all x m ∈ S. Then as in the proof of Theorem 2.5 [13], S is not a dominating set, which is a contradiction.…”

“…Tamizh Chelvam and Asir [13] obtained the value of the domination number of the total graph of a commutative ring R in terms of the cardinality of the maximum annihilator ideal in R. Let n = p α1 1 p α2 2 . .…”

Finite commutative rings with higher genus unit graphs

“…Genus two zero divisor graphs of local rings were investigated by Bloomfield and Wickham in [5]. Recently, Maimani et al [14] determined all isomorphism classes of finite rings whose total graphs have genus at most one, and Tamizh Chelvam and Asir [17] characterized all isomorphism classes of finite rings whose total graphs have genus two. For a finite ring R, the unit graph G(R) is the complement of the total graph of the ring R. In [3, Theorem 5.14], all finite rings having planar unit graphs are completely classified, and in [7] toroidal ones are completely determined.…”

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

“…In the recent past, considerable work was carried out on characterizing rings regarding the genus of the constructed graph (see [6,7,9,13,15]). It can be recalled here that the genus of a graph G, denoted by γ(G), is the smallest nonnegative integer g such that the graph G can be embedded on the surface obtained by attaching g handles to a sphere.…”

“…The graphs of genus 0 and 1 are called planar and toroidal graphs, respectively. Maimani et al [13] characterized all commutative Artinian rings whose total graphs have genus at most one and Tamizh Chelvam et al [15] classified all commutative Artinian rings whose total graphs have genus two.…”