On the genus and crosscap of the total graph of commutative rings with respect to multiplication

Articles in Press

Document Type : Research Paper

Authors

¹ School of Computational Sciences, Faculty of Science and Technology, JSPM University, Pune-412207, India

² Department of Mathematics Faculty of Education Karamanoglu Mehmetbey University Karaman - Turkey

³ Department of Mathematics Aligarh Muslim University Aligarh

⁴ Department of Mathematics, Aligarh Muslim University, Aligarh

Abstract

Let

S

be a commutative ring and

Z (S)

be its zero-divisors set.
The total graph of

S

with respect to multiplication, denoted by

T_{Z (S)} (Γ (S))

, is an undirected graph with vertex set as the ring elements

S

and two distinct vertices

α

and

β

are adjacent if and only if

α β \in Z (S)

.
The graph

T_{Z (S)} (Γ (S^{*}))

is a subgraph of

T_{Z (S)} (Γ (S))

with vertex set

S^{*}

(set of nonzero elements of

S

).
In this paper, we characterize finite rings

S

for which

T_{Z (S)} (Γ (S^{*}))

belongs to some well-known families of graphs. Further, we classify the finite rings

S

for which

T_{Z (S)} (Γ (S^{*}))

is planar, toroidal or double toroidal. Finally, we analyze the finite rings

S

for which the graph

T_{Z (S)} (Γ (S^{*}))

has crosscap at most two.

Keywords