Vertex Domination of the Zero-Divisor Cayley Graph of the Residue Class Ring (

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ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Vertex Domination of the Zero-Divisor Cayley Graph of the Residue Class Ring (๐’ ๐’ , โจ,โŠ™)

Levaku Madhavi

1*

, Jangiti Devendra

2

and Tippaluri Nagalakshumma

3

1

*Assistant Professor,

Department of Applied Mathematics,

Yogi Vemana University, Kadapa-516005, A.P., INDIA.

2

Research Scholar,

Department of Applied Mathematics,

Yogi Vemana University, Kadapa-516005, A.P., INDIA.

3

Research Scholar,

Department of Applied Mathematics,

Yogi Vemana University, Kadapa-516005, A.P., INDIA.

email: lmadhaviyvu@gmail.com, jdevendrayvu@gmail.com, tlakshmiyvu@gmail.com Corresponding Author: Dr. Levaku Madhavi

(Received on: August 3, 2019)

ABSTRACT

The authors have introduced the notion of zero-divisor Cayley graph associated with the set of zero-divisors in the ring (๐‘๐‘›, โจ,โŠ™) of residue classes modulo ๐‘›, ๐‘› โ‰ฅ 1, an integer and studied its properties. In this paper the vertex cover, the vertex dominating set and related properties are studied for this graph.

AMS Subject Classification (2010): 05C25, 05C69.

Keywords: Zero-divisor, Symmetric set, Cayley graph, Zero-divisor Cayley graph.

1. INTRODUCTION

Graph theorists are interested in finding a subset ๐‘ˆ of the vertex set of a graph ๐บ,

which covers every edge of ๐บ, in the sense that, every edge in ๐บ is incident with some vertex

in ๐‘ˆ. Similarly, the notion of edge cover can be defined in a graph. The theory which deals

with vertex cover, edge cover and their related parameters constitute the domination theory of

graphs. The domination theory has lot of applications in science and technology and

engineering, particularly in communication networks.

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Berge

4

and Ore

13

introduced the precise notion of a dominating set and later many graph theorists, to mention some of them, Allan and Laskar

1,2

, Cockayne and Hedetniemi

6

, Haynes et al.,

8

, Kulli and Sigarkant

10

and others have studied various types of domination parameters of graphs. In recent times Madhavi

12

, Maheswari and Madhavi

11

, Sujatha and Madhavi

14

, Swetha et al.,

15

and others have studied certain domination parameters of arithmetic graphs associated with the arithmetic functions namely, the Euler totient function, the quadratic residue modulo a prime and the divisor function.

This paper is devoted for the study of vertex domination in a zero-divisor Cayley graph of the ring (๐‘

๐‘›

, โจ,โŠ™). The notions relating to algebra, graph theory and number theory can be found in

3,5

and

9

.

2. THE ZERO-DIVISOR CAYLEY GRAPH AND ITS PROPERTIES

Consider the ring(๐‘

๐‘›

, โจ,โŠ™)of integers modulo ๐‘›, ๐‘› โ‰ฅ 1, an integer, which is a commutative ring with unity. In

7

, it is established that the set ๐ท

0

of nonzero zero-divisors in the ring (๐‘

๐‘›

, โจ,โŠ™) is a symmetric subset of the group (๐‘

๐‘›

, โจ) and the zero-divisor Cayley graph ๐บ(๐‘

๐‘›

, ๐ท

0

) is the graph whose vertex set is ๐‘

๐‘›

and the edge set is the set of ordered pairs (๐‘ข, ๐‘ฃ)such that ๐‘ข, ๐‘ฃ โˆˆ ๐‘

๐‘›

and either ๐‘ข โˆ’ ๐‘ฃ โˆˆ ๐ท

0

or ๐‘ฃ โˆ’ ๐‘ข โˆˆ ๐ท

0

. This graph (๐‘› โˆ’ ๐œ‘(๐‘›) โˆ’ 1)- regular and its size is

๐‘›2

(๐‘› โˆ’ ๐œ‘(๐‘›) โˆ’ 1).

The graphs ๐บ(๐‘

7

, ๐ท

0

), ๐บ(๐‘

8

, ๐ท

0

) and ๐บ(๐‘

10

, ๐ท

0

) are given below:

๐‘ฎ(๐’๐Ÿ•, ๐‘ซ๐ŸŽ) ๐‘ฎ(๐’๐Ÿ–, ๐‘ซ๐ŸŽ)๐‘ฎ (๐’๐Ÿ๐ŸŽ, ๐‘ซ๐ŸŽ)

We state below the main results that are established in

7

for the zero-divisor Cayley graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Lemma 2.1: (Lemma 2.10, [7]) For a prime ๐‘, the graph ๐บ(๐‘

๐‘

, ๐ท

0

) contains only isolated vertices.

Lemma 2.2: (Theorem 3.7, [7]) For a prime ๐‘ and an integer ๐‘Ÿ > 1, the graph ๐บ(Z

๐‘๐‘Ÿ

, ๐ท

0

)

contains ๐‘ disjoint components, each of which is a complete subgraph of the graph

๐บ(Z

๐‘๐‘Ÿ

, ๐ท

0

).

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Lemma 2.3: (Theorem 4.4, [7]) Let ๐‘› > 1 be an integer, which is not a power of a single prime. Then the graph ๐บ(๐‘

๐‘›

, ๐ท

0

) is a connected graph.

3. VERTEX COVER OF THE ZERO-DIVISOR CAYLEY GRAPH ๐‘ฎ(๐’

๐’

, ๐‘ซ

๐ŸŽ

)

This section is devoted for determining the vertex covering sets, the minimum vertex covering sets and the vertex covering number of the zero-divisor Cayley graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Definition 3.1: A subset ๐‘† of vertices of a graph ๐บ is called a vertex cover of ๐บ, if every edge of ๐บ is incident with some vertex in ๐‘†. A minimum vertex cover is the one with minimum cardinality.

Definition 3.2: The cardinality of a minimum vertex cover of a graph ๐บ is called the vertex covering number and it is denoted by ๐›ฝ(๐บ(๐‘

๐‘›

, ๐ท

0

)).

In determining the vertex cover and the vertex dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

), the smallest prime divisor of ๐‘› plays a key role. In the following Theorems and Lemmas, we

consider the positive integer ๐‘› = โˆ

๐‘Ÿ๐‘–=1

๐‘

๐‘–๐›ผ๐‘–

, where๐‘

1

< ๐‘

2

< โ‹ฏ < ๐‘

๐‘Ÿ

, ๐›ผ

๐‘–

โ‰ฅ 1 and 1 โ‰ค ๐‘– โ‰ค ๐‘Ÿ are integers.

Lemma 3.3: For 1 โ‰ค ๐‘– โ‰ค ๐‘Ÿ, the subset ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is a vertex cover of

1

) โˆ’ 1 the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Proof: The vertex set ๐‘‰ can be written as

๐‘‰ = {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

1

) โˆ’ 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

1

) ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ 1)

1

) + ๐‘™ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…},

since (๐‘› โˆ’ ๐‘

1

) + (๐‘

1

โˆ’ 1) = ๐‘› โˆ’ 1. We shall show that ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is a

1

) โˆ’ 1 vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). To see this let (๐‘ข, ๐‘ฃ) be an edge of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Then, ๐‘ข, ๐‘ฃ โˆˆ ๐‘‰. Our aim is to show that at least one of ๐‘ข and ๐‘ฃ is in ๐‘‰

1

. If possible, assume that ๐‘ข โˆ‰ ๐‘‰

1

and ๐‘ฃ โˆ‰ ๐‘‰

1

. Then, ๐‘ข, ๐‘ฃ โˆˆ ๐‘‰ โˆ’ ๐‘‰

1

, where

๐‘‰ โˆ’ ๐‘‰

1

= {๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’) + 1

1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

1

) + ๐‘™ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}.

1

) + (๐‘

1

โˆ’ 1)

So, ๐‘ข = (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… and ๐‘ฃ = (๐‘› โˆ’ ๐‘

1

) + ๐‘˜ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… for some integers ๐‘˜, ๐‘™, 0 โ‰ค ๐‘˜ < ๐‘™ โ‰ค ๐‘

1

) + ๐‘™

1

โˆ’ 1, and ๐‘ฃ โˆ’ ๐‘ข = (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ’ (๐‘› โˆ’ ๐‘

1

) + ๐‘™ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… = ๐‘™ฬ… โˆ’ ๐‘˜๐‘

1

) + ๐‘˜ ฬ…ฬ…ฬ…ฬ…ฬ… = ๐‘™ โˆ’ ๐‘˜

1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ….

But 0 โ‰ค ๐‘˜ < ๐‘™ โ‰ค ๐‘

1

โˆ’ 1 implies that 0 โ‰ค ๐‘™ โˆ’ ๐‘˜ < ๐‘

1

. Since ๐‘

1

is the least positive integer such that ๐‘ ฬ…ฬ…ฬ… is a zero-divisor in the ring (๐‘

1 ๐‘›

, โจ, โจ€), it follows that ๐‘™ โˆ’ ๐‘˜ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… is not a zero- divisor in the ring (๐‘

๐‘›

, โจ, โจ€) and this shows that (๐‘ข, ๐‘ฃ) is not an edge of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

), which is a contradiction to the fact that (๐‘ข, ๐‘ฃ) is an edge of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). Hence, our assumption that ๐‘ข โˆ‰ ๐‘‰

1

and ๐‘ฃ โˆ‰ ๐‘‰

1

is wrong and at least one of ๐‘ข, or, ๐‘ฃ is in ๐‘‰

1

. This shows that ๐‘‰

1

is a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). โˆŽ

Lemma 3.4: For 2 โ‰ค ๐‘– โ‰ค ๐‘Ÿ, the subset ๐‘‰

๐‘–

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is not a vertex cover

๐‘–

) โˆ’ 1 of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Proof: The vertex set ๐‘‰ is the disjoint union of ๐‘‰

๐‘–

and ๐‘‰ โˆ’ ๐‘‰

๐‘–

, where

๐‘‰ โˆ’ ๐‘‰

๐‘–

= {(๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

๐‘–

) ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

๐‘–

) + 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

๐‘–

) + ๐‘˜ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…,

๐‘–

) + (๐‘

1

โˆ’ 1)

โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

๐‘–

) + ๐‘

1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

๐‘–

) + (๐‘

1

+ 1) ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}.

๐‘–

) + (๐‘

๐‘–

โˆ’ 1)

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Let ๐‘ข = (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… and ๐‘ฃ = (๐‘› โˆ’ ๐‘

๐‘–

) ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…. Then ๐‘ข โˆ‰ ๐‘‰

๐‘–

) + ๐‘

1 ๐‘–

and ๐‘ฃ โˆ‰ ๐‘‰

๐‘–

.

But ๐‘ฃ โˆ’ ๐‘ข = (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ’ (๐‘› โˆ’ ๐‘

๐‘–

) + ๐‘

1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… = ๐‘

๐‘–

) ฬ…ฬ…ฬ… and ๐‘

1

ฬ…ฬ…ฬ… is a zero-divisor in the ring (๐‘

1 ๐‘›

, โจ,โŠ™), so that(๐‘ข, ๐‘ฃ) is an edge of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). That is, the edge (๐‘ข, ๐‘ฃ) is such that ๐‘ข โˆ‰ ๐‘‰

๐‘–

and ๐‘ฃ โˆ‰ ๐‘‰

๐‘–

, so that ๐‘‰

๐‘–

is not a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). โˆŽ Lemma 3.5: If ๐‘† = ๐‘‰

1

โˆ’ {๐‘˜ฬ…}, where ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ ๐‘˜ฬ… โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} and

1

) โˆ’ 1 0 โ‰ค ๐‘˜ โ‰ค (๐‘› โˆ’ ๐‘

1

) โˆ’ 1, then ๐‘† is not a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Proof: By the Lemma 3.3, the subset ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is a vertex cover of the

1

) โˆ’ 1 graph ๐บ(๐‘

๐‘›

, ๐ท

0

). The vertex set ๐‘‰ of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

) can be viewed as the disjoint union of ๐‘‰

1

and ๐‘‰ โˆ’ ๐‘‰

1

, where

๐‘‰ โˆ’ ๐‘‰

1

= {๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

1

) + 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

1

) + ๐‘™ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}.

1

) + (๐‘

1

โˆ’ 1) For 0 โ‰ค ๐‘˜ โ‰ค (๐‘› โˆ’ ๐‘

1

) โˆ’ 1, let ๐‘† = ๐‘‰

1

โˆ’ {๐‘˜ฬ…}. Then ๐‘˜ฬ… โˆ‰ ๐‘†. Further,

๐‘‰ โˆ’ ๐‘† = {๐‘˜ฬ…, ๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

1

) + 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

1

) + ๐‘™ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}.

1

) + (๐‘

1

โˆ’ 1) By the division algorithm applied to ๐‘˜ and ๐‘

1

, there exist integers ๐‘ž and ๐‘ก such that ๐‘˜ = ๐‘

1

๐‘ž + ๐‘ก, where 0 โ‰ค ๐‘ก < ๐‘

1

. Since0 โ‰ค ๐‘ก < ๐‘

1

, (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆˆ ๐‘‰ โˆ’ ๐‘†, so that

1

) + ๐‘ก

(๐‘› โˆ’ ๐‘

1

) + ๐‘ก

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ‰ ๐‘†. That is, ๐‘˜ฬ… โˆ‰ ๐‘† and (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ‰ ๐‘†. Further, since ๐‘›ฬ… = 0,

1

) + ๐‘ก ๐‘˜ฬ… โˆ’ (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… = ๐‘

1

) + ๐‘ก ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ’ (๐‘› โˆ’ ๐‘

1

๐‘ž + ๐‘ก ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… = ๐‘

1

) + ๐‘ก ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ….

1

(1 + ๐‘ž)

Now ๐‘ ฬ…ฬ…ฬ… โˆˆ ๐ท

1 0

implies that ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆˆ ๐ท

1

(1 + ๐‘ž)

0

. That is, there is an edge between, (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…

1

) + ๐‘ก

and ๐‘˜ฬ… and none of these vertices lie in ๐‘†, so that ๐‘† is not a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

โˆŽ

Lemma 3.6: If ๐‘† is any subset of the vertex set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

), such that |๐‘†| < |๐‘‰

1

|, then ๐‘† is not a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Proof: Let ๐‘† be the subset of the vertex set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

) such that |๐‘†| < |๐‘‰

1

|.

Case (i): Let ๐‘† โŠ‚ ๐‘‰

1

. Since |๐‘†| < |๐‘‰

1

|, by the Lemma 3.5, the subset ๐‘† is not a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Case (ii): Let๐‘† โˆฉ ๐‘‰

1

= ๐œ™.

Subcase (i): Let ๐‘† = ๐‘‰ โˆ’ ๐‘‰

1

. The vertex set ๐‘‰ is the disjoint union of ๐‘‰

1

and ๐‘† = ๐‘‰ โˆ’ ๐‘‰

1

, where ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , ๐‘˜ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}, 0 โ‰ค ๐‘˜ โ‰ค (๐‘› โˆ’ ๐‘

1

) โˆ’ 1

1

) โˆ’ 1, and

๐‘† = ๐‘‰ โˆ’ ๐‘‰

1

= {(๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

1

) ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘

1

) + 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, โ€ฆ , ๐‘› โˆ’ 1

1

) + ๐‘™ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}.

Let ๐‘ข = ๐‘› โˆ’ ๐‘˜๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆˆ ๐‘‰

1 1

and ๐‘ฃ = ๐‘› โˆ’ (๐‘˜ + 1)๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆˆ ๐‘‰

1 1

for some ๐‘˜, 0 โ‰ค ๐‘˜ โ‰ค ๐‘› โˆ’ ๐‘

1

.

Then, ๐‘ข โˆ‰ ๐‘† and ๐‘ฃ โˆ‰ ๐‘† and ๐‘ข โˆ’ ๐‘ฃ = [๐‘› โˆ’ ๐‘˜๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…] โˆ’ [๐‘› โˆ’ (๐‘˜ + 1)๐‘

1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…] = ๐‘

1

ฬ…ฬ…ฬ…. Since ๐‘

1

ฬ…ฬ…ฬ… is a zero-

1

divisor in the ring (๐‘

๐‘›

, โจ, โจ€), there is an edge between the vertices ๐‘ข and ๐‘ฃ and none of these vertices lie in ๐‘†, so that ๐‘† is not a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Subcase (ii): Let ๐‘† โŠ‚ ๐‘‰ โˆ’ ๐‘‰

1

. Since ๐‘† โŠ‚ ๐‘‰ โˆ’ ๐‘‰

1

, there exists (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆˆ ๐‘‰ โˆ’ ๐‘‰

1

) + ๐‘™

1

but (๐‘› โˆ’ ๐‘

1

) + ๐‘™

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ‰ ๐‘†. Also, 0 โ‰ค ๐‘™ โ‰ค ๐‘

1

โˆ’ 1 implies that ๐‘™ฬ… โˆˆ ๐‘‰

1

, so that ๐‘™ฬ… โˆ‰ ๐‘‰ โˆ’ ๐‘‰

1

, or, ๐‘™ฬ… โˆ‰ ๐‘†. But (๐‘› โˆ’ ๐‘

1

) + ๐‘™

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ’ ๐‘™ฬ… = (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… and this is a zero-divisor in the ring (๐‘

1

)

๐‘›

, โจ, โจ€), since ๐‘ ฬ…ฬ…ฬ… is so.

1

That is, there is an edge between the vertices ๐‘ข and ๐‘ฃ and none of these vertices lie in ๐‘†, so

that ๐‘† is not a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

(5)

Case(iii): Let |๐‘†| < |๐‘‰

1

| and ๐‘† โˆฉ ๐‘‰

1

โ‰  ๐œ™.

Let ๐‘™ฬ… be the largest integer such that 0 โ‰ค ๐‘™ โ‰ค ๐‘› โˆ’ 1 and such that ๐‘™ฬ… โˆˆ ๐‘†. Then ๐‘™ + 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ‰ ๐‘†.

Consider (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆˆ ๐‘‰. Since(๐‘› โˆ’ ๐‘

๐‘–

) + ๐‘™ + 1

๐‘–

) + ๐‘™ + 1 > ๐‘™, (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ‰ ๐‘†. But

๐‘–

) + ๐‘™ + 1 [(๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…] โˆ’ [๐‘™ + 1

๐‘–

) + ๐‘™ + 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…] = (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… which is a zero-divisor in the ring (๐‘

๐‘–

),

๐‘›

, โจ, โจ€). So

there is an edge between (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… and ๐‘™ + 1

๐‘–

) + ๐‘™ + 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… but (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ‰ ๐‘† and

๐‘–

) + ๐‘™ + 1 ๐‘™ + 1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ‰ ๐‘†, so that ๐‘† is not a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). โˆŽ Theorem 3.7: For ๐‘› > 1, an integer, the vertex cover ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} of the

1

) โˆ’ 1 graph ๐บ(๐‘

๐‘›

, ๐ท

0

) is a minimum vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Proof: By the Lemma 3.3, the subset ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} of vertices is a vertex

1

) โˆ’ 1 cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). Also by the Lemma 3.4, for 2 โ‰ค ๐‘– โ‰ค ๐‘Ÿ, each of the subsets ๐‘‰

๐‘–

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is not a vertex cover of the graph ๐บ(๐‘

๐‘–

) โˆ’ 1

๐‘›

, ๐ท

0

). By the Lemma 3.6, any subset ๐‘† of vertices for which |๐‘†| < |๐‘‰

1

|, is not a vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

So ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is a minimum vertex cover of the graph ๐บ(๐‘

1

) โˆ’ 1

๐‘›

, ๐ท

0

). โˆŽ Example 3.8: Minimum vertex covers of the graphs ๐บ(๐‘

9

, ๐ท

0

), ๐บ(๐‘

10

, ๐ท

0

) and ๐บ(๐‘

15

, ๐ท

0

) are respectively given by {0ฬ…, 1ฬ…, 2ฬ…, 3ฬ…, 4ฬ…, 5ฬ…}, {0ฬ…, 1ฬ…, 2ฬ…, 3ฬ…, 4ฬ…, 5ฬ…, 6ฬ…, 7ฬ…} and {0ฬ…, 1ฬ…, 2ฬ…, 3ฬ…, 4ฬ…, 5ฬ…, 6ฬ…, 7ฬ…, 8ฬ…, 9ฬ…, 10 ฬ…ฬ…ฬ…ฬ…, 11 ฬ…ฬ…ฬ…ฬ…}

and the vertices in the minimum vertex covers are exhibited in the respective graphs by the boldface dots.

๐‘ฎ(๐’๐Ÿ—, ๐‘ซ๐ŸŽ) ๐‘ฎ(๐’๐Ÿ๐ŸŽ, ๐‘ซ๐ŸŽ) ๐‘ฎ(๐’๐Ÿ๐Ÿ“, ๐‘ซ๐ŸŽ)

The following corollary is immediate from the Theorem 3.7.

Corollary 3.9: For ๐‘› > 1, an integer, the vertex covering number ๐›ฝ(๐บ(๐‘

๐‘›

, ๐ท

0

)) of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

) is ๐‘› โˆ’ ๐‘

1

.

Proof: By the Theorem 3.7, the subset ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is a minimum vertex

1

) โˆ’ 1 cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

) and ๐‘‰

1

contains ๐‘› โˆ’ ๐‘

1

vertices. So,

๐›ฝ(๐บ(๐‘

๐‘›

, ๐ท

0

)) = ๐‘› โˆ’ ๐‘

1

. โˆŽ

Remark 3.10: Let ๐‘ be a prime. The edge set of the graph ๐บ(๐‘

๐‘

, ๐ท

0

) is empty, so that the

vertex cover of the graph ๐บ(๐‘

๐‘

, ๐ท

0

) is empty and hence the vertex covering number is zero.

(6)

Remark 3.11: One can easily observe that each of the subsets of the vertex set ๐‘‰ given below is also a minimum vertex cover of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

๐‘‰

0

= {0, ฬ… 1ฬ…, 2ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…},

1

) โˆ’ 1

๐‘‰

1

= {1ฬ…, 2ฬ…, 3ฬ…, โ€ฆ , (๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

1

) โˆ’ 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…},

1

)

โ‹ฎ

๐‘‰

๐‘›โˆ’(๐‘1โˆ’1)

= {(๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

1

) ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

1

) + 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โ€ฆ , (๐‘› โˆ’ ๐‘

1

) + 2 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}.

1

) โˆ’ 1 + (๐‘› โˆ’ ๐‘

1

) Example 3.12: Sets of all possible minimum vertex covers of the graph ๐บ(๐‘

6

, ๐ท

0

), namely, {0ฬ…, 1ฬ…, 2ฬ…, 3ฬ…}, {1ฬ…, 2ฬ…, 3ฬ…, 4ฬ…}, {2ฬ…, 3ฬ…, 4ฬ…, 5ฬ…}, {3ฬ…, 4ฬ…, 5ฬ…, 0ฬ…}, {4ฬ…, 5ฬ…, 0ฬ…, 1ฬ…} and {5ฬ…, 0ฬ…, 1ฬ…, 2ฬ…} are shown in the following figures by the boldface dots.

{0ฬ…, 1ฬ…, 2ฬ…, 3ฬ…} {1ฬ…, 2ฬ…, 3ฬ…, 4ฬ…} {2ฬ…, 3ฬ…, 4ฬ…, 5ฬ…}

{3ฬ…, 4ฬ…, 5ฬ…, 0ฬ…} {4ฬ…, 5ฬ…, 0ฬ…, 1ฬ…} {5ฬ…, 0ฬ…, 1ฬ…, 2ฬ…}

4. VERTEX DOMINATION OF THE ZERO-DIVISOR CAYLEY GRAPH ๐‘ฎ(๐’

๐’

, ๐‘ซ

๐ŸŽ

) In this section the vertex dominating sets, the minimum vertex dominating sets and the vertex domination number of the zero-divisor Cayley graph ๐บ(๐‘

๐‘›

, ๐ท

0

) are determined.

Definition 4.1: A dominating set of a graph ๐บ is a subset ๐ท of the vertex set ๐‘‰ such that each vertex of ๐‘‰ โˆ’ ๐ท is adjacent to at least one vertex of ๐ท. A minimum vertex dominating set is the one with minimum cardinality. The cardinality of a minimum vertex dominating set is called the vertex domination number of the graph and it is denoted by ๐›พ(๐บ).

Lemma 4.2: For ๐‘› > 1, an integer,1 โ‰ค ๐‘– โ‰ค ๐‘Ÿ, the set ๐‘‰

๐‘–

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is a vertex

๐‘–

โˆ’ 1)

dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

(7)

Proof: For 1 โ‰ค ๐‘– โ‰ค ๐‘Ÿ, let ๐‘‰

๐‘–

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , (๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}. Then the vertex set ๐‘‰ of the graph

๐‘–

โˆ’ 1) ๐บ(๐‘

๐‘›

, ๐ท

0

) can be written as the disjoint union of ๐‘‰

๐‘–

and ๐‘‰ โˆ’ ๐‘‰

๐‘–

, where

๐‘‰ โˆ’ ๐‘‰

๐‘–

= {๐‘ ฬ…ฬ…ฬ…, ๐‘

1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โ€ฆ , ๐‘

1

+ 1, ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โ€ฆ ๐‘› โˆ’ 1

1

+ ๐‘˜, ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}.

Let ๐‘ข โˆˆ ๐‘‰ โˆ’ ๐‘‰

๐‘–

. Then ๐‘ข = ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… for some๐‘˜, 0 โ‰ค ๐‘˜ โ‰ค (๐‘› โˆ’ ๐‘

๐‘–

+ ๐‘˜

๐‘–

) โˆ’ 1. By the division algorithm applied to ๐‘

1

and ๐‘˜, there exist integers ๐‘ก and ๐‘  such that ๐‘˜ = ๐‘

1

๐‘ก + ๐‘  and 0 โ‰ค ๐‘  < ๐‘

๐‘–

. From 0 โ‰ค ๐‘  < ๐‘

๐‘–

, it follows that ๐‘  โˆˆ ๐‘‰

๐‘–

. Now

๐‘ข โˆ’ ๐‘  = (๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โˆ’ (๐‘˜ โˆ’ ๐‘

๐‘–

+ ๐‘˜) ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… = ๐‘

๐‘–

๐‘ก) ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…

๐‘–

(๐‘ก + 1)

and this is a zero-divisor in the ring (๐‘

๐‘›

, โจ, โจ€). This shows that every vertex in ๐‘‰ โˆ’ ๐‘‰

๐‘–

is adjacent to atleast one vertex in ๐‘‰

๐‘–

, so that ๐‘‰

๐‘–

is a vertex dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

โˆŽ Lemma 4.3: For ๐‘› > 1, an integer, the subset ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is a vertex dominating

1

โˆ’ 1 set with the minimum cardinality among the vertex dominating sets ๐‘‰

1

, ๐‘‰

2

, โ€ฆ , ๐‘‰

๐‘Ÿ

of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Proof: By the Lemma 4.2, for 1 โ‰ค ๐‘– โ‰ค ๐‘Ÿ, each of the subsets ๐‘‰

๐‘–

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is a

๐‘–

โˆ’ 1 vertex dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). Since ๐‘

1

< ๐‘

2

< โ‹ฏ < ๐‘

๐‘Ÿ

, the cardinality of the subset ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is minimum among the cardinalities of the vertex dominating

1

โˆ’ 1 sets ๐‘‰

1

, ๐‘‰

2

, โ€ฆ , ๐‘‰

๐‘Ÿ

of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). โˆŽ

Lemma 4.4: For ๐‘› > 1, an integer, the subset ๐ท

1

= ๐‘‰

1

โˆ’ {๐‘˜ฬ…}, where 0 โ‰ค ๐‘˜ โ‰ค ๐‘

1

โˆ’ 1, is not a vertex dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Proof: Let ๐‘˜ฬ… โˆˆ ๐‘‰

1

and ๐ท

1

= ๐‘‰

1

โˆ’ {๐‘˜ฬ…}. Then 0 โ‰ค ๐‘˜ โ‰ค ๐‘

1

โˆ’ 1, and ๐‘‰ โˆ’ ๐ท

1

= {๐‘˜ฬ…, ๐‘ ฬ…ฬ…ฬ…, ๐‘

1

ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โ€ฆ , ๐‘

1

+ 1, ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โ€ฆ ๐‘› โˆ’ 1

1

+ ๐‘˜, ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}.

Consider ๐‘˜ฬ… โˆˆ ๐‘‰ โˆ’ ๐ท

1

. For any ๐‘—ฬ… โˆˆ ๐ท

1

, we have ๐‘— โ‰  ๐‘˜ and 0 โ‰ค ๐‘— โ‰ค ๐‘

1

โˆ’ 1. We may assume that 0 โ‰ค ๐‘— < ๐‘˜ โ‰ค ๐‘

1

โˆ’ 1. We have ๐‘˜ โˆ’ ๐‘— < ๐‘

1

, so that ๐‘˜ โˆ’ ๐‘— ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… is not a zero-divisor in the ring (๐‘

๐‘›

, โจ, โจ€). This shows that the vertex ๐‘˜ฬ… โˆˆ ๐‘‰ โˆ’ ๐ท

1

is not adjacent to any vertex ๐‘—ฬ… โˆˆ ๐ท

1

and thus ๐ท

1

is not a vertex dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

). โˆŽ Remark 4.5: As in Lemma 3.6, one can show that, if ๐‘† is any subset of vertices, such that

|๐‘†| < |๐‘‰

1

|, then ๐‘† is not a vertex dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

The following corollary is immediate from the Lemma 4.4.

Theorem 4.6: For ๐‘› > 1, an integer, the subset ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…} is a minimum vertex

1

โˆ’ 1 dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

Proof: The proof follows from Lemma 4.2, Lemma 4.3, Lemma 4.4 and Remark 4.5. โˆŽ Corollary 4.7: For ๐‘› > 1, an integer, the vertex domination number ๐›พ(๐บ(๐‘

๐‘›

, ๐ท

0

)) of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

) is ๐‘

1

.

Proof: By the Theorem 4.6, the subset ๐‘‰

1

= {0ฬ…, 1ฬ…, 2ฬ…, โ€ฆ , ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}of the vertex set ๐‘‰is a

1

โˆ’ 1 minimum vertex dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

) and ๐‘‰

1

contains ๐‘

1

vertices. So

๐›พ(๐บ(๐‘

๐‘›

, ๐ท

0

) = ๐‘

1

. โˆŽ

Corollary 4.8: For ๐‘› > 1, an integer, then ๐›ฝ(๐บ(๐‘

๐‘›

, ๐ท

0

)) + ๐›พ(๐บ(๐‘

๐‘›

, ๐ท

0

) = ๐‘›.

Proof: The proof follows form the Corollaries 3.9, and 4.7. โˆŽ

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Example 4.9: The vertex domination sets of the graphs ๐บ(๐‘

9

, ๐ท

0

), ๐บ(๐‘

10

, ๐ท

0

) and ๐บ(๐‘

15

, ๐ท

0

) are respectively given by {0ฬ…, 1ฬ…, 2ฬ…}, {0ฬ…, 1ฬ…} and {0ฬ…, 1ฬ…, 2ฬ…}. These are exhibited in the respective graphs given below by the boldface dots. The edges that join the vertices in the respective minimum dominating vertex sets with the remaining vertices of the graphs are indicated by dotted lines.

๐‘ฎ(๐’๐Ÿ—, ๐‘ซ๐ŸŽ) ๐‘ฎ(๐’๐Ÿ๐ŸŽ, ๐‘ซ๐ŸŽ) ๐‘ฎ(๐’๐Ÿ๐Ÿ“, ๐‘ซ๐ŸŽ)

Remark 4.10: One can easily observe that each of the subsets of the vertex set ๐‘‰ given below is also a minimum vertex dominating set of the graph ๐บ(๐‘

๐‘›

, ๐ท

0

).

๐‘‰

0

= {0ฬ…, 1 ฬ… , 2ฬ…, โ€ฆ , ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, ๐‘

1

โˆ’ 2 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…},

1

โˆ’ 1 ๐‘‰

1

= {1ฬ…, 2ฬ…, 3ฬ…, โ€ฆ , ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, ๐‘

1

โˆ’ 1 ฬ…ฬ…ฬ…},

1

โ‹ฎ ๐‘‰

๐‘›โˆ’๐‘1

=

{(๐‘› โˆ’ ๐‘ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

1

) ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…, (๐‘› โˆ’ ๐‘

1

) + 1 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… โ€ฆ , (๐‘› โˆ’ ๐‘

1

) + 2 ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…}.

๐‘–

) + (๐‘

1

โˆ’ 1)

Examples 4.11: For the graph ๐บ(๐‘

6

, ๐ท

0

), the minimum vertex dominating sets are given by {0ฬ…, 1ฬ…}, {1ฬ…, 2ฬ…}, {2ฬ…, 3ฬ…}, {3ฬ…, 4ฬ…}, {4ฬ…, 5ฬ…} and {5ฬ…, 0ฬ…} and these exhibited in the figures given below by dotted lines.

{0ฬ…, 1ฬ…} {1ฬ…, 2ฬ…} {2ฬ…, 3ฬ…}

{3ฬ…, 4ฬ…} {4ฬ…, 5ฬ…} {5ฬ…, 0ฬ…}

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5. ACKNOWLEDGMENT

The authors express their thanks to Prof. L. Nagamuni Reddy for his suggestions during the preparation of this paper.

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