Distance labellings of Cayley graphs of semigroups

Kelarev, AV; Ras, Charl J.; Zhou, Sanming

doi:10.1007/s00233-015-9748-7

Cited by 16 publications

(4 citation statements)

References 32 publications

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“…Moreover, the L(j, k)-labelling of Cayley graphs has been studied in [20] and Kelarev, Ras, and Zhou [9] revealed a relationship between a semigroup structure and the minimum distance labelling spans of its Cayley graph.…”

Section: Preliminariesmentioning

confidence: 99%

Lambda number of zero-divisor graphs of finite commutative rings

Ali¹,

Raja²

2022

Preprint

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Let G = (V, E) be a simple graph, an L(2, 1)-labelling of G is assignment of labels from non-negative integers to the vertices of G such that adjacent vertices gets labels which at least differ by two and vertices which are at distance two from each other get different labels. The λ-number of G, denoted by λ(G) is the smallest positive integer such that G has a L(2, 1)labelling with all the labels are members of the set {0, 1, • • • , }. The zero-divisor graph denoted by Γ(R), of a finite commutative ring R with unity is a simple graph with vertices as non-zero zero divisors of R. Two vertices u and v are adjacent in Γ(R) if and only if uv = 0 in R. In this paper, we investigate L(2, 1)-labelling in zero-divisor graphs. We study the partite truncation, a graph operation that reduces a n-partite graph of higher order to a graph of lower order. We establish the relation between λ-numbers of two graphs. We make use of the operation partite truncation to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring. We compute the exact value of λ-numbers of zero-divisor graphs of some classes of local and mixed rings such as Z p n , Z p n × Z q m , and F q × Z p n .

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Section: Preliminariesmentioning

confidence: 99%

Lambda number of zero-divisor graphs of finite commutative rings

Ali¹,

Raja²

2022

Preprint

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“…Let S be a semigroup and T be a nonempty subset of S. The Cayley graph Cay(S, T ) of S relative to T is defined as the graph with vertex set S and edge set E(Cay(S, T )) consisting of those ordered pair (a, b), where a = b and sa = b for some s ∈ T (see, [5,7,14]). The Cayley graphs of semigroups have been investigated by many authors, and a lot of interesting results have been obtained (see, [8][9][10][11][12][13][16][17][18]). The Cayley graphs of semigroups are closely related to finite state automata and have many valuable applications (see, [5-9, 11-13, 18-20]).…”

Section: Introductionmentioning

confidence: 99%

Applications of the Cayley graph of a monoid presentation in characterizations for right type B semigroups

Li,

Huang

2023

Preprint

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Motivated by the rich results on Cayley graphs of semigroups, in this paper we investigate the Cayley graph of a monoid presentation, and get a method which makes it possible to construct a certain type of a monoid (which we call graph expansions) by the Cayley graph of a monoid presentation. After obtaining some basic properties of Cayley graphs of a monoid presentation, we give a new characterization of a right type B semigroup by using a Cayley graph of a monoid presentation of a left cancellative monoid. It is proved that an arbitrary graph expansion for a left cancellative monoid is a proper (F) right type B monoid. AMS Mathematics Subject Classification (2010) 05C25; 20M10

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“…Zhou [34] studied L( j, k)-labelings of Cayley graphs of abelian groups. Kelarev, Ras and Zhou [25] established connections between the structure of a semigroup and the minimum spans of distance labelings of its Cayley graphs. In this paper we study L(2, 1)-labelings of the power graph of a finite group.…”

Section: Introductionmentioning

confidence: 99%

Lambda number of the power graph of a finite group

Feng

Wang

2020

J Algebr Comb

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The power graph G of a finite group G is the graph with the vertex set G, where two distinct elements are adjacent if and only if one is a power of the other. An L(2, 1)-labeling of a graph is an assignment of labels from nonnegative integers to all vertices of such that vertices at distance two get different labels and adjacent vertices get labels that are at least 2 apart. The lambda number of , denoted by λ( ), is the minimum span or range over all L(2, 1)-labelings of . In this paper, we obtain bounds for λ( G ) and give necessary and sufficient conditions when the bounds are attained. As applications, we compute the exact value of λ( G ) if G is a dihedral group, a generalized quaternion group, a P-group or a cyclic group of order pq n , where p and q are distinct primes and n is a positive integer.

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Distance labellings of Cayley graphs of semigroups

Cited by 16 publications

References 32 publications

Lambda number of zero-divisor graphs of finite commutative rings

Lambda number of zero-divisor graphs of finite commutative rings

Applications of the Cayley graph of a monoid presentation in characterizations for right type B semigroups

Lambda number of the power graph of a finite group

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