Structural Properties of Absorption Cayley Graphs

Sinha, Deepa; Sharma, Deepakshi

doi:10.18576/amis/100626

Cited by 4 publications

(7 citation statements)

References 16 publications

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“…In 2015, Suksumran and Panma [16] proposed some concepts on connected Cayley graphs of semigroups. Later in 2016, Afkhami et al [1] constructed a new class of Cayley graphs and studied their structural properties similar to the research presented by Sinha and Sharma [15] in the same year. Furthermore, in 2018, Panda and Krishna [12] investigated the connectedness of power graphs of finite groups.…”

Section: Introductionmentioning

confidence: 83%

On Connectedness and Completeness of Cayley Digraphs of Transformation Semigroups With Fixed Sets

Nupo¹,

Pookpienlert²

2020

International Electronic Journal of Algebra

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Let Fix(X, Y) be a semigroup of full transformations on a set X in which elements in a nonempty subset Y of X are fixed. In this paper, we construct the Cayley digraphs of Fix(X, Y) and study some structural properties of such digraphs such as the connectedness and the completeness. Further, some prominent results of Cayley digraphs of Fix(X, Y) relative to minimal idempotents are verified. In addition, the characterization of an equivalence digraph of the Cayley digraph of Fix(X, Y) is also investigated.

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Section: Introductionmentioning

confidence: 83%

On Connectedness and Completeness of Cayley Digraphs of Transformation Semigroups With Fixed Sets

Nupo¹,

Pookpienlert²

2020

International Electronic Journal of Algebra

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“…The absorption Cayley graph of the ring Z n was introduced and studied in [274,275]. As the name conveys, this variant of Cayley graph was defined based on the absorption property of the elements in the ring, as given below, following which an example of an absorption Cayley graph is given in Figure 14.…”

Section: Absorption Cayley Graphsmentioning

confidence: 99%

“…As the graph is defined on the subset formed by all the elements of the ring that absorbs some other element of the ring, the properties of this set were first discussed in [275]. The cardinality of this set and the properties of the elements in the set were discussed, and it was found that, for n = 2k, where k is odd, this subset S ⊆ Z n coincides with the set of zero-divisors of the ring.…”

Section: Absorption Cayley Graphsmentioning

confidence: 99%

“…An interesting relation was seen between the adjacency matrix of the absorption Cayley graph on Z n and the Cayley table of Z n : if each element a ∈ S is replaced with 1 in the Cayley table and all the other elements, including the diagonals, are given 0, the adjacency matrix for the absorption Cayley graph on Z n can be obtained. As the absorption Cayley graph is defined based on the sum of two elements belonging to the symmetric subset, an interesting relation between the unitary addition Cayley graphs and the absorption Cayley graphs was given in [275], as follows.…”

Section: Absorption Cayley Graphsmentioning

confidence: 99%

“…An important question that arises for defining a new algebraic graph is the realisation of a given graph as the defined algebraic graph, that is, in this context, the question is, when can a graph of order n be realised as an absorption Cayley graph of order n? This was answered in [274,275] as follows.…”

Section: Absorption Cayley Graphsmentioning

confidence: 99%

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Graphs Defined on Rings: A Review

Madhumitha,

Naduvath

2023

Mathematics

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The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area.

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On Another Class of Strongly Perfect Graphs

et al. 2022

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For a commutative ring R with unity, the associate ring graph, denoted by AG(R), is a simple graph with vertices as nonzero elements of R and two distinct vertices are adjacent if they are associates. The graph AG(R) contains components equal in number to the number of distinct orbits, except for the orbit of an element 0. Moreover, each component is a complete graph. An important finding is that this is a class of strongly perfect graphs. In this article we describe the structure of the associate ring graph of the ring of integers modulo n, denoted by AG(Zn). We carried out computer experiments and provide a program for the same. We further characterize cases in which AG(Zn), its complement AG(Zn)¯, and their line graphs are planar, ring graphs, and outerplanar. We also discuss the properties of the associate ring graph of a commutative ring R with unity.

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Structural Properties of Absorption Cayley Graphs

Cited by 4 publications

References 16 publications

On Connectedness and Completeness of Cayley Digraphs of Transformation Semigroups With Fixed Sets

On Connectedness and Completeness of Cayley Digraphs of Transformation Semigroups With Fixed Sets

Graphs Defined on Rings: A Review

On Another Class of Strongly Perfect Graphs

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