On the trace graph of matrices

Sivagami, M.; Chelvam, T. Tamizh

doi:10.1007/s10474-019-00918-5

Cited by 12 publications

(3 citation statements)

References 5 publications

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“…Moreover, Nupo and Panma [14] investigated the independent domination number in Cayley digraphs of rectangular groups. Recently, in 2019, Sivagami and Chelvam [17] considered the domination number of the trace graph of matrices. In addition, Ye et al [18] provided more results on the domination number of the Cartesian product of two directed cycles.…”

Section: Introductionmentioning

confidence: 99%

Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

Nupo¹,

Pookpienlert²

2021

Turk J Math

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For a nonempty subset Y of a nonempty set X , denote by F ix(X, Y ) the semigroup of full transformations on the set X in which all elements in Y are fixed. The Cayley digraph Cay(F ix(X, Y ), A) of F ix(X, Y ) with respect to a connection set A ⊆ F ix(X, Y ) is defined as a digraph whose vertex set is F ix(X, Y ) and two vertices α, β are adjacent in sense of drawing a directed edge (arc) from α to β if there exists µ ∈ A such that β = αµ . In this paper, we determine domination parameters of Cay(F ix(X, Y ), A) where A is a subset of F ix(X, Y ) related to minimal idempotents and permutations in F ix(X, Y ).

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Section: Introductionmentioning

confidence: 99%

Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

Nupo¹,

Pookpienlert²

2021

Turk J Math

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“…The trace graph of the matrix ring M n (R), denoted by Γ t (M n (R)), is the simple undirected graph with vertex set {A ∈ M n (R) * : there exists B ∈ M n (R) * such that Tr(AB) = 0} and two distinct vertices A and B are adjacent if and only if Tr(AB) = 0. Further study on the trace graph of matrices was done by authors [10].…”

Section: Introductionmentioning

confidence: 99%

Ideal based trace graph of matrices

Chelvam

Manikandan

2020

Hacettepe Journal of Mathematics and Statistics

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Let R be a commutative ring and M n (R) be the set of all n × n matrices over R where n ≥ 2. The trace graph of the matrix ring M n (R) with respect to an ideal I of R, denoted by Γ I t (M n (R)), is the simple undirected graph with vertex set M n (R) \ M n (I) and two distinct vertices A and B are adjacent if and only if Tr(AB) ∈ I. Here Tr(A) represents the trace of the matrix A. In this paper, we exhibit some properties and structure of Γ I t (M n (R)).

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“…Another very general problem is to examine the induced subgraphs of various generalizations of zero-divisor graphs, such as the extended zerodivisor graphs and the trace graphs [2,24,25,26,27].…”

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confidence: 99%

Induced subgraphs of zero-divisor graphs

G.¹,

Cameron²,

Kavaskar³

et al. 2022

Preprint

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The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with a and b adjacent if ab = 0. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divisor graph. This remains true for various restricted classes of rings, including boolean rings, products of fields, and local rings. But in more restricted classes, the zero-divisor graphs do not form a universal family. For example, the zero-divisor graph of a local ring whose maximal ideal is principal is a threshold graph; and every threshold graph is embeddable in the zero-divisor graph of such a ring. More generally, we give necessary and sufficient conditions on a non-local ring for which its zero-divisor graph to be a threshold graph. In addition, we show that there is a countable local ring whose zero-divisor graph embeds the Rado graph, and hence every finite or countable graph, as induced subgraph. Finally, we consider embeddings in related graphs such as the 2-dimensional dot product graph.

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On the trace graph of matrices

Cited by 12 publications

References 5 publications

Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

Ideal based trace graph of matrices

Induced subgraphs of zero-divisor graphs

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