Notes on simplicial rook graphs

Brouwer, Andries E.; Cioabă, Sebastian M.; Haemers, Willem H.; Vermette, Jason R.

doi:10.1007/s10801-015-0633-y

Cited by 5 publications

(5 citation statements)

References 7 publications

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“…In [2] it is shown that the automorphism group of SR(m, n) is S m when n > 3, where S m stands for the permutation group on m elements. Also it is S m × Z 2 when n = 3.…”

Section: Theoremmentioning

confidence: 99%

“…In [2] it is shown that the automorphism group of SR(m, n) is S m for n > 3, and is S m ×Z 2 for n = 3, where S m stands for the permutation group on m elements. Finally, in this section we study the automorphism group of CSR(m, n).…”

Section: Automorphism Group Of Csr(m N)mentioning

confidence: 99%

“…calculating the independence number of SR(3, n)) was independently investigated by several authors. Nivasch and Lev [7], Blackburn et al [1] and Vaderlind et al [8] independently proved that the independence number of SR (3, n) is equal to 1+⌊ 2 3 n⌋. Recently, Brouwer et al calculated the independence number of SR(m, 3) [2].…”

Section: Introductionmentioning

confidence: 99%

“…Nivasch and Lev [7], Blackburn et al [1] and Vaderlind et al [8] independently proved that the independence number of SR (3, n) is equal to 1+⌊ 2 3 n⌋. Recently, Brouwer et al calculated the independence number of SR(m, 3) [2]. In this work we present tight lower and upper bounds for the independence number of SR(m, n) for any m and n. For more details about the connection between the independence number of SR(m, n) and the maximum number of non attacking rooks please see [6].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the maximum number of non attacking rooks on a high-dimensional simplicial chessboard

Ahadi¹,

Mollahajiaghaei²,

Dehghan³

2021

Preprint

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The simplicial rook graph SR(m, n) is the graph whose vertices are vectors in N m such that for each vector the summation of its coordinates is n and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. Martin and Wagner (Graphs Combin. (2015) 31:1589-1611) asked about the independence number of SR(m, n) that is the maximum number of non attacking rooks which can be placed on a (m−1)-dimensional simplicial chessboard of side length n+1. In this work, we solve this problem and show that α(SR(m, n)) = 1 − o(1). We also prove that for the domination number of rook graphs we have γ(SR(m, n)) = Θ(n m−2 ). Moreover we show that these graphs are Hamiltonian.The cyclic simplicial rook graph CSR(m, n) is the graph whose vertices are vectors in Z m n such that for each vector the summation of its coordinates modulo n is 0 and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. In this work we determine several properties of these graphs such as independence number, chromatic number and automorphism group. Among other results, we also prove that computing the distance between two vertices of a given CSR(m, n) is NPhard in terms of n and m.

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“…In [2] it is shown that the automorphism group of SR(m, n) is S m when n > 3, where S m stands for the permutation group on m elements. Also it is S m × Z 2 when n = 3.…”

Section: Theoremmentioning

confidence: 99%

Section: Automorphism Group Of Csr(m N)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the maximum number of non attacking rooks on a high-dimensional simplicial chessboard

Ahadi¹,

Mollahajiaghaei²,

Dehghan³

2021

Preprint

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“…The eigenvalues of the Johnson graph are known (see [6,10]). Using Loday [25], one can also observe that the graph A n is an induced subgraph of the simplicial rook graph SR n − 2, n−1 2 introduced by Martin and Wagner [28] (see also [5]). We have not been able to use these facts to calculate the eigenvalues of A n .…”

mentioning

confidence: 98%

A lower bound for the smallest eigenvalue of a graph and an application to the associahedron graph

Cioabă¹,

Gupta²

2022

Preprint

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In this paper, we obtain a lower bound for the smallest eigenvalue of a regular graph containing many copies of a smaller fixed subgraph. This generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a triangle. We apply our results to obtain a lower bound on the smallest eigenvalue of the associahedron graph, and we prove that this bound gives the correct order of magnitude of this eigenvalue. We also survey what is known regarding the second-largest eigenvalue of the associahedron graph.

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On the Maximum Number of Non-attacking Rooks on a High-Dimensional Simplicial Chessboard

Ahadi

Mollahajiaghaei²,

Dehghan

2022

Graphs and Combinatorics

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Notes on simplicial rook graphs

Cited by 5 publications

References 7 publications

On the maximum number of non attacking rooks on a high-dimensional simplicial chessboard

On the maximum number of non attacking rooks on a high-dimensional simplicial chessboard

A lower bound for the smallest eigenvalue of a graph and an application to the associahedron graph

On the Maximum Number of Non-attacking Rooks on a High-Dimensional Simplicial Chessboard

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