The restricted edge-connectivity and restricted connectivity of augmented<i>k</i>-ary<i>n</i>-cubes

Lin, Ruizhi; Zhang, Heping

doi:10.1080/00207160.2015.1067690

Cited by 18 publications

(4 citation statements)

References 31 publications

(4 reference statements)

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“…n is an n-dimensional hypercube. There are various works concerning various topological and structural properties of this class of graphs in the literature, and some of the recent papers include [7,8,10,11,18,21].…”

Section: Some Results Which Follow From Theorem 32mentioning

confidence: 99%

On The automorphism groups of $us$-Cayley graphs

Mirafzal¹

2019

Preprint

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Let G be a finite abelian group written additively with identity 0, and Ω be an inverse closed generating subset of G such that 0 / ∈ Ω. We say that Ω has the property "us" (unique summation), whenever for every 0 = g ∈ G if there are s 1 , s 2 , s 3 , s 4 ∈ Ω such that s 1 + s 2 = g = s 3 + s 4 , then we have {s 1 , s 2 } = {s 3 , s 4 }. We say that a Cayley graph Γ = Cay(G; Ω) is a us-Cayley graph, whenever Γ is an abelian Cayley graph and the generating subset Ω has the property "us". In this paper, we show that if Γ = Cay(G; Ω) is a us-Cayley graph, then Aut(Γ) = L(G) A, where and A is the group of all automorphisms θ of the group G such that θ(Ω) = Ω. Then, we explicitly determine the automorphism groups of some classes of graphs, including Möbius ladders and k-array n-cubes.

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Section: Some Results Which Follow From Theorem 32mentioning

confidence: 99%

On The automorphism groups of $us$-Cayley graphs

Mirafzal¹

2019

Preprint

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“…An augmented k-ary n-cube AQ n,k is extended from a k-ary n-cube Q k n in a manner analogous to the extension of an n-dimensional hypercube Q n to an n-dimensional augmented AQ n [4] and has many triangles. Some results about topological properties and routing problems on augmented k-ary n-cube can be found in [5,10,19,20] etc. In this paper, by exploring and utilizing the structural properties of AQ n,k , we show that AQ n,3 (resp.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 1. ( [10]) Let u and v be two distinct vertices in AQ i n,k such that u and v have common neighbors in…”

mentioning

confidence: 99%

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Strong Menger connectedness of augmented $k$-ary $n$-cubes

Gu¹,

Chang²,

Hao³

2019

Preprint

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A connected graph G is called strongly Menger (edge) connected if for any two distinct vertices x, y of G, there are min{deg G (x), deg G (y)} vertex(edge)-disjoint paths between x and y. In this paper, we consider strong Menger (edge) connectedness of the augmented k-ary n-cube AQ n,k , which is a variant of k-ary n-cube Q k n . By exploring the topological proprieties of AQ n,k , we show that AQn,3 for n ≥ 4 (resp. AQ n,k for n ≥ 2 and k ≥ 4) is still strongly Menger connected even when there are 4n − 9 (resp. 4n − 8) faulty vertices and AQ n,k is still strongly Menger edge connected even when there are 4n − 4 faulty edges for n ≥ 2 and k ≥ 3. Moreover, under the restricted condition that each vertex has at least two fault-free edges, we show that AQ n,k is still strongly Menger edge connected even when there are 8n − 10 faulty edges for n ≥ 2 and k ≥ 3. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

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