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Nowhere-zero 3-flows in cayley graphs on generalized dihedral group and generalized quaternion group

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Frontiers of Mathematics in China 2015年第2期10卷 293-302页

作者： Liangchen LI Xiangwen LI Department of Mathematics Huazhong Normal University Wuhan 430079 China Department of Mathematics Luoyang Normal University Luoyang 471022 China

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111： 416-419]. We also generalizes an early result of M. Nanasiova and M. Skoviera [J. Algebraic Combin., 2009, 30： 103-110].

关键词： Nowhere-zero 3-flow cayley graph generalized dihedral group,generalized quaternion group

Automorphism Groups of a Class of Cubic cayley graphs on Symmetric Groups

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Algebra Colloquium 2017年第4期24卷 541-550页

作者： Xueyi Huang Qiongxiang Huang Lu Lu College of Mathematics and Systems Science Xinjiang University Urumqi Xinjiang 830046 China

Let Sn denote the symmetric group of degree n with n 〉 3, S = {cn = （1 2…n）, cn^-1, （1 2）} and Fn=Cay（Sn, S） be the cayley graph on Sn with respect to S. In this paper, we show that Fn （n 〉 13） is a normal cayley graph, and that the full automorphism group of Fn is equal to Aut（Гn） = R（Sn）（Inn（C））≌- Sn×Z2, where R（Sn） is the right regular representation of Sn,φ = （1 2）（3 n）（4 n-1）（5 n-2）... （∈ Sn）, and Inn（C） is the inner isomorphism of Sn induced by φ .

关键词： cayley graph normal automorphism group

Conjecture of Li and Praeger concerning the isomorphisms of cayley graphs of A_5

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Science China Mathematics 2001年第12期44卷 1502-1508页

作者：徐明曜方新贵沈孝燮白永吉 1. School of Mathematical Sciences Peking University 100871 Beijing China 2. Division of Mathematical Sciences Pukyong National University 608-737 Pusan Republic of Korea

LetG be a finite group, andS a subset ofG \ |1| withS =S ?1. We useX = Cay(G,S) to denote the cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ?1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA 5 is a 4-Cl-group, which was a conjecture posed by Li and Praeger.

关键词： cayley graph CI-subset m-CI-group Normal cayley graph

Enumeration of Cubic cayley graphs on Dihedral Groups

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Acta Mathematica Sinica,English Series 2017年第7期33卷 996-1010页

作者： Xue Yi HUANG Qiong Xiang HUANG Lu LU College of Mathematics and Systems Science Xinjiang UniversityUrumqi 830046P.R.China

Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic cayley graphs on D2p up to isomorphism by means of spectral method. By the way, we show that two cubic cayley graphs on D2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic cayley graphs on D2 by using Gauss＇ celebrated law of quadratic reciprocity.

关键词： cayley graph dihedral group cospectral isomorphic classes quadratic reciprocity

On the Metric Dimension of Barycentric Subdivision of cayley graphs

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Acta Mathematicae Applicatae Sinica 2016年第4期32卷 1067-1072页

作者： Muhammad IMRAN Department of Mathematics School of Natural Sciences National University of Sciences and Technology Sector H-12 Islamabad Pakistan

In a connected graph G, the distance d（u, v） denotes the distance between two vertices u and v of G. Let W = {w1, w2,……, wk} be an ordered set of vertices of G and let v be a vertex of G. The representation r（v1W） of v with respect to W is the k-tuple （d（v, w1）, d（v, w2）,…, d（v, wk））. The set W is called a resolving set or a locating set if every vertex of G is uniquely identified by its distances from the vertices of W, or equivalently, if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension of G, denoted by β（G）. Metric dimension is a generalization of affine dimension to arbitrary metric spaces （provided a resolving set exists）. In this paper, we study the metric dimension of barycentric subdivision of cayley graphs Cay （Zn Z2）. We prove that these subdivisions of cayley graphs have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of barycentric subdivision of cayley graphs Cay （Zn Z2）.

关键词： metric dimension basis resolving set barycentric subdivision cayley graph

One-regular Normal cayley graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

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Acta Mathematica Sinica,English Series 2006年第5期22卷 1305-1320页

作者： JinHoKWAK JuMokOH Mathematics Pohang University of Science and Technology Pohang 790-784 Korea

A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（Г, S）. Xu, M. Y., Xu, J. （Southeast Asian Bulletin of Math., 25, 355-363 （2001）） classified one-regular cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. （Croat. Chemica Acta, 73, 969-981 （2000）） classified cubic one-regular cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal cayley graphs G on a dihedral group whose vertex stabilizers in Aut（G） are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.

关键词： one-regular graph cayley graph dihedral group half-transitive graph

Two sufficient conditions for non-normal cayley graphs and their applications

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Science China Mathematics 2007年第2期50卷 201-216页

作者： Jin-xin ZHOU Yan-quan FENG Department of Mathematics Beijing Jiaotong UniversityBeijing 100044China

A cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G, S). In this paper, two sufficient conditions for non-normal cayley graphs are given and by using the conditions, five infinite families of connected non-normal cayley graphs are constructed. As an application, all connected non-normal cayley graphs of valency 5 on A 5 are determined, which generalizes a result about the normality of cayley graphs of valency 3 or 4 on A 5 determined by Xu and Xu. Further, we classify all non-CI cayley graphs of valency 5 on A 5, while Xu et al. have proved that A 5 is a 4-CI group.

关键词： cayley graph normal cayley graph arc-transitive graph 05C25 20B25

On Connected Tetravalent cayley graphs of a Non-abelian Group of Order 3p2

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Algebra Colloquium 2017年第3期24卷 467-480页

作者： M.R. Darafsheh M. Abdollahi School of Mathematics Statistics and Computer Science College of Science University of Tehran Tehran Iran

In this paper we determine all tetravalent cayley graphs of a non-abelian group of order 3p2, where p is a prime number greater than 3, and with a cyclic Sylow p-subgroup. We show that all of these tetravalent cayley graphs are normal. The full automorphism group of these cayley graphs is given and the half-transitivity and the arc-transitivity of these graphs are investigated. We show that this group is a 5-CI-group.

关键词： cayley graph automorphism group tetravalent graph half-transitive arctransitive m-CI-group

Perfect State Transfer on Weighted Abelian cayley graphs

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Chinese Annals of Mathematics,Series B 2021年第4期42卷 625-642页

作者： Xiwang CAO Keqin FENG Ying-Ying TAN Department of Mathematics Nanjing University of Aeronautics and Astronau-ticsNanjing 210016China Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China School of Mathematics and Physics Anhui Jianzhu UniversityHefei 230601China

Recently,there are extensive studies on perfect state transfer(PST for short)on graphs due to their significant applications in quantum information processing and quantum ***,there is not any general characterization of graphs that have PST in *** this paper,the authors present a depiction on weighted abelian cayley graphs having *** give a unified approach to describe the periodicity and the existence of PST on some specific graphs.

关键词： Perfect state transfer cayley graph Eigenvalues of a graph Weighted graph Random walk