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Isomorphisms of Finite Semi-cayley graphs

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Acta Mathematica Sinica,English Series 2015年第4期31卷 715-730页

作者： Majid AREZOOMAND Bijan TAERI Department of Mathematical Sciences Isfahan University of Technology

Let G be a finite group. A cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A cayley graph is called a CI-graph （cayley iso- morphism） if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a cayley graph F of G is a CI-graph if and only if all regular subgroups of Aut（F） isomorphic to G are conjugate in Aut（F）. A semi-cayley graph （also called bi-cayley graph by some authors） over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits （of equal size）. In this paper, we introduce the concept of SCI-graph （semi-cayley isomorphism） and prove a Babai type theorem for semi-cayley graphs. We prove that every semi-cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-cayley graphs.

关键词： Semi-cayley graph cayley graph CI-graph semiregular subgroup

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

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

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

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

Tetravalent Edge-transitive cayley graphs of PGL(2, p)

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Acta Mathematicae Applicatae Sinica 2013年第4期29卷 837-842页

作者： Xiao-hui HUA Shang-jin XU Yun-ping DENG College of Mathematics and Information Science Henan Normal University School of Mathematics and Information Science Guangxi University Department of Mathematics Shanghai Jiaotong University

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.

关键词： cayley graph normal bi-normal simple group

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

Isomorphisms of Cubic cayley graphs on Dihedral Groups and Sparse Circulant Matrices

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Acta Mathematica Sinica,English Series 2023年第4期39卷 618-632页

作者： Istvan KOVACS UP IAM&FAMNIT University of PrimorskaMuzejski trg 26000 KoperSlovenia

We show that,up to isomorphism,there is a unique non-CI connected cubic cayley graph on the dihedral group of order 2n for each even number n≥*** answers in the negative the question of Li whether all connected cubic cayley graphs are CI-graphs(Discrete Math.,256,301-334(2002)).As an application,a formula is derived for the number of isomorphism classes of connected cubic cayley graphs on dihedral groups,which generalises the earlier formula of Huang et *** with the particular case when n is a prime(Acta ***.,***.,33,996-1011(2017)).As another application,a short proof is also given for a result on sparse circulant matrices obtained by Wiedemann and Zieve(arXiv preprint,(2007)).

关键词： cayley graph graph isomorphism dihedral group circulant matrix

Reliability Analysis of the cayley graphs of Dihedral Groups

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Tsinghua Science and Technology 2011年第1期16卷 36-40页

作者：宋淑娇王殿军 Department of Mathematical Sciences Tsinghua University

cayley graphs have many good properties as models of communication networks. This study analyzes the reliability of the cayley graph based on the dihedral graph. graph theory and analyses show that almost all cayley graphs of the dihedral graph D2n are optimal super-λ. The number Ni（G） of cutsets of size i, λ≤ i≤λ＇ is given as Ni（G） = n[^（n-1）δ i-δ].

关键词： super-λ reliability cayley graph dihedral group