Semi-Cayley图的匹配可扩性和谱
|
摘要
连通图r称为κ-可扩的,如果|V(Γ)|≥2κ+2,且r的每个大小为κ的匹配均可以扩充为r的一个完美匹配.图r的谱是r的邻接矩阵A(r)的特征值以及它们的重数.
于青林等人刻画了有限阿贝尔群上的Cayley图的2-可扩性,且提出了两个公开问题：(1)刻画有限阿贝尔群上的Cayley图的3-可扩性和κ-可扩性；(2)刻画任意群上的Cayley图的1-可扩性和2-可扩性.Semi-Cayley图是Cayley图的自然推广.本文主要研究群上的Semi-Cayley图的匹配可扩性.作为应用,我们刻画了一类非阿贝尔群上的Cayley图的2-可扩性.另外,给出了有限阿贝尔群上的Semi-Cayley图的谱.全文共分为五章.
在第一章中,我们首先介绍了本文所需要的基本概念,术语和记号,然后指出本文所研究问题的背景,进而综述了该领域的研究进展和本文所得到的主要结论.
Bi-Cayley图是特殊的Semi-Cayley图.在第二章中,我们研究了有限阿贝尔群上的Bi-Cayley图的匹配可扩性.特别地,分别刻画了有限阿贝尔群上的Bi-Cayley图的2-可扩性和3-可扩性.
在第三章中,我们研究了有限非阿贝尔群上的Bi-Cayley图的匹配可扩性.特别地,分别刻画了任意有限群上的Bi-Cayley图的1-可扩性和二面体群上的Bi-Cayley图的2-可扩性.
第四章我们研究了二面体群Dn上的Semi-Cayley图SC(Dn；R,R,T)的1-可扩性和2-可扩性.作为应用,刻画了群Dn×Z2上的Cayley图的2-可扩性.
在第五章中,我们给出了有限阿贝尔群上的Semi-Cayley图的谱公式,证明了二面体群和双循环群上的Cayley图是有限阿贝尔群上的Semi-Cayley图,从而给出了二面体群和双循环群上的Cayley图的谱公式.
A connected graphΓis called k-extendable if|V(Γ)|≥2k+2 and every matching of size k inΓcan be extended to a perfect matching ofΓ. The spectrum of a graphΓis the set of numbers which are eigenvalues of A(Γ), together with their multiplicities.
Yu et al characterized the 1-extendability and 2-extendability of Cayley graphs over abelian groups and posed two open questions:(1) Characterize 3-extendable abelian Cayley graphs and, in general, k-extendable abelian Cayley Graphs; (2) Characterize 1-extendable and 2-extendable Cayley graphs. Semi-Cayley graphs are generalization of Cayley graphs. In this thesis, we mainly study the extendability of Semi-Cayley graphs over groups. As applications, we characterize the 2-extendability of a Cayley graph over a non-abelian group. In addition, we give a formula of the spectrum of semi-Cayley graphs over finite abelian groups. This thesis consists of five chapters.
In Chapter 1, we first introduce some basic concepts, terminology and notations. Then we point out the research backgrounds. Finally, we survey the research developments in this area and make a brief introduction to the main results obtained in the thesis.
Bi-Cayley graphs are special Semi-Cayley graphs. In Chapter 2, we explore the extendability of Bi-Cayley graphs over finite abelian groups. In particular, we character the 2-extendable and 3-extendable Bi-Cayley graphs over finite abelian groups, respectively.
In Chapter 3, we study the extendability of Bi-Cayley grpahs over finite non-abelian groups. In particular, we characterize the 1-extendability of Bi-Cayley graphs over finite groups and the 2-extendability of Bi-Cayley graphs over dihedral groups.
In Chapter 4, we classify the 1-extendable and 2-extendable Semi-Cayley graphs SC(Dn; R, R,T) over dihedral groups, respectively. As an application, we characterize the 2-extendable Cayley graph over the group Dn x Z2.
In Chapter 5, we give a formula of the spectrum of Semi-Cayley graphs over finite abelian groups. In particular, we give the spectrum of Cayley graphs over dihedral groups and dicyclic groups, respectively.
于青林等人刻画了有限阿贝尔群上的Cayley图的2-可扩性,且提出了两个公开问题：(1)刻画有限阿贝尔群上的Cayley图的3-可扩性和κ-可扩性；(2)刻画任意群上的Cayley图的1-可扩性和2-可扩性.Semi-Cayley图是Cayley图的自然推广.本文主要研究群上的Semi-Cayley图的匹配可扩性.作为应用,我们刻画了一类非阿贝尔群上的Cayley图的2-可扩性.另外,给出了有限阿贝尔群上的Semi-Cayley图的谱.全文共分为五章.
在第一章中,我们首先介绍了本文所需要的基本概念,术语和记号,然后指出本文所研究问题的背景,进而综述了该领域的研究进展和本文所得到的主要结论.
Bi-Cayley图是特殊的Semi-Cayley图.在第二章中,我们研究了有限阿贝尔群上的Bi-Cayley图的匹配可扩性.特别地,分别刻画了有限阿贝尔群上的Bi-Cayley图的2-可扩性和3-可扩性.
在第三章中,我们研究了有限非阿贝尔群上的Bi-Cayley图的匹配可扩性.特别地,分别刻画了任意有限群上的Bi-Cayley图的1-可扩性和二面体群上的Bi-Cayley图的2-可扩性.
第四章我们研究了二面体群Dn上的Semi-Cayley图SC(Dn；R,R,T)的1-可扩性和2-可扩性.作为应用,刻画了群Dn×Z2上的Cayley图的2-可扩性.
在第五章中,我们给出了有限阿贝尔群上的Semi-Cayley图的谱公式,证明了二面体群和双循环群上的Cayley图是有限阿贝尔群上的Semi-Cayley图,从而给出了二面体群和双循环群上的Cayley图的谱公式.
A connected graphΓis called k-extendable if|V(Γ)|≥2k+2 and every matching of size k inΓcan be extended to a perfect matching ofΓ. The spectrum of a graphΓis the set of numbers which are eigenvalues of A(Γ), together with their multiplicities.
Yu et al characterized the 1-extendability and 2-extendability of Cayley graphs over abelian groups and posed two open questions:(1) Characterize 3-extendable abelian Cayley graphs and, in general, k-extendable abelian Cayley Graphs; (2) Characterize 1-extendable and 2-extendable Cayley graphs. Semi-Cayley graphs are generalization of Cayley graphs. In this thesis, we mainly study the extendability of Semi-Cayley graphs over groups. As applications, we characterize the 2-extendability of a Cayley graph over a non-abelian group. In addition, we give a formula of the spectrum of semi-Cayley graphs over finite abelian groups. This thesis consists of five chapters.
In Chapter 1, we first introduce some basic concepts, terminology and notations. Then we point out the research backgrounds. Finally, we survey the research developments in this area and make a brief introduction to the main results obtained in the thesis.
Bi-Cayley graphs are special Semi-Cayley graphs. In Chapter 2, we explore the extendability of Bi-Cayley graphs over finite abelian groups. In particular, we character the 2-extendable and 3-extendable Bi-Cayley graphs over finite abelian groups, respectively.
In Chapter 3, we study the extendability of Bi-Cayley grpahs over finite non-abelian groups. In particular, we characterize the 1-extendability of Bi-Cayley graphs over finite groups and the 2-extendability of Bi-Cayley graphs over dihedral groups.
In Chapter 4, we classify the 1-extendable and 2-extendable Semi-Cayley graphs SC(Dn; R, R,T) over dihedral groups, respectively. As an application, we characterize the 2-extendable Cayley graph over the group Dn x Z2.
In Chapter 5, we give a formula of the spectrum of Semi-Cayley graphs over finite abelian groups. In particular, we give the spectrum of Cayley graphs over dihedral groups and dicyclic groups, respectively.
引文
[1]O. Ahmadi, N. Alon, I.F. Blake, I.E. Shparlinski, Graphs with integral spectrum, Linear Algebra Appl.430 (2009),547-552.
[2]A. Ahmadi, R. Belk, C. Tamon, C.Wendler, On mixing of continuous-time quantum walks on some circulant graphs, Quant. Inform. Comput. 3 (2003),611-618.
[3]N.L. Biggs, Albebraic Graph Theorey, second ed., Cambridge University Press, Cambridge,1993.
[4]L. Babai, Spectra of Cayley graphs, J. Combin. Theorey Ser. B 27 (1979) 180-189.
[5]A.E. Brouwer, Small integral trees, Electron. J. Combin.15 (Note 1) (2008),1-8.
[6]A.E. Brouwer,W.H. Haemers, The integral trees with spectral radius 3, Linear Algebra Appl.429(11-12) (2008),2710-2718.
[7]F.C. Bussemaker, D. Cvetkovic, There are exactly 13 connected, cubic, integral graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544 (1976),43-48.
[8]F.K. Bell, D. Cvetkovic, P. Rowlinson, S. K. Simc, Graphs for which the least eigenvalue is minimal, Ⅱ, Linear Algebra Appl.429 (2008), 2168-2179
[9]K. Balinska, D. Cvetkovic, Z. Radosavljevic, S. Simic, D. Stevanovic, A survey on integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.13 (2003),42-65.
[10]R.A. Brualdi, A.J. Hoffman, On the spectral radius of (0,1) matrices, Linear Algebra Appl.65 (1985),133-146.
[11]A.E. Brouwer, W.H. Haemers, The integral trees with spectral radius 3, Linear Algebra Appl.429 (2008),2710-2718.
[12]J.A. Bondy, U.S.R. Murty, Graph Theory, Springer,2008.
[13]L.W. Beineke, R.J. Wilson, P.J. Cameron, Topic in Algebraic Graph Theory, Cambridge University Press, Cambridge,2004.
[14]A. Berman, X.D. Zhang, On the spectral radius of graphs with cut ver-tices, Journal of combinatorial theory, Ser. B 83 (2001),233-240.
[15]C.C. Chen, On the classfication of 2-extendable Cayley graphs on dihe-dral groups, Australasian Journal of Combinatories 6 (1992) 209-219.
[16]D.M. Cvetkovic, Graphs and their spectra, Univ. Beograd. Publ. Elek-trotehn. Fak. Ser. Mat. Fiz.354-356 (1971),1-50.
[17]O. Chan, C.C. Chen, Q.L. Yu, On 2-extendable abelian Cayley graphs, Discrete Math.146 (1995),19-32.
[18]M. Christandl, N. Datta, A. Ekert, A. Kay, A.J. Landahl, Perfect transfer of arbitrary states in quantum spin networks, Phys. Rev. A 71 (2005), 032312.
[19]M. Christandl, N. Datta, A. Ekert, A.J. Landahl, Perfect state transfer in quantum spin networks, Phys. Rev. Lett.92 (2004),187902.
[20]D.M. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs Theory and Application, second ed., Berlin,1993.
[21]D.M. Cvetkovic, M.Doob, I.Gutman, A.Yorgasev. Recent results in the theory of graph spectra, NorthHolland, Amsterdam,1988.
[22]H.S.M. Coxetex, W.O.J. Moser, Generators and relations for discrete groups, Springer, Berlin,1957.
[23]C.C. Chen, N. Quimpo, On strongly connected abelian group graphs, Combinatorial Math. VIII, Springer, Lecture Notes Series, Vol.884, Springer, Berlin,1981.
[24]D.M. Cardoso, P.Rama, Spectral results on graphs with regularity con-straints, Linear Algebra Appl.423 (2007),90-98.
[25]L. Collatz, U. Singoantz, Spektren and lichor grafen, Abn. Math, Sem. Univ. Hamburg,21 (1957),63-71.
[26]E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl.373 (2003),241-272.
[27]S.F. Du, D. Marusic, An infinite family of biprimitive semisymmetric graphs, J. Graph Theory 32 (1999),217-228.
[28]S.F. Du, D. Marusic, Biprimitive graphs of smallest order, J. Algebr. Combin.9 (1999),151-156.
[29]S.F. Du, A. Malnic, D. Marusic, Classification of 2-arc-transitive dihe-drants, Journal of Combinatorial Theory Ser. B 986(2008),1349-1372.
[30]Y. Dong, H.P. Zhang,2-extendability of toroidal polyhexes and Klein-bottle polyhexes, Discrete Applied Mathematics 157 (2009),292-299.
[31]R. Frucht, Herstellung von Graphen mit vorgegbenel abstrakten Gruppe, Compositio Math.6 (1938),239-250.
[32]S. Friendland, The maximal eigenvalue of 0-1 matrices with prescribed number of l's, Linear Algebra Appl.69 (1988),33-69.
[33]O. Favaron, M. Maheo, J.F. Sacle, Some eigenvalue properties in graphs, Discrete Math. 111 (1993),197-220.
[34]C. Facer, J. Twamley, J.D. Cresser, Quantum Cayley networks of the hypercube, Phys. Rev. A 77 (2008),012334.
[35]C. Godsil, G. Royle, Algebraic Graph Theory, Springer-Verlag, New York,2001.
[36]X. Gao, W.W Liu, On the extendability of Semi-Cayley graphs of finite abelian groups, submitted to Discrete Math..
[37]G. Hetyei, Rectangular configurations which can be covered by 2 X 1 rectangles, Pecsi Tan. Foisk. Kozl.8 (1964),351-367.
[38]M. Hofmeister, Spectral radius and degreee sequence, Math. Noelr.139 (1988),37-44.
[39]Y. Hong, The k-th largest eigenvalue of a tree, Linear Algebra Appl.73 (1986),151-155.
[40]Y. Hong, Tree-width, clique-minors, and eigenvalues, Discrete Math.274 (2004),281-287.
[41]R.A. Horn, C.R. Johnson, Topics in Matrix analysis, Cambridge Univer-sity Press, Cambridge,1991.
[42]D.A. Holton, D.J. Lou, M.D. Plummer, On the 2-extendability of planar graphs, Discrete Math.96 (1991),81-99.
[43]P. Hic, R. Nedela, Balanced integral trees, Math. Slavaca 48 (1998), 429-445.
[44]F. Harary, A. Schwenk, Which graphs have integral spectra?, in:R. Bari, F. Harary (Eds.), Graphs and Combinatorics, Springer, Berlin,1974, pp. 45.
[45]Y. Hong, J.L. Shu, Sharp lower bounds of the least eigenvalue of planar graphs, Linear Algebra Appl.296 (1999),227-232.
[46]W. Imrich, S. Klavzar, Product Graphs, John Wiley& Sons, New York, 2000.
[47]G. Indulal, A. Vijayakumar, Some new integral graphs, Appl. Anal. Dis-crete Math.1 (2007),420-426.
[48]J.H. Kwak, Y.S. Kwon, J.M. On, Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency, Journal of Combi-natorial Theory, Ser. B 98 (2008),585-598.
[49]I. Kovacs a, A. Malnic, D. Marusic, S. Miklavic, One-matching bi-Cayley graphs over abelian groups, European J. Combin.30 (2009),602-616.
[50]K. Kutnar, D. Marusic,S. Miklavic, P. Sparl, Strongly regular tri-Cayley graphs, European J. Combin.30 (2009),822-832.
[51]C.H. Li, The finite vertex-primitive and vertex-biprimitive s-transitive graphs for s≥4, Trans. Amer. Math. Soc.353 (2001),3511-3529.
[52]C.H. Li, The finite primitive permutation groups containing an abelian regular subgroup, Proc. London Math. Soc.87 (2003),725-747.
[53]C.H. Li, Permutation groups with a cyclic regular subgroup and arc transitive circulants, J. Algebraic Combin.21 (2005),131-36.
[54]C.H.C. Little, D.D. Grant, D.A. Holton, On defect-d matching in graphs, Discrete Math.13 (1975),41-54.
[55]L. Lovasz, On the structure of factorizable graphs, Acta Math. Acad. Sci. Hunger.23 (1972),179-195.
[56]L. Lovaisz, Matching structure and the matching lattice, J. Combin. Theory Ser.(B) 43 (1987),187-222.
[57]L. Lovasz, M.D. Plummer, Matching Theory, North-Holland, Amster-dam, New York,1986.
[58]Z.P. Lu, On the Automorphism Groups of Bi-Cayley graphs. Acta.Sci.Natu. Universitatis Pekinensis 39(1) (2003),1-5.
[59]Z.P. Lu, C.Q. Wang, M.Y. Xu, Semisymmetric cubic graphs constructed from bi-Cayley graphs of An, Ars Combin.80 (2006),177-187.
[60]K.H. Leung, S.L. Ma, Partial difference triples, J. Algebr. Combin.2 (1993),397-409.
[61]D.J. Lou, Q.L. Yu, Connectivity of k-extendable graphs with large k, Discrete Applied Mathematics 136 (2004),55-61.
[62]D. Marusic, Strongly regular bicirculants and tricirculants, Ars Combin. 25 (1988),11-15.
[63]H. Minc, Nonnegative matrices Willey, New York,1988 (Chap2. pp.67).
[64]J. Meier, Groups, Graphs and Trees, Cambridge University Press, New York,2008.
[65]M. Muzychuk, On normal quotients of transitive graphs, Ars Combin. 59 (2001),171-180.
[66]M. Muzychuk, A solution of the isomorphism problem for circulant graphs, Proc. London Math. Soc.88 (2004) 1-41.
[67]B.D. McKay, M. Miller, J. Siran, A note on large graphs of diameter two and given maximum degree, Journal of Combinatorial Theory Ser. B 74 (1998),110-118.
[68]S. Miklavic, P. Potocnik, Distance-regular Cayley graphs on dihedral groups, Journal of Combinatorial Theory Ser. B 97 1(2007) 14-33.
[69]A. Malnic, D. Marusic, P. sparl, On strongly regular bicirculants, Euro-pean J. Combin.28 (2007),891-900.
[70]E. Nosal, Eigenvalues of graphs, Master's thesis, University of Calgary, 1970.
[71]D.L. Powers, Graph partitioning by eigenectors, Linear Algebra Appl. 101 (1988) 121-133.
[72]M.D. Plummer, Matching extension in bipartite graphs, Proceedings of the 17th Southeastern Conference on Combinatorics, Graph Theory and Computering Congress Numer.54, Utilitas Math., Winnipeg,1986 pp. 245-258.
[73]M.D. Plummer, On n-extendable graphs, Discrete Math.31 (1980) 201-210.
[74]M.D. Plummer, Matching extension and the genus of a graph, J. Combin. Theory Ser. B 44 (1988),329-337.
[75]M.D. Plummer, Extending matchings in graphs:A survey, Discrete Math.127 (1994),277-292.
[76]M.D. Plummer, Extending matchings in graphs:an update, Congr. Nu-mer.116 (1996),3-32.
[77]M.J.D. Resmi, D. Jungnickel, Strongly regular Semi-Cayley graphs, Journal of Algebraic Combinatorics 1 (1992),171-195.
[78]A.J. Schwenk, Almost all trees are cospectral, in:F. Harary (Ed.), New Directions in the Theory of Graphs, Academic Press, New York,1973.
[79]G. Sabidussi, On a class of fixed-point-free graphs, Proc. Amer. Math. Soc.9 (1958),800-804.
[80]R.A. Stone, On 1-factorizability of Cayley graphs, J. Combi. Theory, Ser. (B) 39 (1985),298-307.
[81]R.P. Stanley, Abound on the spectral radius of graphs with edges, Linear Algebra Appl.87 (1987),267-269.
[82]W. So, Integral circulant graphs, Discrete Math.306 (2005),153-158.
[83]D. Stevanovic, N.M.M. de Abreu, M.A.A. de Freitas, R. Del-Vecchio, Walks and regular integral graphs, Linear Algebra Appl.423 (2007), 119-135.
[84]G. Schrag, L. Cammack, On the 2-extendability of the generalized Pe-tersen graphs, Discrete Math.78 (1989),167-177.
[85]W.T. Tutte, A family of cubical graphs, proc. Camb. philos, soc.43 (1947),459-474.
[86]T.H. Hungerford, Algebra, Springer-Verlag, New York,1974.
[87]H.S. Wilf, The eigenvalues of a graph and its chromatic number, J. Lon-don Math. Soe,42 (1967),330-332.
L.Wang, X. Li, Integral trees with diameters 5 and 6, Discrete Math. 297 (2005),128-143.
[89]R.J. Wilson, Introduction to Graph Theory, third ed., Longman Inc., New York,1982.
[90]Q.L. Yu, Classifying two-extendable generalized Petersen graphs, Dis-crete Math.103 (1992),209-220.
[91]Q.L. Yu, A note on n-extendable graphs, J. Graph Theory 16 (1992), 349-353.
[92]Q.L. Yu, G.Z. Liu, Graph Factors and Matching Extensions, Higher Education Press,2009.
[93]X.L. Zhang, H.P. Zhang, Some graphs determined by their spectra, Lin-ear Algebra Appl.431 (2009),1443-1454.
[94]梁晓东,Bi-Cayley图与半传递图的连通性,博士学位论文,新疆大学.
[95]柳泊濂,组合矩阵,高等教育出版社,2005.
[96]王爱民,孟吉翔,有限交换群上Bi-Cayley图的Hamilton性,新疆大学学报(自然科学版),23(2)(2006),156-158.
[97]许英,阿贝尔群上的Cayley图的谱,硕士学位论文,新疆大学,2007.
[98]徐尚进,郭松涛,刘君,双Cayley图的Hamilton性,广西大学学报(自然科学版),34(2)2009,211-215.
[99]徐曜,有限群导引(第二版),科学出版社,1999.
[100]邹华,孟吉翔,Bi-Cayley图的一些代数性质,数学学报,50(5)(2007),1075-1080.
[2]A. Ahmadi, R. Belk, C. Tamon, C.Wendler, On mixing of continuous-time quantum walks on some circulant graphs, Quant. Inform. Comput. 3 (2003),611-618.
[3]N.L. Biggs, Albebraic Graph Theorey, second ed., Cambridge University Press, Cambridge,1993.
[4]L. Babai, Spectra of Cayley graphs, J. Combin. Theorey Ser. B 27 (1979) 180-189.
[5]A.E. Brouwer, Small integral trees, Electron. J. Combin.15 (Note 1) (2008),1-8.
[6]A.E. Brouwer,W.H. Haemers, The integral trees with spectral radius 3, Linear Algebra Appl.429(11-12) (2008),2710-2718.
[7]F.C. Bussemaker, D. Cvetkovic, There are exactly 13 connected, cubic, integral graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544 (1976),43-48.
[8]F.K. Bell, D. Cvetkovic, P. Rowlinson, S. K. Simc, Graphs for which the least eigenvalue is minimal, Ⅱ, Linear Algebra Appl.429 (2008), 2168-2179
[9]K. Balinska, D. Cvetkovic, Z. Radosavljevic, S. Simic, D. Stevanovic, A survey on integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.13 (2003),42-65.
[10]R.A. Brualdi, A.J. Hoffman, On the spectral radius of (0,1) matrices, Linear Algebra Appl.65 (1985),133-146.
[11]A.E. Brouwer, W.H. Haemers, The integral trees with spectral radius 3, Linear Algebra Appl.429 (2008),2710-2718.
[12]J.A. Bondy, U.S.R. Murty, Graph Theory, Springer,2008.
[13]L.W. Beineke, R.J. Wilson, P.J. Cameron, Topic in Algebraic Graph Theory, Cambridge University Press, Cambridge,2004.
[14]A. Berman, X.D. Zhang, On the spectral radius of graphs with cut ver-tices, Journal of combinatorial theory, Ser. B 83 (2001),233-240.
[15]C.C. Chen, On the classfication of 2-extendable Cayley graphs on dihe-dral groups, Australasian Journal of Combinatories 6 (1992) 209-219.
[16]D.M. Cvetkovic, Graphs and their spectra, Univ. Beograd. Publ. Elek-trotehn. Fak. Ser. Mat. Fiz.354-356 (1971),1-50.
[17]O. Chan, C.C. Chen, Q.L. Yu, On 2-extendable abelian Cayley graphs, Discrete Math.146 (1995),19-32.
[18]M. Christandl, N. Datta, A. Ekert, A. Kay, A.J. Landahl, Perfect transfer of arbitrary states in quantum spin networks, Phys. Rev. A 71 (2005), 032312.
[19]M. Christandl, N. Datta, A. Ekert, A.J. Landahl, Perfect state transfer in quantum spin networks, Phys. Rev. Lett.92 (2004),187902.
[20]D.M. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs Theory and Application, second ed., Berlin,1993.
[21]D.M. Cvetkovic, M.Doob, I.Gutman, A.Yorgasev. Recent results in the theory of graph spectra, NorthHolland, Amsterdam,1988.
[22]H.S.M. Coxetex, W.O.J. Moser, Generators and relations for discrete groups, Springer, Berlin,1957.
[23]C.C. Chen, N. Quimpo, On strongly connected abelian group graphs, Combinatorial Math. VIII, Springer, Lecture Notes Series, Vol.884, Springer, Berlin,1981.
[24]D.M. Cardoso, P.Rama, Spectral results on graphs with regularity con-straints, Linear Algebra Appl.423 (2007),90-98.
[25]L. Collatz, U. Singoantz, Spektren and lichor grafen, Abn. Math, Sem. Univ. Hamburg,21 (1957),63-71.
[26]E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl.373 (2003),241-272.
[27]S.F. Du, D. Marusic, An infinite family of biprimitive semisymmetric graphs, J. Graph Theory 32 (1999),217-228.
[28]S.F. Du, D. Marusic, Biprimitive graphs of smallest order, J. Algebr. Combin.9 (1999),151-156.
[29]S.F. Du, A. Malnic, D. Marusic, Classification of 2-arc-transitive dihe-drants, Journal of Combinatorial Theory Ser. B 986(2008),1349-1372.
[30]Y. Dong, H.P. Zhang,2-extendability of toroidal polyhexes and Klein-bottle polyhexes, Discrete Applied Mathematics 157 (2009),292-299.
[31]R. Frucht, Herstellung von Graphen mit vorgegbenel abstrakten Gruppe, Compositio Math.6 (1938),239-250.
[32]S. Friendland, The maximal eigenvalue of 0-1 matrices with prescribed number of l's, Linear Algebra Appl.69 (1988),33-69.
[33]O. Favaron, M. Maheo, J.F. Sacle, Some eigenvalue properties in graphs, Discrete Math. 111 (1993),197-220.
[34]C. Facer, J. Twamley, J.D. Cresser, Quantum Cayley networks of the hypercube, Phys. Rev. A 77 (2008),012334.
[35]C. Godsil, G. Royle, Algebraic Graph Theory, Springer-Verlag, New York,2001.
[36]X. Gao, W.W Liu, On the extendability of Semi-Cayley graphs of finite abelian groups, submitted to Discrete Math..
[37]G. Hetyei, Rectangular configurations which can be covered by 2 X 1 rectangles, Pecsi Tan. Foisk. Kozl.8 (1964),351-367.
[38]M. Hofmeister, Spectral radius and degreee sequence, Math. Noelr.139 (1988),37-44.
[39]Y. Hong, The k-th largest eigenvalue of a tree, Linear Algebra Appl.73 (1986),151-155.
[40]Y. Hong, Tree-width, clique-minors, and eigenvalues, Discrete Math.274 (2004),281-287.
[41]R.A. Horn, C.R. Johnson, Topics in Matrix analysis, Cambridge Univer-sity Press, Cambridge,1991.
[42]D.A. Holton, D.J. Lou, M.D. Plummer, On the 2-extendability of planar graphs, Discrete Math.96 (1991),81-99.
[43]P. Hic, R. Nedela, Balanced integral trees, Math. Slavaca 48 (1998), 429-445.
[44]F. Harary, A. Schwenk, Which graphs have integral spectra?, in:R. Bari, F. Harary (Eds.), Graphs and Combinatorics, Springer, Berlin,1974, pp. 45.
[45]Y. Hong, J.L. Shu, Sharp lower bounds of the least eigenvalue of planar graphs, Linear Algebra Appl.296 (1999),227-232.
[46]W. Imrich, S. Klavzar, Product Graphs, John Wiley& Sons, New York, 2000.
[47]G. Indulal, A. Vijayakumar, Some new integral graphs, Appl. Anal. Dis-crete Math.1 (2007),420-426.
[48]J.H. Kwak, Y.S. Kwon, J.M. On, Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency, Journal of Combi-natorial Theory, Ser. B 98 (2008),585-598.
[49]I. Kovacs a, A. Malnic, D. Marusic, S. Miklavic, One-matching bi-Cayley graphs over abelian groups, European J. Combin.30 (2009),602-616.
[50]K. Kutnar, D. Marusic,S. Miklavic, P. Sparl, Strongly regular tri-Cayley graphs, European J. Combin.30 (2009),822-832.
[51]C.H. Li, The finite vertex-primitive and vertex-biprimitive s-transitive graphs for s≥4, Trans. Amer. Math. Soc.353 (2001),3511-3529.
[52]C.H. Li, The finite primitive permutation groups containing an abelian regular subgroup, Proc. London Math. Soc.87 (2003),725-747.
[53]C.H. Li, Permutation groups with a cyclic regular subgroup and arc transitive circulants, J. Algebraic Combin.21 (2005),131-36.
[54]C.H.C. Little, D.D. Grant, D.A. Holton, On defect-d matching in graphs, Discrete Math.13 (1975),41-54.
[55]L. Lovasz, On the structure of factorizable graphs, Acta Math. Acad. Sci. Hunger.23 (1972),179-195.
[56]L. Lovaisz, Matching structure and the matching lattice, J. Combin. Theory Ser.(B) 43 (1987),187-222.
[57]L. Lovasz, M.D. Plummer, Matching Theory, North-Holland, Amster-dam, New York,1986.
[58]Z.P. Lu, On the Automorphism Groups of Bi-Cayley graphs. Acta.Sci.Natu. Universitatis Pekinensis 39(1) (2003),1-5.
[59]Z.P. Lu, C.Q. Wang, M.Y. Xu, Semisymmetric cubic graphs constructed from bi-Cayley graphs of An, Ars Combin.80 (2006),177-187.
[60]K.H. Leung, S.L. Ma, Partial difference triples, J. Algebr. Combin.2 (1993),397-409.
[61]D.J. Lou, Q.L. Yu, Connectivity of k-extendable graphs with large k, Discrete Applied Mathematics 136 (2004),55-61.
[62]D. Marusic, Strongly regular bicirculants and tricirculants, Ars Combin. 25 (1988),11-15.
[63]H. Minc, Nonnegative matrices Willey, New York,1988 (Chap2. pp.67).
[64]J. Meier, Groups, Graphs and Trees, Cambridge University Press, New York,2008.
[65]M. Muzychuk, On normal quotients of transitive graphs, Ars Combin. 59 (2001),171-180.
[66]M. Muzychuk, A solution of the isomorphism problem for circulant graphs, Proc. London Math. Soc.88 (2004) 1-41.
[67]B.D. McKay, M. Miller, J. Siran, A note on large graphs of diameter two and given maximum degree, Journal of Combinatorial Theory Ser. B 74 (1998),110-118.
[68]S. Miklavic, P. Potocnik, Distance-regular Cayley graphs on dihedral groups, Journal of Combinatorial Theory Ser. B 97 1(2007) 14-33.
[69]A. Malnic, D. Marusic, P. sparl, On strongly regular bicirculants, Euro-pean J. Combin.28 (2007),891-900.
[70]E. Nosal, Eigenvalues of graphs, Master's thesis, University of Calgary, 1970.
[71]D.L. Powers, Graph partitioning by eigenectors, Linear Algebra Appl. 101 (1988) 121-133.
[72]M.D. Plummer, Matching extension in bipartite graphs, Proceedings of the 17th Southeastern Conference on Combinatorics, Graph Theory and Computering Congress Numer.54, Utilitas Math., Winnipeg,1986 pp. 245-258.
[73]M.D. Plummer, On n-extendable graphs, Discrete Math.31 (1980) 201-210.
[74]M.D. Plummer, Matching extension and the genus of a graph, J. Combin. Theory Ser. B 44 (1988),329-337.
[75]M.D. Plummer, Extending matchings in graphs:A survey, Discrete Math.127 (1994),277-292.
[76]M.D. Plummer, Extending matchings in graphs:an update, Congr. Nu-mer.116 (1996),3-32.
[77]M.J.D. Resmi, D. Jungnickel, Strongly regular Semi-Cayley graphs, Journal of Algebraic Combinatorics 1 (1992),171-195.
[78]A.J. Schwenk, Almost all trees are cospectral, in:F. Harary (Ed.), New Directions in the Theory of Graphs, Academic Press, New York,1973.
[79]G. Sabidussi, On a class of fixed-point-free graphs, Proc. Amer. Math. Soc.9 (1958),800-804.
[80]R.A. Stone, On 1-factorizability of Cayley graphs, J. Combi. Theory, Ser. (B) 39 (1985),298-307.
[81]R.P. Stanley, Abound on the spectral radius of graphs with edges, Linear Algebra Appl.87 (1987),267-269.
[82]W. So, Integral circulant graphs, Discrete Math.306 (2005),153-158.
[83]D. Stevanovic, N.M.M. de Abreu, M.A.A. de Freitas, R. Del-Vecchio, Walks and regular integral graphs, Linear Algebra Appl.423 (2007), 119-135.
[84]G. Schrag, L. Cammack, On the 2-extendability of the generalized Pe-tersen graphs, Discrete Math.78 (1989),167-177.
[85]W.T. Tutte, A family of cubical graphs, proc. Camb. philos, soc.43 (1947),459-474.
[86]T.H. Hungerford, Algebra, Springer-Verlag, New York,1974.
[87]H.S. Wilf, The eigenvalues of a graph and its chromatic number, J. Lon-don Math. Soe,42 (1967),330-332.
L.Wang, X. Li, Integral trees with diameters 5 and 6, Discrete Math. 297 (2005),128-143.
[89]R.J. Wilson, Introduction to Graph Theory, third ed., Longman Inc., New York,1982.
[90]Q.L. Yu, Classifying two-extendable generalized Petersen graphs, Dis-crete Math.103 (1992),209-220.
[91]Q.L. Yu, A note on n-extendable graphs, J. Graph Theory 16 (1992), 349-353.
[92]Q.L. Yu, G.Z. Liu, Graph Factors and Matching Extensions, Higher Education Press,2009.
[93]X.L. Zhang, H.P. Zhang, Some graphs determined by their spectra, Lin-ear Algebra Appl.431 (2009),1443-1454.
[94]梁晓东,Bi-Cayley图与半传递图的连通性,博士学位论文,新疆大学.
[95]柳泊濂,组合矩阵,高等教育出版社,2005.
[96]王爱民,孟吉翔,有限交换群上Bi-Cayley图的Hamilton性,新疆大学学报(自然科学版),23(2)(2006),156-158.
[97]许英,阿贝尔群上的Cayley图的谱,硕士学位论文,新疆大学,2007.
[98]徐尚进,郭松涛,刘君,双Cayley图的Hamilton性,广西大学学报(自然科学版),34(2)2009,211-215.
[99]徐曜,有限群导引(第二版),科学出版社,1999.
[100]邹华,孟吉翔,Bi-Cayley图的一些代数性质,数学学报,50(5)(2007),1075-1080.