On the spectrum of Cayley graphs via character table

Document Type : Original Article

Authors

Department of Mathematics, Faculty of Science, Shahid RajaeeTeacher Training University, Tehran, 16785 – 136, I R. Iran

Abstract

A Cayley graph can be consider as an interpretation of group representation. This means that one can construct the Cayley graph Г = Cay(G, S) by having the vertex set to be the elements of G with g~h if and only if hg-1 ε S. In this paper, we focus our attention on Cayley graphs with a normal symmetric subset. We also determine all eigenvalues of Buckminster fullerene by constructing Cay(G,S) where G is the alternating group A5.

Graphical Abstract

On the spectrum of Cayley graphs via character table

Keywords


[1] A. R. Abdollahi, A. Loghman, Cayley graphs isomorphic to the product of two Cayley graphs, Ars Combin., in press.
[2] A. R. Ashrafi, M. Hamadanian, The full non-rigid group theory for tetra ammine platinum(II), Croatica Chemica Acta, 76 (2003) 299-303.
[3] N. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974.
[4] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs-Theory and Application, Deutscher Verlag der Wissenschaften, Berlin, Academic Press, third edition, Johann Ambrosius Barth Verlag, 1995.
[5] M. DeVos, L. Goddyn, B. Mohar, R. Šámal, Cayley sum graphs and eigenvalues of (3,6)-fullerenes, J. Comb. Theory Series B, 99 (2009) 358-369.
[6] P. Diaconis, M. Shahshahani, Generating a random permutation with random transposition, Zeit. für Wahrscheinlichkeitstheorie verw. Gebiete, 57 (1981) 159-179.
[7] M. Deza, P. W Fowler, A. Rassat, K. M Rogers, Fullerenes as tilings of surfaces, J. Chem. Inf. Comp. Sci., 40 (2000) 550–558.
[8] C. Domb, On the theory of cooperative phenomena in crystals, Adv. in Phys., 9 (1960) 149–361.
[9] E. Estrada, Characterization of 3D molecular structure, Chem. Phys. Lett., 319 (2000) 713-718.
[10] I. Gutman, The energy of a graph, Ber. Math.Statist. Sekt. Forschungsz. Graz, 103 (1978) 1-22.
[11] M. Ghorbani, E. Bani-Asadi, Remarks on characteristic coefficients of fullerene graphs, Appl. Math.Comput., 230 (2014) 428–435.
[12] M. Ghorbani, E. Naserpour, The Clar Number of Fullerene C24n and Carbon Nanocone CNC4[n] , Iranian J. Math. Chem., 1 (2011) 53-59.
[13] A. Ilić, The energy of unitary cayley graphs, Linear Algebra and its Applications, 431 (2009) 1881–1889.
[14] H. N. Ramaswamy, C. R. Veena, On the energy of unitary Cayley graphs, Electron. J. Combin. 16 (2009) #N24.
[15] G. James, M. Liebeck, Representation and characters of groups, Cambridge University Press, Cambridge, 1993.
[16] H. W. Kroto, J. E. Fichier, D. E Cox, The Fulerene, Pergamon Press, New York, 1993.
[17] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, R.E. Smalley, Nature, 318 (1985) 162.
[18] X. Meng, Q. Huang, Z. Zhang, Fullerenes which are Cayley Graphs, MATCH Commun. Math. Comput. Chem., 56 (2006) 493-500.
[19] D. E Manolopoulos, D. R Woodall, P.W. Fowler, Electronic stability of fullerenes eigenvalue theorems for leapfrog carbon clusters, J. Chem. Soc. Faraday Trans., 88 (1992) 2427–2435.
[20] M. Schönert, H. U. Besche, Th. Breuer, F. Celler, B.Eick, V. Felsch, A. Hulpke, J. Mnich, W. Nickel, G. Pfeiffer, U. Polis, H. Theißen, A. Niemeyer, GAP, Groups, Algorithms and Programming, Lehrstuhl De für Mathematik, RWTH, Aachen, 1995.
Volume 4, 1-2
June 2014
Pages 1-11
  • Receive Date: 06 July 2013
  • Revise Date: 11 February 2014
  • Accept Date: 01 June 2014
  • Publish Date: 01 June 2014