Graphs with integral spectrum

Abstract It is shown that only a fraction of 2 - Ω ( n ) of the graphs on n vertices have an integral spectrum. Although there are several explicit constructions of such graphs, no upper bound for their number has been known. Graphs of this type play an important role in quantum networks supporting the so-called perfect state transfer.

[1]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[2]  Willem H. Haemers,et al.  The Integral Trees With Spectral Radius 3 , 2007 .

[3]  Mirko Lepovic.  On integral graphs which belong to the class αKa,a,...,a,b,b,...,b , 2006 .

[4]  Andries E. Brouwer,et al.  A new infinite series of regular uniformly geodetic code graphs , 1993, Discret. Math..

[5]  Matthias Christandl,et al.  Perfect Transfer of Arbitrary States in Quantum Spin Networks , 2005 .

[6]  Jason Twamley,et al.  Quantum Cayley networks of the hypercube , 2008 .

[7]  G. Indulal,et al.  SOME NEW INTEGRAL GRAPHS , 2007 .

[8]  Michael Doob,et al.  Spectra of graphs , 1980 .

[9]  N. Alon,et al.  On the concentration of eigenvalues of random symmetric matrices , 2000, math-ph/0009032.

[11]  Wasin So,et al.  Integral circulant graphs , 2006, Discret. Math..

[12]  Andries E. Brouwer,et al.  Small Integral Trees , 2008, Electron. J. Comb..

[13]  Christino Tamon,et al.  On mixing in continuous-time quantum walks on some circulant graphs , 2003, Quantum Inf. Comput..

[14]  Xueliang Li,et al.  Integral trees with diameters 5 and 6 , 2005, Discret. Math..

[15]  Mirko Lepovic.  On integral graphs which belong to the class alphaKa cup betaKb, b , 2004, Discret. Math..

[16]  Chris D. Godsil,et al.  Periodic Graphs , 2008, Electron. J. Comb..

[17]  T. Tao,et al.  RANDOM MATRICES: THE CIRCULAR LAW , 2007, 0708.2895.

[18]  T. Tao,et al.  On the singularity probability of random Bernoulli matrices , 2005, math/0501313.

[19]  W. Haemers,et al.  Which graphs are determined by their spectrum , 2003 .

[20]  T. Tao,et al.  On random ±1 matrices: Singularity and determinant , 2006 .

[21]  Terence Tao,et al.  The condition number of a randomly perturbed matrix , 2007, STOC '07.

[22]  Xueliang Li,et al.  Two Classes of Integral Regular Graphs , 2005, Ars Comb..

[23]  János Komlós,et al.  The eigenvalues of random symmetric matrices , 1981, Comb..

[24]  Maria Aguieiras A. de Freitas,et al.  Walks and regular integral graphs , 2007 .

[25]  Matthias Christandl,et al.  Perfect state transfer in quantum spin networks. , 2004, Physical review letters.

[26]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

[27]  Hou Yao-ping.  On Integral Graphs which Belong to the Class , 2008 .

[28]  S. Perlis,et al.  Theory of Matrices , 1953 .

[29]  Simone Severini,et al.  Parameters of Integral Circulant Graphs and Periodic Quantum Dynamics , 2007 .

[30]  Dragoš Cvetković,et al.  A SURVEY ON INTEGRAL GRAPHS , 2002 .

[31]  Kevin P. Costello,et al.  Random symmetric matrices are almost surely nonsingular , 2005, math/0505156.

[32]  Frank Harary,et al.  Which graphs have integral spectra , 1974 .