scholarly journals Quantum Walks on Regular Graphs and Eigenvalues

10.37236/652 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Chris Godsil ◽  
Krystal Guo
Keyword(s):  
Transition Matrix ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular

We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of $S^+(U^3)$, a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We find the eigenvalues of $S^+(U)$ and $S^+(U^2)$ for regular graphs and show that $S^+(U^2) = S^+(U)^2 + I$.

scholarly journals Quantum Walks on Generalized Quadrangles

10.37236/5982 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Chris Godsil ◽  
Krystal Guo ◽  
Tor G. J. Myklebust
Keyword(s):  
Transition Matrix ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Generalized Quadrangles ◽  
Intersection Graphs ◽  
Strongly Regular

We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of $S^+(U^3)$, a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We probabilistically compute the spectrum of the line intersection graphs of two non-isomorphic generalized quadrangles of order $(5^2,5)$ under this matrix and thus provide strongly regular counter-examples to the conjecture.

scholarly journals Uniform Mixing and Association Schemes

10.37236/4745 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Chris Godsil ◽  
Natalie Mullin ◽  
Aidan Roy
Keyword(s):  
Continuous Time ◽  
Hadamard Matrices ◽  
Quantum Walks ◽  
Association Schemes ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Paley Graph ◽  
Time Quantum ◽  
Strongly Regular ◽  
Distance Regular Graphs

We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing.First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph $X$ with $n$ vertices, then $n$ is divisible by four. We also prove that if $X$ is bipartite and regular, then $n$ is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on $C_{2m}$ for $m \geq 3$. Using a result of Haagerup, we show that uniform mixing does not occur on $C_p$ for any prime $p$ such that $p \geq 5$. In contrast to this result, we see that $\epsilon$-uniform mixing occurs on $C_p$ for all primes $p$.

On extensions of small strongly regular graphs with eigenvalue 3

2015 ◽  
Vol 92 (1) ◽  
pp. 482-486
Author(s):  
A. A. Makhnev ◽  
D. V. Paduchikh
Keyword(s):  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular

On triple systems and strongly regular graphs

2012 ◽  
Vol 119 (7) ◽  
pp. 1414-1426 ◽  
Author(s):  
Majid Behbahani ◽  
Clement Lam ◽  
Patric R.J. Östergård
Keyword(s):  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Triple Systems ◽  
Strongly Regular
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