Strongly regular and distance-regular graphs

2016 ◽  
pp. 39-60
Author(s):  
Bart De Bruyn
Keyword(s):  
Regular Graphs ◽  
Strongly Regular ◽  
Distance Regular Graphs

scholarly journals Using Algebraic Properties of Minimal Idempotents for Exhaustive Computer Generation of Association Schemes

10.37236/754 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
K. Coolsaet ◽  
J. Degraer
Keyword(s):  
Positive Semidefinite ◽  
Association Schemes ◽  
Regular Graphs ◽  
Fano Plane ◽  
A Value ◽  
First Case ◽  
Strongly Regular ◽  
Computer Classification ◽  
Distance Regular Graphs ◽  
Algebraic Constraints

During the past few years we have obtained several new computer classification results on association schemes and in particular distance regular and strongly regular graphs. Central to our success is the use of two algebraic constraints based on properties of the minimal idempotents $E_{i}$ of these association schemes : the fact that they are positive semidefinite and that they have known rank. Incorporating these constraints into an actual isomorph-free exhaustive generation algorithm turns out to be somewhat complicated in practice. The main problem to be solved is that of numerical inaccuracy: we do not want to discard a potential solution because a value which is close to zero is misinterpreted as being negative (in the first case) or nonzero (in the second). In this paper we give details on how this can be accomplished and also list some new classification results that have been recently obtained using this technique: the uniqueness of the strongly regular $(126,50,13,24)$ graph and some new examples of antipodal distance regular graphs. We give an explicit description of a new antipodal distance regular $3$-cover of $K_{14}$, with vertices that can be represented as ordered triples of collinear points of the Fano plane.

scholarly journals On distance-regular graphs in which neighborhoods of vertices are strongly regular

2013 ◽  
Vol 88 (2) ◽  
pp. 532-536
Author(s):  
A. L. Gavrilyuk ◽  
A. A. Makhnev ◽  
D. V. Paduchikh
Keyword(s):  
Regular Graphs ◽  
Strongly Regular ◽  
Distance Regular Graphs

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 $p$.

scholarly journals Quasi-Spectral Characterization of Strongly Distance-Regular Graphs

10.37236/1529 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
M. A. Fiol
Keyword(s):  
Harmonic Mean ◽  
Spectral Characterization ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Distance Regular Graphs

A graph $\Gamma$ with diameter $d$ is strongly distance-regular if $\Gamma$ is distance-regular and its distance-$d$ graph $\Gamma _d$ is strongly regular. The known examples are all the connected strongly regular graphs (i.e. $d=2$), all the antipodal distance-regular graphs, and some distance-regular graphs with diameter $d=3$. The main result in this paper is a characterization of these graphs (among regular graphs with $d$ distinct eigenvalues), in terms of the eigenvalues, the sum of the multiplicities corresponding to the eigenvalues with (non-zero) even subindex, and the harmonic mean of the degrees of the distance-$d$ graph.

Some Spectral Characterizations of Strongly Distance-Regular Graphs

2001 ◽  
Vol 10 (2) ◽  
pp. 127-135 ◽  
Author(s):  
M. A. FIOL
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Strongly Regular ◽  
Distance Regular Graphs ◽  
Spectral Characterizations

A graph Γ with diameter d is strongly distance-regular if Γ is distance-regular and its distance-d graph Γd is strongly regular. Some known examples of such graphs are the connected strongly regular graphs, with distance-d graph Γd = Γ (the complement of Γ), and the antipodal distance-regular graphs. Here we study some spectral conditions for a (regular or distance-regular) graph to be strongly distance-regular. In particular, for the case d = 3 the following characterization is proved. A regular (connected) graph Γ, with distinct eigenvalues λ0 > λ1 > λ2 > λ3, is strongly distance-regular if and only if λ2 = −1, and Γ3 is k-regular with degree k satisfying an expression which depends only on the order and the different eigenvalues of Γ.

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