scholarly journals Parallelogram-free distance-regular graphs having completely regular strongly regular subgraphs

2009 ◽  
Vol 30 (3) ◽  
pp. 401-413 ◽  
Author(s):  
Hiroshi Suzuki
Keyword(s):  
Regular Graphs ◽  
Completely Regular ◽  
Free Distance ◽  
Strongly Regular ◽  
Distance Regular Graphs

scholarly journals Families of nested completely regular codes and distance-regular graphs

2015 ◽  
Vol 9 (2) ◽  
pp. 233-246 ◽  
Author(s):  
Joaquim Borges ◽  
◽  
Josep Rifà ◽  
Victor A. Zinoviev ◽  
◽  
...  
Keyword(s):  
Regular Graphs ◽  
Completely Regular ◽  
Distance Regular Graphs ◽  
Regular Codes ◽  
Completely Regular Codes

scholarly journals Some Families of Orthogonal Polynomials of a Discrete Variable and their Applications to Graphs and Codes

10.37236/172 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
M. Cámara ◽  
J. Fàbrega ◽  
M. A. Fiol ◽  
E. Garriga
Keyword(s):  
Orthogonal Polynomials ◽  
Regular Graphs ◽  
Discrete Variable ◽  
Completely Regular ◽  
Local Distance ◽  
Distance Regular Graphs ◽  
Regular Codes ◽  
Completely Regular Codes ◽  
Orthogonal Systems ◽  
Spectral Characterizations

We present some related families of orthogonal polynomials of a discrete variable and survey some of their applications in the study of (distance-regular) graphs and (completely regular) codes. One of the main peculiarities of such orthogonal systems is their non-standard normalization condition, requiring that the square norm of each polynomial must equal its value at a given point of the mesh. For instance, when they are defined from the spectrum of a graph, one of these families is the system of the predistance polynomials which, in the case of distance-regular graphs, turns out to be the sequence of distance polynomials. The applications range from (quasi-spectral) characterizations of distance-regular graphs, walk-regular graphs, local distance-regularity and completely regular codes, to some results on representation theory.

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 Triangle- and pentagon-free distance-regular graphs with an eigenvalue multiplicity equal to the valency

2005 ◽  
Vol 94 (2) ◽  
pp. 245-258 ◽  
Author(s):  
Aleksandar Jurišić ◽  
Jack Koolen ◽  
Štefko Miklavič
Keyword(s):  
Regular Graphs ◽  
Free Distance ◽  
Distance Regular Graphs

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 Kite-free distance-regular graphs

1995 ◽  
Vol 16 (4) ◽  
pp. 405-414 ◽  
Author(s):  
Paul Terwilliger
Keyword(s):  
Regular Graphs ◽  
Free Distance ◽  
Distance Regular Graphs
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