scholarly journals A general method to obtain the spectrum and local spectra of a graph from its regular partitions

2020 ◽  
Vol 36 (36) ◽  
pp. 446-460
Author(s):  
Cristina Dalfó ◽  
Miquel Àngel Fiol
Keyword(s):  
Adjacency Matrix ◽  
Regular Partition ◽  
Quotient Graph ◽  
Completely Regular ◽  
General Part ◽  
Regular Codes ◽  
Completely Regular Codes ◽  
Regular Partitions ◽  
General Method

It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, a method that gives all the spectrum, and also the local spectra, of a graph from the quotient matrices of some of its regular partitions, is proposed. Moreover, from such partitions, the $C$-local multiplicities of any class of vertices $C$ is also determined, and some applications of these parameters in the characterization of completely regular codes and their inner distributions are described. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.  

scholarly journals Arithmetic completely regular codes

2016 ◽  
Vol Vol. 17 no. 3 (PRIMA 2013) ◽  
Author(s):  
Jacobus Koolen ◽  
Woo Sun Lee ◽  
William Martin ◽  
Hajime Tanaka
Keyword(s):  
Cartesian Products ◽  
Regular Partition ◽  
Completely Regular ◽  
Completely Regular Code ◽  
Open Questions ◽  
Hamming Graphs ◽  
International Audience ◽  
Distance Regular Graphs ◽  
Regular Codes ◽  
Completely Regular Codes

International audience In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.

scholarly journals Combinatorial vs. Algebraic Characterizations of Completely Pseudo-Regular Codes

10.37236/309 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
M. Cámara ◽  
J. Fàbrega ◽  
M. A. Fiol ◽  
E. Garriga
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Regular Graphs ◽  
Completely Regular ◽  
Completely Regular Code ◽  
Positive Eigenvector ◽  
Vertex Set ◽  
Regular Codes ◽  
Simple Connected Graph

Given a simple connected graph $\Gamma$ and a subset of its vertices $C$, the pseudo-distance-regularity around $C$ generalizes, for not necessarily regular graphs, the notion of completely regular code. We then say that $C$ is a completely pseudo-regular code. Up to now, most of the characterizations of pseudo-distance-regularity has been derived from a combinatorial definition. In this paper we propose an algebraic (Terwilliger-like) approach to this notion, showing its equivalence with the combinatorial one. This allows us to give new proofs of known results, and also to obtain new characterizations which do not depend on the so-called $C$-spectrum of $\Gamma$, but only on the positive eigenvector of its adjacency matrix. Along the way, we also obtain some new results relating the local spectra of a vertex set and its antipodal. As a consequence of our study, we obtain a new characterization of a completely regular code $C$, in terms of the number of walks in $\Gamma$ with an endvertex in $C$.

scholarly journals Erratum to: On Completely Regular Codes

2019 ◽  
Vol 55 (3) ◽  
pp. 298-298
Author(s):  
J. Borges ◽  
J. Rifà ◽  
V. A. Zinoviev
Keyword(s):  
Completely Regular ◽  
Regular Codes ◽  
Completely Regular Codes

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 Dual distances of completely regular codes

1991 ◽  
Vol 89 (1) ◽  
pp. 7-15 ◽  
Author(s):  
B. Courteau ◽  
A. Montpetit
Keyword(s):  
Completely Regular ◽  
Regular Codes ◽  
Completely Regular Codes
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