Distance mean-regular graphs

We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let Γ be a graph with vertex set V , diameter D , adjacency matrix A , and adjacency algebra A . Then, Γ is distance mean-regular when, for a given u...

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Bibliographic Details
Published in:Designs, codes, and cryptography Vol. 84; no. 1-2; pp. 55 - 71
Main Authors: Diego, V., Fiol, M. A.
Format: Journal Article Publication
Language:English
Published: New York Springer US 01.07.2017
Springer Nature B.V
Subjects:
05 Combinatorics
05C Graph theory
05E Algebraic combinatorics
Adjacency Algebra
Algebra
Circuits
Classificació AMS
Coding and Information Theory
Combinacions (Matemàtica)
Combinatorial analysis
Combinatòria
Computation
Computer Science
Cryptology
Data Structures and Information Theory
Discrete Mathematics in Computer Science
Distance mean-regular graph
Distance-regular graph
Grafs, Teoria de
Graph theory
Graphs
Information and Communication
Interlacing
Intersection mean-matrix
Matemàtica discreta
Matemàtiques i estadística
Spectrum
Teoria de grafs
Vertex-transitive graph
Àrees temàtiques de la UPC
05E30
Vertex-transitive graph
05C50
Interlacing
Adjacency Algebra
Distance-regular graph
Intersection mean-matrix
Distance mean-regular graph
Spectrum
ISSN:0925-1022, 1573-7586
Online Access:Get full text
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Summary:We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let Γ be a graph with vertex set V , diameter D , adjacency matrix A , and adjacency algebra A . Then, Γ is distance mean-regular when, for a given u ∈ V , the averages of the intersection numbers p i j h ( u , v ) = | Γ i ( u ) ∩ Γ j ( v ) | (number of vertices at distance i from u and distance j from v ) computed over all vertices v at a given distance h ∈ { 0 , 1 , … , D } from u , do not depend on u . In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of Γ and, hence, they generate a subalgebra of A . Some other algebras associated to distance mean-regular graphs are also characterized.
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ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-016-0208-5