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 G be a graph with vertex set V , diameter D, adjacency matrix A, and adjacency algebra A. Then, G is distance mean-regular when, for a given u ¿ V...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2016 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/91052 |
| Acceso en línea: | https://hdl.handle.net/2117/91052 https://dx.doi.org/10.1007/s10623-016-0208-5 |
| Access Level: | acceso abierto |
| Palabra clave: | Graph theory Combinatorial analysis Distance-regular graph Vertex-transitive graph Distance mean-regular graph Intersection mean-matrix Adjacency Algebra Spectrum Interlacing Grafs, Teoria de Combinacions (Matemàtica) Classificació AMS::05 Combinatorics::05C Graph theory Classificació AMS::05 Combinatorics::05E Algebraic combinatorics Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria |
| Sumario: | 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 G be a graph with vertex set V , diameter D, adjacency matrix A, and adjacency algebra A. Then, G is distance mean-regular when, for a given u ¿ V , the averages of the intersection numbers p h ij (u, v) = |Gi(u) n Gj (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 G and, hence, they generate a subalgebra of A. Some other algebras associated to distance mean-regular graphs are also characterized. |
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