The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues
Let G be a distance-regular graph with diameter d and Kneser graph K=Gd, the distance-d graph of G. We say that G is partially antipodal when K has fewer distinct eigenvalues than G. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the...
| Autor: | |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2015 |
| 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:2117/85047 |
| Acceso en línea: | https://hdl.handle.net/2117/85047 https://dx.doi.org/10.1016/j.endm.2015.06.064 |
| Access Level: | acceso abierto |
| Palabra clave: | Graph theory Combinatorial analysis Distance-regular graph Kneser graph Partial antipodality Predistance polynomials Spectrum Teoria de grafs Combinatòria Classificació AMS::05 Combinatorics::05C Graph theory Classificació AMS::05 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: | Let G be a distance-regular graph with diameter d and Kneser graph K=Gd, the distance-d graph of G. We say that G is partially antipodal when K has fewer distinct eigenvalues than G. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with d distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular. |
|---|