Name
Abstract | ||
|---|---|---|
Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study 'almost distance-regular graphs'. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called m-walk-regularity. Another studied concept is that of m-partial distance-regularity or, informally, distance-regularity up to distance m. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of (@?,m)-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.1016/j.jcta.2010.10.005 | Journal of Stroke & Cerebrovascular Diseases |
Keywords | DocType | Volume |
key concept,punctual distance-regularity,predistance polynomial,eigenvalues,walk-regular graph,distance-regular graph,m-partial distance-regularity,punctually distance-regular graph,symmetric cubic graph,common generalization,spectral excess theorem,punctual walk-regularity,whole distance-regular,local multiplicities,association scheme,distance regular graph,algebraic combinatorics,cubic graph,regular graph | Journal | 118 |
Issue | ISSN | Citations |
3 | Journal of Combinatorial Theory, Series A | 10 |
PageRank | References | Authors |
0.86 | 11 | 5 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Cristina Dalfó | 1 | 46 | 9.47 |
| E. R. van Dam | 2 | 50 | 8.37 |
| M. A. Fiol | 3 | 816 | 87.28 |
| E. Garriga | 4 | 164 | 19.92 |
| Bram L. Gorissen | 5 | 42 | 3.22 |