Timeline for Graphs which are "distance-regular" with respect to a vertex (but not distance-regular)
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 7, 2014 at 15:32 | vote | accept | Ken W. Smith | ||
| Jun 7, 2014 at 15:31 | answer | added | Ken W. Smith | timeline score: 1 | |
| Mar 26, 2012 at 14:33 | comment | added | Ken W. Smith | A follow-up -- much of the theory I need seems to be collected in section 9.3 (Equitable Partitions) of Algebraic Graph Theory by Chris Godsil and Gordon Royle (of course!) -- which I own and is sitting on my desk! I thank you both and I apologize for not putting these ideas together.... | |
| Mar 26, 2012 at 12:38 | comment | added | Ken W. Smith | Thanks to both of you! "Equitable partition" is the key idea I am after, I think. I'll go ahead and get the JCT(B) article through interlibrary loan. | |
| Mar 26, 2012 at 3:02 | comment | added | Chris Godsil | @Ken: If you have an equitable partition and replace the subgraph induced by one cell with some other regular graph of the same valency, then your partition is still equitable. So the eigenvalues of the quotient are eigenvalues of the modified graph. Your example with the Petersen graph is this operation applied to the distance partition with respect to a vertex. This explains the divisibility by the minimal polynomial in your last paragraph. And the result that Gordon cites is also due (independently) to Delorme. | |
| Mar 26, 2012 at 1:00 | comment | added | Gordon Royle | Not an answer, but there has been some work done on graphs that are distance-regular from EVERY vertex. Godsil and Shawe-Taylor show that these are either distance-regular (as usual) or distance-biregular (bipartite with each side of the bipartition having its own set of intersection numbers.) See sciencedirect.com/science/article/pii/009589568790027X | |
| Mar 25, 2012 at 19:30 | history | asked | Ken W. Smith | CC BY-SA 3.0 |