Timeline for Is this Graph Iteration Already Known?
Current License: CC BY-SA 4.0
9 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Jun 10, 2018 at 7:04 | comment | added | Manfred Weis | @GerhardPaseman the motivation for the "useless" $\omega^0$ is to have a recursion basis for the even-odd alternation between vertex weights and edge weights and apart from that, it allows on to stay within the number field at hand. In a more general setting the $\omega^0$ would be the neutral element of addition and $\omega^1$ the unit of addition, but that is owed only to formal beauty. | |
| Jun 10, 2018 at 6:55 | comment | added | Manfred Weis | @GerhardPaseman thank you for the interesting feedback. That gives me some directions for further search. | |
| Jun 10, 2018 at 6:52 | history | edited | Manfred Weis | CC BY-SA 4.0 |
added a caveat about the practical applicability and corrected a wrong claim about the number of constraints
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| Jun 9, 2018 at 23:11 | comment | added | Gerhard Paseman | Finally, approaching but not answering your question, the edge version of your matrix is (E + 2I), which is a square matrix indexed by pairs of edges, with the sum having 2's on the diagonal, 1's when two different edges share a vertex, and 0 otherwise. This may relate to dual graphs or line graphs, but I am guessing here. However, behaviours of powers of matrices like (E+2I) have been analyzed, and suggest to me the topic of incidence structures for your situation. Gerhard "Hopefully Those Are Enough Guesses" Paseman, 2018.06.09. | |
| Jun 9, 2018 at 23:00 | comment | added | Gerhard Paseman | Indeed, if you change w0 to be the vector with coefficients 1/2 instead of 0, you can start the recursion from this constant nonzero vector. Gerhard "Not Seeing W0 Used Otherwise" Paseman, 2018.06.09. | |
| Jun 9, 2018 at 22:56 | comment | added | Gerhard Paseman | Upon further consideration, one of the products above should be the adjacency matrix of the (I assume simple loopless and undirected) graph plus a diagonal matrix D with the jth diagonal entry the degree of vertex v_j. So the w2k functions on the vertices considered as vectors should be powers of (A+D) on the vector of vertex degrees, and w2k+1 easily derived from these. I know nothing about graph Laplacians or similar constructs derived from adjacency matrices, but I would look through that literature for connections to your dynamic. Gerhard "Did Not Leave It Alone" Paseman, 2018.06.09. | |
| Jun 9, 2018 at 21:04 | comment | added | Gerhard Paseman | If you represent the omega functions by vectors, and call by A an edge-vertex incidence matrix, and B its transpose, then each function is of a power of BA (or AB) times either a unit vector or vector of degrees of the appropriate length. In this formulation I imagine it has been explored, but I have no references for you. Gerhard "Leaves The Analysis To You" Paseman, 2018.06.09. | |
| Jun 9, 2018 at 17:17 | history | asked | Manfred Weis | CC BY-SA 4.0 |