Timeline for Examples of Binary Functions that Yield Regular Graphs with Invertible Adjacency Matrix
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 23, 2018 at 16:03 | comment | added | Puck Rombach | In that case I think you should rephrase the question here to include explicitly the properties that you need your matrix to have. | |
| Dec 23, 2018 at 15:46 | comment | added | Manfred Weis | @PuckRombach I guess I have already mentioned that these cyclically wrapped band matrices don't always work. See my question mathoverflow.net/questions/318761/… take $k=2$ and $n$ even | |
| Dec 23, 2018 at 15:27 | comment | added | Puck Rombach | Does $f(i,j)=\begin{cases} 1,\;\; \mbox{ if } 1\leq i-j \pmod{n} \leq k,\\ 0,\;\; \mbox{otherwise,} \end{cases}$ do the trick? | |
| Dec 23, 2018 at 15:24 | comment | added | Manfred Weis | @PuckRombach Initially I would be happy with a function that works for $k=2$ and arbitrary $n\ge 4$ | |
| Dec 23, 2018 at 15:21 | comment | added | Puck Rombach | It would be $k=n-1$, but we can also let every column have $k$ 1s starting below the diagonal, for example. Isn't the matrix equivalent to the kind of function you want? | |
| Dec 23, 2018 at 15:21 | history | edited | Manfred Weis | CC BY-SA 4.0 |
clarified the question in reply to @PuckRombach's comment
|
| Dec 23, 2018 at 15:03 | comment | added | Manfred Weis | @PuckRombach your first example amounts to $k=n$, which implies a dependency of $k$ on $n$, a case that I thought I had ruled out; I will edit accordingly. To your second question: I am not so much interested in the number of such matrices or in concrete examples of such matrices, but rather in functions that yield such matrices. | |
| Dec 23, 2018 at 14:05 | comment | added | Puck Rombach | These exist for every $n$: if you take a matrix that has 0s on the diagonal and 1s everywhere else it is invertible. Does your question come down to: how many matrices in $GL(n,\mathbb{Z}_2)$ have constant row/column sums and 0s on the diagonal? | |
| Dec 23, 2018 at 12:02 | history | asked | Manfred Weis | CC BY-SA 4.0 |