All Questions
7 questions
26
votes
2
answers
2k
views
Function of $(x_1,x_2,x_3,x_4)$ that factors in two ways as $\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$
Suppose we have a function $f(x_1 ,x_2 ,x_3 ,x_4).$ We know that we can factor it in two ways as $f(x_1 ,x_2 ,x_3 ,x_4)=\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$
Show that ...
- 519
6
votes
2
answers
237
views
Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$
This question relates to Realizing groups as automorphism groups of graphs.
Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?
- 48.8k
6
votes
3
answers
854
views
Which graphs are zero-divisor graphs for some ring?
Given a (non commutative) ring $R$, we construct a (directed) graph $G_0(R)$ with vertex set $Z(R)\backslash \{0\}$, the zero divisors of $R$ except for $0$. And an edge from $x$ to $y$ whenever $xy=0$...
- 85.5k
6
votes
1
answer
135
views
Automorphisms of special egg-box diagrams
By a egg-box diagram I will simply mean a (possibly infinite) rectangular array of holes, with some of the holes containing an egg (denoted by a filled-in circle) and the rest of the holes are empty (...
- 18.7k
5
votes
1
answer
564
views
When is the adjacency algebra of a graph an association scheme?
The adjacency algebra of a graph is the algebra consisting of all polynomials in the adjacency matrix of the graph. An association scheme is a commutative matrix algebra containing the identity and ...
- 2,339
3
votes
0
answers
170
views
Question about circulants and association schemes
Suppose $X$ and $Y$ are two $n$-circulants (Cayley graphs for $\mathbb{Z}_n$) with adjacency matrices $A_X$ and $A_Y$. Since they are circulants, both $X$ and $Y$ lie in some symmetric association ...
- 2,339
2
votes
0
answers
102
views
When do two path algebras share an underlying graph?
Suppose $Q$ and $Q'$ are two quivers. I am curious as to what relation $\mathbb{C}Q$ bears to $\mathbb{C}Q'$ when $Q$ and $Q'$ share the same underlying graph and only differ by direction.
Since ...
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