All Questions
Tagged with algebraic-combinatorics algebraic-graph-theory
22
questions
3
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answers
42
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Exercise 2.2 In Stanley's Algebraic Combinatorics
This is Exercise $2.2$ In Stanley's Algebraic Combinatorics. I don't have much work to show because despite being stuck on this problem for a long time, I haven't got a clue how to start.
$\mathcal{C}...
- 3,100
0
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0
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23
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Affiness, $U_{2,4}$ and $M(K_4).$
I do not know why $M(K_4)$ is not affine over $GF(2)$ or $GF(3)$ but it is affine over all fields with more than 3 elements. I proved that $U_{2,4}$ is $\mathbb F$-representable iff $|\mathbb F| \geq ...
- 3,139
0
votes
0
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20
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Lift and frame matroids.
I want to read more about lift matroid and frame matroid and their flats and relations to signed graphs, do you know any basic resources for this?
- 2,087
0
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90
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Hyperplane Areangements and contraction.
I am trying to understand an idea presented in McNulty book, matriods a geometric introduction about the new hyperplane arrangement $\mathcal{A}^{''} = \{ H \cap H_x | H \in \mathcal{A}\}$ where $\...
- 3,139
0
votes
1
answer
46
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The basis of a regular matroid.
I know that a regular matroid is one that can be represented by a totally unimodular matrix.
I also know that a rank r totally unimodular matrix is a matrix over $\mathbb R$ for which every submatrix ...
- 3,139
1
vote
1
answer
56
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affine geometries that are self-dual matroids.
I want to know which of the affine geometries $AG(n,q)$ are self-dual matroids?
I have proved before that uniform matroids $U_{n,m}$ that are self duals are those who satisfies that $m = 2n.$ I am ...
- 3,139
3
votes
1
answer
101
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Understanding how to find the dual of a matroid.
I am trying to understand the following picture of a matroid and its dual but I found myself not understanding exactly what we are doing to find the dual:
]1
Roughly speaking, according to some ...
- 95
2
votes
1
answer
94
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Show that $C^*(e,B)$ is the unique cocircuit that is disjoint from $B - e.$
I want to prove the following question:
Let $B$ be a basis of a matroid $M.$ If $e \in B,$ denote $C_{M^*}(e, E(M) - B)$ by $C^*(e,B)$ and call it the fundamental cocircuit of $e$ with respect to $B.$\...
- 3,139
0
votes
1
answer
68
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if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis.
here is the question I am trying to solve:
In a matroid $M,$ if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis.
I know how to prove that a set ...
- 3,139
0
votes
1
answer
36
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Let $A\subset B$ be flats of a matroid $M$ such that $r(A)=r(B)<\infty$. Then $A = B$.
I want to prove the following lemma:
Let $r$ denotes the rank.
Lemma. Let $A\subset B$ be any flats of a matroid $M$ such that $r(A)=r(B)<\infty$. Then $A = B$.
My thoughts are:
I know that $cl(A) =...
- 3,139
0
votes
1
answer
82
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Proving that $ \beta(M) = \beta (M - e) + \beta (M /e).$
Here is the statement I am trying to prove:
If $e \in E$ is neither a loop nor an isthmus, then $$ \beta(M) = \beta (M - e) + \beta (M /e).$$
Here are all the properties I know about the Crapo's beta ...
- 3,139
0
votes
1
answer
71
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Why always the Crapo beta invariant value greater than or equal zero?
Here are the definitions of the Crapo beta invariant I know:
My definition of the Crapo's beta invariant of a matroid from the book "Combinatorial Geometries" from page 123 and 124 is as ...
- 3,139
2
votes
1
answer
69
views
Sum of squares of chromatic roots of a bipartite graph
Given a graph $G = (V, E)$, we can calculate its chromatic polynomial $P(G, k)$, and it has $n$ (complex) roots, also known as chromatic roots. It is a well-known fact that the sum of chromatic roots ...
- 77
2
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47
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Can we characterize the “associate classes” of a unipotent quasi-commutative quasigroup as some combinatorial design?
$I_n$ is the $n \times n$ or order $n$ identity matrix, $J_n$ is the order $n$ matrix of all ones, and $n \in \mathbb{Z}^+$.
We define a Latin square $\mathcal{L_n}$ to be a set of $n$ permutation ...
- 317
2
votes
2
answers
90
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In a primitive symmetric association scheme, why does $E_j$ occur in some power of $E_i$ for each $i,j$?
I am having some trouble in the proof of the Absolute Bound Condition for primitive symmetric association Schemes in the book Algebraic Combinatorics I by Bannai and Ito (Chapter 2, Section 4, Theorem ...
- 508