All Questions
Tagged with algebraic-combinatorics graph-theory
50
questions
1
vote
0
answers
30
views
Exercise 11.1 in Algebraic Combinatorics by Stanley
Let $\mathcal{C}_{D} = \mathcal{C}$ denote the cycle space, or the space of all circulations, on some digraph $D$. Let $C_n$ denote the $n$-dimensional hypercube, and $\mathfrak{o}$ denote some ...
3
votes
0
answers
42
views
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
votes
1
answer
36
views
Exercise 9.6 in Algebraic Combinatorics by Stanley
Exercise 6 in chapter 9 of Algebraic Combinatorics by Stanley: Let $G$ be a finite graph on $p$ vertices with Laplacian matrix $L(G)$. Let $G'$ be obtained from $G$ by adding a new vertex $v$ and ...
1
vote
1
answer
38
views
Exercise 7.2 in Algebraic Combinatorics by Stanley
This is exercise 2 in chapter 7 of Algebraic Combinatorics by Stanley.
For part (a), I first found the entire automorphism group. By labeling the root 1, and then numbering off the remaining vertices ...
0
votes
1
answer
60
views
Contraction, loops and flats.
This idea is being used a lot, but I cannot justify why it is correct:
If M is a matroid and $T$ a subset of $E(M).$ Then $M/T$ has no loops iff $T$ is a flat of $M.$
I know how to proof that in a ...
- 3,139
1
vote
1
answer
56
views
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
views
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
-1
votes
1
answer
52
views
what will happen if we contract an element in a uniform matroid? [closed]
Are the parallel elements in a matroid just behaving like loops? If so, why?
For example, in $U_{2,3}$ if we contract an element what will happen? In $U_{2,2}$ if we contract an element what Will ...
- 95
2
votes
1
answer
87
views
Partition lattice properties and an invariant.
I am trying to guess the value of the beta invariant of the partition lattice $\pi_4$ if I know the following information:
For any matroid $M,$ I know that
1- $\beta(M) \geq 0.$
2- $\beta(M) > 0$ ...
- 95
1
vote
1
answer
82
views
what will happen to the uniform matroid $U_{2,m}$ if we remove an element from it?
I am trying to figure out what will happen to the uniform matroid $U_{2,m}$ if we remove an element e from it, where e is neither a coloop nor a loop. I am guessing that it will become disconnected ...
- 95
2
votes
1
answer
94
views
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
2
votes
1
answer
82
views
Showing the isomorphism between (the geometric lattice)$\pi_n$ and the lattice $\mathcal{L}(M(K_n)).$
Here is the question I am trying to solve:
Show that the lattice of flats of $M(K_n)$ is isomorphic to the partition lattice $\pi_n$. Definition: $\pi_n$ is the set of partitions of $[n]$, partially ...
- 3,139
1
vote
1
answer
111
views
Philip Hall's theorem
Here is the theorem I want to prove:
For $x,y \in P$ and $i \geq 0,$ let $c_i(x,y)$ be the number of chains $x=x_0 < x_1 < \dots < x_i = y$ of length $i$ from $x$ to $y.$ Let $$\phi (x,y) = ...
- 3,139
2
votes
1
answer
61
views
an $\mathbb F$- representation of the matroid $M_1 \oplus M_2.$
Suppose that $A_1$ and $A_1$ are $\mathbb F$- representations of the matroids $M_1$ and $M_2$ respectively, show that
$$\begin{matrix}
A_1& 0\\
0 & A_2
\end{matrix}$$
is an $\mathbb F$- ...
- 3,139
1
vote
1
answer
75
views
Trace of power of matrix (arising from number of closed walks)
Let $G_n$ be the graph obtained from the $n-$cube graph $C_n$ by adding one extra edge between each vertex and its antipode (vertex whose label has all $0$'s and $1$'s switched) Note: The $C_n$ graph ...
- 326