All Questions
9 questions
6
votes
1
answer
746
views
Relationship between spectral gaps of adjacency and Laplacian matrices of graphs
Let $G$ be an undirected simple graph on $n$ vertices, with self-loops allowed, and with arbitrary positive edge weights $w_{u,v}$ (which is $0$ if there is no edge between $u$ and $v$).
Let $A$ be ...
- 150
13
votes
0
answers
237
views
A Dynkin type classification result in linear algebra
Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...
- 26.5k
1
vote
0
answers
126
views
Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
- 444
3
votes
0
answers
105
views
Can one do better than using general purpose determinant algorithms when using the Fisher-Kasteleyn-Temperley method for perfect matchings?
Questions.
(numerical.generalPfaffian) Is it proved anywhere that in general it is not easier0 to calculate the determinant (over $\mathbb{Q}$) of the skew-symmetric signed adjacency matrix defined ...
- 6,051
0
votes
1
answer
229
views
Reference request: Strong Connectivity and the Incidence Matrix
Question: What would be a good reference for characterizations of strong connectivity of a digraph in terms of its incidence matrix?
Details: Consider a digraph $(V, E)$ with vertex set
$$V = \{v_1,...
- 123
1
vote
1
answer
1k
views
Does this linear algebra construction based on a graph have a name, and where has it been studied?
In the paper Kochen-Specker set with seven contexts by Lisonek, Badziag, Portillo and Cabello, the following construction is used :
Question : Have such constructions been used elsewhere, and if so ...
- 2,901
2
votes
1
answer
520
views
Neighborhood overlap matrix for a bipartite graph
Let $G$ be an undirected, simple, bipartite graph with parts $V$ (having $n$ vertices) and $W$ (having $m$ vertices). Define the following $n$-by-$n$ matrix: for any $i,j \in V$, $$a_{ij} = |N_i \cap ...
- 1,068
3
votes
1
answer
166
views
The spectral radius of a modified graph
Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...
- 6,962
4
votes
3
answers
755
views
Is this statement about the real edge space of a graph known or trivial?
The statement is:
($u$ is a fixed node in a fixed graph $G$)
$G$ is 3-connected
if and only if
the set of u-cycles span $\mathbb{R}^{E(G)}$.
A u-cycle is a simple (no vertex repetitions) cycle in G ...
- 406