# Tagged Questions

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### Extended Hypercube Graph

Definition 1. The $n$-hypercube graph has vertices which are the elements of the set $\lbrace 0,1\rbrace^n$ of $n$-bit binary strings, and an edge is drawn between each pair of vertices representing a ...

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465 views

### “Trellis graph” is it standard term in graph theory ? What are its properties ?

In coding theory (convolutional codes) the graph called "trellis diagramm" is used
to visualize something.
I wonder is it a standard term in graph theory? Corresponding Wikipedia article is not
...

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240 views

### The chromatic number of a Hamming-related graph

For integer $1\le k\le n$, let ${\overline H}_n^k$ denote the complement of
the $k$-th power of the Hamming graph on the vertex set ${\mathbb
F}_2^n$; that is, two vectors from ${\mathbb F}_2^n$ are ...

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**1**answer

273 views

### Is there any relationship between a tree(graph theory) and semi-metric?

suppose we have a tree(undirected) with $n$ vertices.The edges are weighted(distances). Is it possible to impose a semi-metric structure on the graph using these distances and adjacency matrix?

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221 views

### Ax=0, estimate min(Hamming(x)) ? Equivalently: Bipartite graph. How to find (estimate) minimal number of vertices1 which are connected with EVEN number of vertices2 ? Equivalently: estimate minimal weight of error correcting code ?

Consider system of linear equations Ax=0 over $F_2$ (field with two elements {0,1}).
Where number of variables is bigger than equations - so we have many solutions $x$.
Question How to estimate ...

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### If graph is tree what can be said about its adjacency matrix ?

Question If graph is tree what can be said about its adjacency matrix ? And vice versa ?
Especially I am interested in case when graph is bipartite graph.
Such graphs are related to ...

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279 views

### Adjacency matrices of graphs as parity check matrices of error correcting codes

Consider bipartite graph.
Consider its adjacency matrix.
It will have a form
0 A^t
A 0
Take matrix $A$.
Consider the null-space $L$ of $A$ over $F_2^N$.
Question Can we say something about the ...

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464 views

### Hamming graphs and power series

Let $i$ and $h$ be two adjacent nodes in a Hamming graph and let $a$ be any positive real. Let us denote by $d_{ij}$ the distance between node $i$ and node $j$ in the graph.
I'm trying to find a ...

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1k views

### “sparse graphs are locally tree-like”

I would like to be able to state with confidence that sparse graphs (graphs with small numbers of edges) are locally tree-like (they have few short cycles). Apparently "Sparse graphs are locally tree ...