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## A numerical approach for the filtered generalized Cech complex

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- Written by Jesus F. Espinoza
- Category: Research
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In this page you can download the source file `Cech.scale.cpp`

(first version), complementary to the paper *A numerical approach for the filtered generalized Cech complex*. The script is also available in https://github.com/gcs-unison/Cech-scale.

The paper is available online in arXiv.

## Small simple connected graphs

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- Written by Héctor Hernández
- Category: Research
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On this page you can find a repository of programs written in C/C++ as well as compressed data that generates these programs.

The source programs in C/C++ are written in standard C, without using additional data structure, with a understandable style.

The main idea is can compile and run programs using a standar personal computer.

**Connected graphs**. The script`Cnt_Graphs_1_9.cpp`

generate connected graphs of 1, 2, ..., 9 vertices. The files generated are`CG_num.zip`

each num represents a graph, that can changed in a adjacency matrix throught`Ch_num_to_AM_1_9.cpp`

with output`CG_AM.zip`

. Also`Ch_num_to_CF_AM_1_9.cpp`

change each num to the canonical form of the adjacency matrix. Here are the first 400 geometrical representations of connected graphs in`ggb`

(Geogebra).- The script
`CanonicalForm.cpp`

takes a graph from the file`graph.txt`

and return its canonical form. Both`Cnt_Graphs_1_9.cpp`

and`CanonicalForm.cpp`

are using a fast technique called Canonical Labeling, written exclusively to solve the problem of isomorphism of small graphs. **Contractible graphs**. Is an interesting family of graphs, because its characteristics in the homology calculation. The script`G_I.cpp`

calculate if`graph.txt`

belongs or not to this family. There is another family called collapsible graphics. The script`G_coll.cpp`

performs the corresponding work.- Also
`CG_to_G_Ivach_Coll_AM.cpp`

takes each graph from`CG_num.zip`

and filter which are contractible, in the Ivashchenko sense; at the same time verify the conjecture that they are also collapsible graphs. The ouput are files with adjacency matrix. While`CG_to_G_Ivach_n_G_coll.cpp`

show the number of graphs. - A graph belong to the contractible graphs family if can be reduced to K(1) using contractible transformations, in particular deleting vertices. A many graph dont belong to this family, but can be reduced throught this operations. This helps to reduce graphs and facilities homology calculation. See other main link.

Many graphs are not in the family contractible graphs; however, it is desirable to determine how far it is possible to eliminate vertices from the graph. The script `Ivach.Red.cpp`

uses the algorithm `contractible.graph`

to eliminate vertices when possible. The following two videos represent a filtration of graphs (actually we can think as the Vietoris-Rips filtration) and the sequence of reduced graphs, with the same persistent homology groups: