[Inquiry] Re: Peirce's Logic Of Information

[Jon Awbrey]

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PLOI.  Note 10

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Though the most careful among us will occasionally slip up and use
the terms interchangeably, there is a slight but significant nuance
of distinction between "adjacency matrices" and "incidence matrices".
The incidence matrix of our labeled representative of K_2,3 uses the
underlying partition of points to economize the dimensions of matric
materiel, tantamount to treating the two parts as if they contained
different types of points, thus allowing the following compact form:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` 2 ` ` 3 ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` 4 | ` 1 ` ` 1 ` ` 1 ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` 5 | ` 1 ` ` 1 ` ` 1 ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The economy of the incidence matrix, in those cases where it's available,
will be evident if we compare it with the corresponding adjacency matrix:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` 1 ` ` 2 ` ` 3 ` ` 4 ` ` 5 ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` o-------------------------------o ` ` ` ` ` ` `
` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` `
` ` ` ` ` ` 1 | ` 0 ` ` 0 ` ` 0 ` ` 1 ` ` 1 ` | ` ` ` ` ` ` `
` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` `
` ` ` ` ` ` 2 | ` 0 ` ` 0 ` ` 0 ` ` 1 ` ` 1 ` | ` ` ` ` ` ` `
` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` `
` ` ` ` ` ` 3 | ` 0 ` ` 0 ` ` 0 ` ` 1 ` ` 1 ` | ` ` ` ` ` ` `
` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` `
` ` ` ` ` ` 4 | ` 1 ` ` 1 ` ` 1 ` ` 0 ` ` 0 ` | ` ` ` ` ` ` `
` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` `
` ` ` ` ` ` 5 | ` 1 ` ` 1 ` ` 1 ` ` 0 ` ` 0 ` | ` ` ` ` ` ` `
` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` `
` ` ` ` ` ` ` o-------------------------------o ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The adjacency matrix exhibits a "1" at the intersection of row j and column k
if and only point j and point k are "adjacent", that is, connected by a line,
and it exhibits a "0" otherwise.  Since we are presently discussing graphs,
where each line runs in two directions, that is, not directed graphs, or
"digraphs", the adjacency matrix is symmetric about the main diagonal.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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