Access to a 1932 paper by R. M. Foster

[Elijah Beregovsky]

Hey seqfans!

Does anyone have access to this paper: https://doi.org/10.1109/T-AIEE.1932.5056068? It is mentioned in A002631, but there is no unpaywalled pdf of the full paper anywhere I can see only the abstract, so I cannot quite understand what Foster means by “geometrical circuit”. I’d like to attempt extending the sequence, but to do that I need to know what is being counted 😁

Thanks in advance!

Elijah

[Sean A. Irvine]

I have sent Elijah the requested paper off list.

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[Christopher Landauer]

Hihi -
I do not have the paper, but I can read it;
here is my personal interpretation of the definitions
(and I think that it is important to have the necessary definitions more readily accessible on the sequence page, indeed, on every sequence page;
in fact, i do not believe that any of the definitions should be hidden away at all; results, methods, exposition yes, ok, but not the definitions)

A geometric circuit is an undirected not necessarily connected graph
(loops and multiple edges allowed),
with two additional equivalence conditions based on abstracted properties of electric circuitry
(illustrated by nice hand-drawn pictures),
and extended by transitive closure

(1) A graph whose vertices fall into two subsets A and B with exactly one vertex v in common and no edges from any vertex in A \ {v} to any vertex in B \ {v} is equivalent ``by separation’' to a graph with two disconnected parts having vertices A and B respectively, but with two separate copies of v
(there is no assumption about whether these two parts are themselves connected, but both A \ {v} or B \ {v} must be non-empty or there are not two separate parts);
An example is that a graph with edges {u, v} and {v, w} is equivalent to a graph with edges {u, v'} and {v'’, w} for distinct ``new’’ vertices v’ and v''

(2) A graph with three vertices u, v, w for which there is no other edge incident to v except those that are also incident to u or w is equivalent ``by series’’ to a graph with any {u, v} edge moved to {v, w}, provided there is at least one {u, v} edge remaining
An example is that a graph with multiset of edges {u, v}, {u, v}, {v, w} is equivalent to a graph with multiset of edges {u, v}, {v, w}, {v, w}

There is a third equivalence ``by deformation’’,
but that is subsumed in the definition of a graph;
his use of the term ``geometric’’ seems to be referring to mappings of the graph into a euclidean space with all vertex and edge images distinct
(and it seems that he uses the term ``branch’’ for ``edge’')

Nullity is the number of edges minus the number of vertices plus the number of connected components; it can be seen that each of these equivalence conditions preserves that number

It is not actually clear from the definitions and pictures whether the ``no other edge'' condition on vertex v in equivalence condition (2) is required, since even without it, nullity is preserved
(you should probably just try it both ways and see which one matches the existing values)

Hope this helps,
chris

Dr. Chris Landauer
Topcy House Consulting