Clearly, a set of hypertext pages and the set of links that connect some of those pages can also be represented as a graph. Where pages become vertices and links become edges.

A tree is a special case of a graph that has no self-referential loops.
(Collins dictionary of mathematics 1989)

The directory structures of MS-DOS and MS-Windows are trees.

The librarian's problem
One version of the librarian's problem is that a single book can occupy more than one place in an ordered list (i.e. that book could be shelved in more than one location).

The usual way to solve this problem is to shelve "a book dealing with two or more interrelated subjects with the one that receives the chief emphasis." (Dewey decimal classification and relative index, 9th abridged version, 1965, p.25)

Translating the librarian's problem into graph theory
The librarian's problem can be translated into graph theory. In graph theory, an ordered set has to be a tree. If the book-subject relations can be converted into a graph, then this graph can be measured to see whether there is a unique tree that covers all edges.

As I have a partial bibliography on my webpages, I will assume that the librarian's problem can be converted into hypertext. Umberto Eco highlights links between Leibniz and the "hypertext" of John Wilkins (1993:1995, pp.258-9,279).

The web cube is a simple graph represented as hypertext.

General properties of linked graphs
(This defines my use of the word "linked". I'm not sure if this is the same as connected?)

For any linked graph, g, one in which there is a walk between any two vertices, the minimum number of edges forms a tree tree.

Therefore the minimum number of edges, Emin, of a linked graph with v vertices is given by:

Emin = v - 1 (Eq 1.1)

The maximum number of edges, Emax, of a linked graph with v vertices is clearly given by the combination:

Emax = C(v,2) (Eq 1.2)

(introducing some text-friendly notation)

Where C(n,r) = n!/((n-r)!.r!) (Eq 1.3)

Therefore

Emax = C(v,2) = v!/((v-2)!.2!) = v(v-1)/2 (Eq 1.4)

Clearly as v teads to infinity, the ratio Emax/Emin also tends to infinity.

The librarian's problem can therefore be expressed as:

For a linked graph, G, of v, books or subjects, with E, links or cross-references between them, what is the value of E?

• If E = Emin, then the graph, G, is a tree. Therefore the solution to the librarian's problem for that graph is "Yes, the books can be ordered with unique positions on a shelf".

• If E > Emin, then there is no tree that can travel all edges. Therefore, more than one tree exists for the graph, G. Combining trees will result in there being more than one order for some vertices. Therefore the answer to the librarian's problem is "No, some books need to be shelved in two separate locations".

• If E = Emax, then any tree will be a tree of graph, G, and the librarian is not needed.

• If E < Emax, then not all trees will be trees of graph, G, and the librarian's task is to find out which trees are which.

Diversion: Does E > Emin mean that we cannot do research?
(Note: I think I'm going to move this to part I conclusions.)

How do values of E relate to Mies & Shiva's methodology of ecofeminist research (1993, p.10-13)? How do these values relate to their definitions of, on one hand, cultural imperialism and, on the other hand, cultural relativism? How do these values relate to their departure from this binary opposition?

I would suggest that:

• Extreme cultural imperialism assumes E=Emin
• Extreme cultural relativism assumes E=Emax

Here, I would suggest that interconnectedness can be represented by a graph and that diversities can be represented by different graphs. And that ,for connected graphs of these links, Emin < E < Emax.

A practical application of this would be to convert Shiva's comparisons of biomass contributions to rural-life support systems (1993, p.27-39) into graphs. Two examples she gives (ibid., Figures 2. & 3., pp.37-38) are already graphs. For figure 2., a generalisation of native tree diversity, Emin < E < Emax. For figure 3., the local contributions of a eucalyptus monculture, E = Emin.

• Do encyclopedias assume that Emin <= E <= Emax?
• Is an encyclopedia an example of good research?

Applications to translations between biology and sociology
Criticisms of biology by sociologists may make the assumption that scientists assume that E = Emin in all cases. Criticisms that present biology as essentialist may make this assumption. However, biologists reserve the essentialist position for creationism (Dennett) and reject this criticism as being trivial.

Criticisms of sociology by biologists may make the assumption that sociologists assume that E = Emax in all cases. This is

Derrida's Circle of Meaning can be converted into graph theory by stating that at least one loop must exist in the graph. The smallest loop allowed in graph theory is the maximally-linked, three-vertice graph. In this case, E > Emin, but also E = Emax. This may have been taken by some scientists of mean that Derrida offers no place for librarians. However this is only true for three vertice graphs. Loops can be present in any graph with more that two vertices. For all graphs with more than three vertices, there can exist values of E, such that E > Emin and E < Emax.

Eco's conclusions to the "The Search for the Perfect Language" suggest that E > Emin and that by using a plurality of languages, ie. a plurality of trees, we can advance our understanding of the world. This is the approach of intertextuality, the approach that compares the links between ordered narratives.

Ersnt Mayr, in "This is biology", suggests that a philosophy of biology must be one that allows for plurality (pp.67-68). However, he also thinks that literary criticism one of the subjects with the least to say about biology (ibid. p.37). Mayr's discussion of multiple causality in biology may allow links between biology and literary criticism to be found.

The difficulty of assigning decomposers a place within energy "pyramids" would suggest that this is one area where sociological approaches could help biology.

Conclusions
By translating the librarian's problem into graph theory it has been possible to distinguish four general cases of graphs. It has been shown that these cases make it possible for biologists and sociologists to both mis-represent each other's arguements and to agree with each other's arguements or results. The translation problem then becomes: which specific graph are we dealing with in each specific biological or sociological example? Furthermore, is that graph one that allows agreement and therefore is that graph one that allows translations to be made?

• Cube 1

• Translations between biology and sociology
  • Part I: initial points of translation.
    • Initial points of translation from biology
    • Formalism: a linking theme between mathematics, computation, and cultural theory
    • Initial points of translation from formalism
    • Conclusion: Possible grounds for translation
  • Part II: What questions does sociology ask
  • Part III: What are the roles of calcium in the genotype - phenotype argument?

• Directories, hierarchies and set theory. Some differences between the ways in which computers like to represent information and the ways in which information finds itself organised.

• Links between formalism in mathematics and linguistics

• Genotype and phenotype
• Hierarchies in biology
• The librarian's problem