Канеюку

A remark

Dear colleagues,

The graphs are many-sided formations, they have a lot of “hidden sides”, for researching (ascertaining) of which new methods and algorithms are needed. Graphs were discovered, and these have been repeatedly rediscovered in the course of solving problems with paths and cycles. This has left a strong mark to graph theory and, in many cases, it limits its development. To hidden remains also the problems of graph’s structure. The graph structure is a complete invariant of graph, i. e. such a graph representation attribute, which stay permanent for a class of isomorphic graphs, and only for them.

One of the important attributes of the graph structure are its orbits, i. e. equivalence classes of vertices and pairs of vertices. Orbits and the graph structure as a whole is useful to ascertain in the form of a model, what obtained in the way of deep-measuring and decomposition of the graph (where the obtained orbits coincide with the orbits of AutG). This finding to cause confusion, but is not able to find counterexamples.

Graphs with the equivalent models are isomorphic. The structural equivalence of two graphs is one-to-one correspondence between their orbits of vertices and vertex-pairs. The isomorphism is such one-to-one correspondence between corresponding vertices, which does not recognize the orbits. The model of graph’s structure is a complete invariant of the graph (counterexamples cannot to find).

Ascertaining the orbits of the pairs of vertices (binary orbits) in the graph’s model make possible to form the systems of graphs. The system of graphs is a set of graphs with n vertices, where are fixed the relationships between the elements, i. e. graphs. To these relationships are the relations (called morphisms) between a graph G and its largest subgraphs (G\e), and its smallest supergraph (Ge). To each binary orbit corresponds one morphism (counterexamples cannot to find). For example, starting from a complete graph (or empty graph) is generated by corresponding algorithm all the 572 morphisms (and their probabilities) between 1144 binary orbits of 156 graphs (structures) with 6-vertices.

Thus, here exist three conjectures: 1. The graph orbits are ascertainable on the basis of deep-identification of vertex pairs. 2. The structural model of a graph is its complete invariant. 3. The graph systems can be ascertainable by generating the structural models of the graphs.

For interested party is the base material available http://www.graphs.ee (see version June, 18, 2012). If you wish, you can prove or disprove these theoretical opinions! The first is simpler.

The graphs are a fundamental phenomenon, which does not fit into the existing attributes of discrete mathematics.

Sincerely, John-Tagore Tevet

Dear eager gentlemen. You delete my article "Semiotics of the structure". Uses the rights of “convicted person” I have the last word. Read your scenario was been useful. Correctly argues Thuvko: ”This is not really material that can be reviewed by mathematicians”. It seems that a meaningful discussion about did not happen. Mainly considered external signs of that from time to time become like a farce. I’m, obviously, at half older than you and it reminded me of the 1950 years when cybernetics was characterized as “capitalistic pseudo-science” and genetics as “scientifically senseless”, and so on. Of course, I do not think that I've invented a “new cybernetics or genetics”, but an otherwise approach to the graphs – yes. Right has also David Epstein about references. Some of the references in my article were for me only confused details. Enough, if I refer only to: 1) a thorough textbook on graph theory (basic information about the graphs), 2) a source of basic knowledge of semiotics, 3) a handbook on the philosophy outlined where the definition of terms used, as well as an apology, and, excuse me, 4) some reference to edition of the author. The only purely mathematical problem here is the proof that the structural “positions“ coincide with the orbits (domains of automorphisms). This can be achieved in collaboration with some group theorists. It is of course very difficult, but possible. One approach to this is available.