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The canonical form of a graph is an example of a complete graph invariant: every two isomorphic graphs have the same canonical form, and every two non-isomorphic graphs have different canonical forms. [1] [2] Conversely, every complete invariant of graphs may be used to construct a canonical form. [3]
A canonical form is a labeled graph Canon(G) that is isomorphic to G, such that every graph that is isomorphic to G has the same canonical form as G. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism : to test whether two graphs G and H are isomorphic, compute their canonical forms ...
The original formulation is based on graph canonization, a normal form for graphs, while there is also a combinatorial interpretation in the spirit of color refinement and a connection to logic. There are several versions of the test (e.g. k-WL and k-FWL) referred to in the literature by various names, which easily leads to confusion.
The Jordan form is used to find a normal form of matrices up to conjugacy such that normal matrices make up an algebraic variety of a low fixed degree in the ambient matrix space. Sets of representatives of matrix conjugacy classes for Jordan normal form or rational canonical forms in general do not constitute linear or affine subspaces in the ...
In morphology and lexicography, a lemma is the canonical form of a set of words. In English, for example, run, runs, ran, and running are forms of the same lexeme, so we can select one of them; ex. run, to represent all the forms. Lexical databases such as Unitex use this kind of representation.
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata.These automata represent the Cayley graph of the group. That is, they can tell whether a given word representation of a group element is in a "canonical form" and can tell whether two elements given in canonical words differ by a generator.
The De Morgan dual is the canonical conjunctive normal form , maxterm canonical form, or Product of Sums (PoS or POS) which is a conjunction (AND) of maxterms. These forms can be useful for the simplification of Boolean functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular.
The left figure below shows a binary decision tree (the reduction rules are not applied), and a truth table, each representing the function (,,).In the tree on the left, the value of the function can be determined for a given variable assignment by following a path down the graph to a terminal.