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An example for an undirected Graph with a vertex r and its corresponding level structure For the concept in algebraic geometry, see level structure (algebraic geometry) In the mathematical subfield of graph theory a level structure of a rooted graph is a partition of the vertices into subsets that have the same distance from a given root vertex.
The automorphism group of the free group with ordered basis [ x 1, …, x n] is generated by the following 4 elementary Nielsen transformations: Switch x 1 and x 2; Cyclically permute x 1, x 2, …, x n, to x 2, …, x n, x 1. Replace x 1 with x 1 −1; Replace x 1 with x 1 ·x 2; These transformations are the analogues of the elementary row ...
Sometimes graph grammar is used as a synonym for graph rewriting system, especially in the context of formal languages; the different wording is used to emphasize the goal of constructions, like the enumeration of all graphs from some starting graph, i.e. the generation of a graph language – instead of simply transforming a given state (host ...
The image under a Nielsen transformation (elementary or not, regular or not) of a generating set of a group G is also a generating set of G. Two generating sets are called Nielsen equivalent if there is a Nielsen transformation taking one to the other (beware this is not an equivalence relation). If the generating sets have the same size, then ...
The extremal number (,) is the maximum number of edges in an -vertex graph containing no subgraph isomorphic to . is the complete graph on vertices. (,) is the Turán graph: a complete -partite graph on vertices, with vertices distributed between parts as equally as possible.
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems.. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations.
The vertex set of an n-vertex graph may be identified with the integers from 1 to n, and using such an identification a canonical form of a graph may also be described as a permutation of its vertices. Canonical forms of a graph are also called canonical labelings, [4] and graph canonization is also sometimes known as graph canonicalization.
Its authors have divided Elementary Number Theory, Group Theory and Ramanujan Graphs into four chapters. The first of these provides background in graph theory, including material on the girth of graphs (the length of the shortest cycle), on graph coloring, and on the use of the probabilistic method to prove the existence of graphs for which both the girth and the number of colors needed are ...