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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 ...
In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points ) which are connected by edges (also called arcs , links or lines ).
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. [1]
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
Graph_level_structure.pdf (685 × 283 pixels, file size: 65 KB, MIME type: application/pdf) This is a file from the Wikimedia Commons . Information from its description page there is shown below.
The cube of every connected graph necessarily contains a Hamiltonian cycle. [10] It is not necessarily the case that the square of a connected graph is Hamiltonian, and it is NP-complete to determine whether the square is Hamiltonian. [11] Nevertheless, by Fleischner's theorem, the square of a 2-vertex-connected graph is always Hamiltonian. [12]
Petersen graph as Kneser graph ,. The Petersen graph is the complement of the line graph of .It is also the Kneser graph,; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an edge if and only if the corresponding 2-element subsets are disjoint from each other.
Equivalently, a tournament is an orientation of an undirected complete graph. (However, as directed graphs, tournaments are not complete: complete directed graphs have two edges, in both directions, between each two vertices. [1]) Equivalently, a tournament is a complete asymmetric relation. [2] [3]