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This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.
In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges, vertices and by contracting edges.
In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart. A graph is perfect when these numbers are equal, and remain equal after the deletion of arbitrary subsets of vertices.
In graph theory, the perfect graph theorem of László Lovász states that an undirected graph is perfect if and only if its complement graph is also perfect. This result had been conjectured by Berge, and it is sometimes called the weak perfect graph theorem to distinguish it from the strong perfect graph theorem characterizing perfect graphs by their forbidden induced subgraphs.
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).
In graph theory, the strong perfect graph theorem is a forbidden graph characterization of the perfect graphs as being exactly the graphs that have neither odd holes nor odd antiholes. It was conjectured by Claude Berge in 1961. A proof by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas was announced in 2002 and published by them in 2006.
In the mathematical field of graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two distinct vertices of H are adjacent if and only if they are not adjacent in G. That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there.
In graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K1 and is closed under complementation and disjoint union.
In the mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.
In graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge. The neighbourhood of a vertex v in a graph G is the subgraph of G induced by all vertices adjacent to v, i.e., the graph composed of the vertices adjacent to v and all edges connecting vertices adjacent to v.
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of Wilhelm Ackermann. The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
In graph theory, a starSk is the complete bipartite graph K1,k : a tree with one internal node and k leaves. Alternatively, some authors define Sk to be the tree of order k with maximum diameter 2; in which case a star of k > 2 has k − 1 leaves.
In mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the theory of graph minors and topological embeddings. The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and Paul Seymour. Its proof is very long and involved. Kawarabayashi & Mohar (2007) and Lovász (2006) are surveys accessible to nonspecialists, describing the theorem and its consequences.
In the mathematical field of graph theory, the Clebsch graph is either of two complementary graphs on 16 vertices, a 5-regular graph with 40 edges and a 10-regular graph with 80 edges. The 80-edge graph is the dimension-5 halved cube graph; it was called the Clebsch graph name by Seidel (1968) because of its relation to the configuration of 16 lines on the quartic surface discovered in 1868 by the German mathematician Alfred Clebsch. The 40-edge variant is the dimension-5 folded cube graph; it is also known as the Greenwood–Gleason graph after the work of Robert E. Greenwood and Andrew M. Gleason, who used it to evaluate the Ramsey number R(3,3,3) = 17.
Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group G there exist infinitely many non-isomorphic simple connected graphs such that the automorphism group of each of them is isomorphic to G.
In the mathematical field of graph theory, the Schläfli graph, named after Ludwig Schläfli, is a 16-regular undirected graph with 27 vertices and 216 edges. It is a strongly regular graph with parameters srg(27, 16, 10, 8).
In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit–evasion games on the graph, or as topological ends of topological spaces associated with the graph.
Italo Jose Dejter is an Argentine-born American mathematician, a retired professor of mathematics and computer science from the University of Puerto Rico, and a researcher in algebraic topology, differential topology, graph theory, coding theory and combinatorial designs. He obtained a Licentiate degree in mathematics from University of Buenos Aires in 1967, arrived at Rutgers University in 1970 by means of a Guggenheim Fellowship and obtained a Ph.D. degree in mathematics in 1975 under the supervision of Professor Ted Petrie, with support of the National Science Foundation. He was a professor at the Federal University of Santa Catarina, Brazil, from 1977 to 1984, with grants from the National Council for Scientific and Technological Development, (CNPq).
In graph theory, a quotient graphQ of a graph G is a graph whose vertices are blocks of a partition of the vertices of G and where block B is adjacent to block C if some vertex in B is adjacent to some vertex in C with respect to the edge set of G. In other words, if G has edge set E and vertex set V and R is the equivalence relation induced by the partition, then the quotient graph has vertex set V/R and edge set {([u]R, [v]R) | (u, v) ∈ E(G)}.
In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P3-free graphs. They are the complement graphs of the complete multipartite graphs and the 2-leaf powers. The cluster graphs are transitively closed, and every transitively closed undirected graph is a cluster graph.
References
- Buczak, J. M. J. (1980), Finite Group Theory, Ph.D. thesis, Oxford University. As cited by Devillers (2002).
- Cameron, Peter Jephson (1980), "6-transitive graphs", Journal of Combinatorial Theory , Series B, 28: 168–179, doi: 10.1016/0095-8956(80)90063-5 . As cited by Devillers (2002).
- Devillers, Alice (2002), Classification of some homogeneous and ultrahomogeneous structures, Ph.D. thesis, Université Libre de Bruxelles.
- Gardiner, A. (1976), "Homogeneous graphs", Journal of Combinatorial Theory , Series B, 20 (1): 94–102, doi:10.1016/0095-8956(76)90072-1, MR 0419293 .
- Gardiner, A. (1978), "Homogeneity conditions in graphs", Journal of Combinatorial Theory , Series B, 24 (3): 301–310, doi: 10.1016/0095-8956(78)90048-5 , MR 0496449 .
- Gray, R.; Macpherson, D. (2010), "Countable connected-homogeneous graphs", Journal of Combinatorial Theory , Series B, 100 (2): 97–118, doi: 10.1016/j.jctb.2009.04.002 , MR 2595694 .
- Lachlan, A. H.; Woodrow, Robert E. (1980), "Countable ultrahomogeneous undirected graphs", Transactions of the American Mathematical Society , 262 (1): 51–94, doi: 10.2307/1999974 , MR 0583847 .
- Ronse, Christian (1978), "On homogeneous graphs", Journal of the London Mathematical Society , Second Series, 17 (3): 375–379, doi:10.1112/jlms/s2-17.3.375, MR 0500619 .