In graph theory, the degree diameter problem is the problem of finding the largest possible graph G (in terms of the size of its vertex set V) of diameter k such that the largest degree of any of the vertices in G is at most d. The size of G is bounded above by the Moore bound; for 1 < k and 2 < d only the Petersen graph, the Hoffman-Singleton graph, and possibly one more graph (not yet proven to exist) of diameter k = 2 and degree d = 57 attain the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound.
Let be the maximum possible number of vertices for a graph with degree at most d and diameter k. Then , where is the Moore bound:
This bound is attained for very few graphs, thus the study moves to how close there exist graphs to the Moore bound. For asymptotic behaviour note that .
Define the parameter . It is conjectured that for all k. It is known that and that .
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes.
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let r and s be any two positive integers. Ramsey's theorem states that there exists a least positive integer R(r, s) for which every blue-red edge colouring of the complete graph on R(r, s) vertices contains a blue clique on r vertices or a red clique on s vertices. (Here R(r, s) signifies an integer that depends on both r and s.)
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring.
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter of a set by its volume ,
The Turán graph, denoted by , is a complete multipartite graph; it is formed by partitioning a set of vertices into subsets, with sizes as equal as possible, and then connecting two vertices by an edge if and only if they belong to different subsets. Where and are the quotient and remainder of dividing by , the graph is of the form , and the number of edges is
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that does not have a given subgraph.
In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.
In graph theory, a strongly regular graph (SRG) is defined as follows. Let G = (V, E) be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers λ and μ such that:
In graph theory, the Kneser graphK(n, k) (alternatively KGn,k) is the graph whose vertices correspond to the k-element subsets of a set of n elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1956.
In the mathematical field of graph theory, the Hoffman–Singleton graph is a 7-regular undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with parameters (50,7,0,1). It was constructed by Alan Hoffman and Robert Singleton while trying to classify all Moore graphs, and is the highest-order Moore graph known to exist. Since it is a Moore graph where each vertex has degree 7, and the girth is 5, it is a (7,5)-cage.
In graph theory, a Moore graph is a regular graph whose girth is more than twice its diameter. If the degree of such a graph is d and its diameter is k, its girth must equal 2k + 1. This is true, for a graph of degree d and diameter k, if and only if its number of vertices equals
In the mathematical area of graph theory, the Mycielskian or Mycielski graph of an undirected graph is a larger graph formed from it by a construction of Jan Mycielski (1955). The construction preserves the property of being triangle-free but increases the chromatic number; by applying the construction repeatedly to a triangle-free starting graph, Mycielski showed that there exist triangle-free graphs with arbitrarily large chromatic number.
In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth.
In graph theory, boxicity is a graph invariant, introduced by Fred S. Roberts in 1969.
In graph theory, the degree diameter problem is the problem of finding the largest possible graph for a given maximum degree and diameter. The Moore bound sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer. The table below gives current progress on this problem.
In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph , find the maximal number of edges an -vertex graph can have such that it does not have a subgraph isomorphic to . In this context, is called a forbidden subgraph.
In graph theory, a branch of mathematics, the cop number or copnumber of an undirected graph is the minimum number of cops that suffices to ensure a win in a certain pursuit–evasion game on the graph.
In graph theory, a locally linear graph is an undirected graph in which every edge belongs to exactly one triangle. Equivalently, for each vertex of the graph, its neighbors are each adjacent to exactly one other neighbor, so the neighbors can be paired up into an induced matching. Locally linear graphs have also been called locally matched graphs.
In graph theory, the McKay–Miller–Širáň graphs are an infinite class of vertex-transitive graphs with diameter two, and with a large number of vertices relative to their diameter and degree. They are named after Brendan McKay, Mirka Miller, and Jozef Širáň, who first constructed them using voltage graphs in 1998.
In graph theory, a geodetic graph is an undirected graph such that there exists a unique (unweighted) shortest path between each two vertices.
