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Meredith Graph
In the mathematical field of graph theory, the Meredith graph is a 4- regular undirected graph with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973. The Meredith graph is 4- vertex-connected and 4- edge-connected, has chromatic number 3, chromatic index 5, radius 7, diameter 8, girth 4 and is non-Hamiltonian. It has book thickness 3 and queue number 2. Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian. However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian.Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969. The characteristic polynomial of the Meredith graph is (x-4) (x-1)^ x^ (x+1)^ (x+3) (x^2-13) (x^6-26 x^4+3 x^3+169 x^2-39 x-45)^4. Gallery Image:Meredith graph 3COL.svg, The chromatic number of the Meredith graph is 3. Image:Meredith graph 5color edge.svg, The chromatic index In graph theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meredith Graph
In the mathematical field of graph theory, the Meredith graph is a 4- regular undirected graph with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973. The Meredith graph is 4- vertex-connected and 4- edge-connected, has chromatic number 3, chromatic index 5, radius 7, diameter 8, girth 4 and is non-Hamiltonian. It has book thickness 3 and queue number 2. Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian. However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian.Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969. The characteristic polynomial of the Meredith graph is (x-4) (x-1)^ x^ (x+1)^ (x+3) (x^2-13) (x^6-26 x^4+3 x^3+169 x^2-39 x-45)^4. Gallery Image:Meredith graph 3COL.svg, The chromatic number of the Meredith graph is 3. Image:Meredith graph 5color edge.svg, The chromatic index In graph theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Index
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 different colors, for a given value of , 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. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree or . For some graphs, such as bipartite graphs and high-degree planar graphs, the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crispin Nash-Williams
Crispin St John Alvah Nash-Williams FRSE (19 December 1932 – 20 January 2001) was a British mathematician. His research interest was in the field of discrete mathematics, especially graph theory. Biography Nash-Williams was born on 19 December 1932 in Cardiff, Wales. His father, Victor Erle Nash-Williams ( Williams), was an archaeologist at University College Cardiff, and his mother had studied classics at Oxford. As a small boy, Nash-Williams attended Christ Church Cathedral School in Oxford, which was then headed by Wilfrid Oldaker. A biographer has said that Oldaker was a formative influence on Nash-Williams.D. J. A. Welsh"Crispin St J. A. Nash-Williams (1932–2001)"in ''Bulletin of the London Mathematical Society'', Vol. 35, Issue 6, November 2003, Pages 829–844 After studying mathematics at the University of Cambridge, earning the title of Senior Wrangler in 1953, he remained at Cambridge for his graduate studies, under the supervision of Shaun Wylie and David Rees. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Queue Number
In the mathematical field of graph theory, the queue number of a graph is a graph invariant defined analogously to stack number (book thickness) using first-in first-out (queue) orderings in place of last-in first-out (stack) orderings. Definition A queue layout of a given graph is defined by a total ordering of the vertices of the graph together with a partition of the edges into a number of "queues". The set of edges in each queue is required to avoid edges that are properly nested: if and are two edges in the same queue, then it should not be possible to have in the vertex ordering. The queue number of a graph is the minimum number of queues in a queue layout.. Equivalently, from a queue layout, one could process the edges in a single queue using a queue data structure, by considering the vertices in their given ordering, and when reaching a vertex, dequeueing all edges for which it is the second endpoint followed by enqueueing all edges for which it is the first en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Book Thickness
In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings into a ''book'', a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie on this boundary line, called the ''spine'', and the edges are required to stay within a single half-plane. The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph. Book thickness is also called pagenumber, stacknumber or fixed outerthickness. Book embeddings have also been used to define several other graph invariants including the pagewidth and book crossing number. Every graph with vertices has book thickness at most \lceil n/2\rceil, and this formula gives the exact book thickness for complete graphs. The graphs with book thickness one are the outerplanar graphs. The graphs with book thickness at most two are the subhamiltonian graphs, which are always planar; more generally, ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Graph
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eulerian Graph
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this: :Given the graph in the image, is it possible to construct a path (or a cycle; i.e., a path starting and ending on the same vertex) that visits each edge exactly once? Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem: :A connected g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-edge-connected Graph
In graph theory, a connected graph is -edge-connected if it remains connected whenever fewer than edges are removed. The edge-connectivity of a graph is the largest for which the graph is -edge-connected. Edge connectivity and the enumeration of -edge-connected graphs was studied by Camille Jordan in 1869. Formal definition Let G = (V, E) be an arbitrary graph. If subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is ''k''-edge-connected. The edge connectivity of is the maximum value ''k'' such that ''G'' is ''k''-edge-connected. The smallest set ''X'' whose removal disconnects ''G'' is a in ''G''. The edge connectivity version of provi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected graph is said to be -vertex-connected (or -connected) if it has more than vertices and remains connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest for which the graph is -vertex-connected. Definitions A graph (other than a complete graph) has connectivity ''k'' if ''k'' is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with ''n'' vertices has connectivity ''n'' − 1, as implied by the first definition. An equivalent definition is that a graph with at least two vertices is ''k''-connected if, for every pair of its vertices, it is possible to find ''k'' vertex-independent paths connecting these vertices; see Menger's theorem . This definition produces the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
