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Harries Graph
In the mathematical field of graph theory, the Harries graph or Harries (3-10)-cage is a 3- regular, undirected graph with 70 vertices and 105 edges. The Harries graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian. It is also a 3- vertex-connected and 3- edge-connected, non-planar, cubic graph. It has book thickness 3 and queue number 2. The characteristic polynomial of the Harries graph is : (x-3) (x-1)^4 (x+1)^4 (x+3) (x^2-6) (x^2-2) (x^4-6x^2+2)^5 (x^4-6x^2+3)^4 (x^4-6x^2+6)^5. \, History In 1972, A. T. Balaban published a (3-10)-cage graph In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an is defined to be a graph in which each vertex has exactly neighbors, and in which the shortest cycle has le ..., a cubic graph that has as few vertices as possible for girth 10. It was the first (3-10)-cage discovered but it was not unique. The comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harries Graph
In the mathematical field of graph theory, the Harries graph or Harries (3-10)-cage is a 3- regular, undirected graph with 70 vertices and 105 edges. The Harries graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian. It is also a 3- vertex-connected and 3- edge-connected, non-planar, cubic graph. It has book thickness 3 and queue number 2. The characteristic polynomial of the Harries graph is : (x-3) (x-1)^4 (x+1)^4 (x+3) (x^2-6) (x^2-2) (x^4-6x^2+2)^5 (x^4-6x^2+3)^4 (x^4-6x^2+6)^5. \, History In 1972, A. T. Balaban published a (3-10)-cage graph In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an is defined to be a graph in which each vertex has exactly neighbors, and in which the shortest cycle has le ..., a cubic graph that has as few vertices as possible for girth 10. It was the first (3-10)-cage discovered but it was not unique. The comp ... [...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] |
Spectral Graph Theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one. Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière number. Cospectral graphs Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs are isospectral, that is, if the adjacency matrices have equal multisets of eigenvalues. Cospectral graphs need not be isomorphic, but isomorphic graphs a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harries–Wong Graph
In the mathematical field of graph theory, the Harries–Wong graph is a 3- regular undirected graph with 70 vertices and 105 edges. The Harries–Wong graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian. It is also a 3- vertex-connected and 3- edge-connected non-planar cubic graph. It has book thickness 3 and queue number 2. The characteristic polynomial of the Harries–Wong graph is : (x-3) (x-1)^4 (x+1)^4 (x+3) (x^2-6) (x^2-2) (x^4-6x^2+2)^5 (x^4-6x^2+3)^4 (x^4-6x^2+6)^5. \, History In 1972, A. T. Balaban published a (3-10)-cage graph, a cubic graph that has as few vertices as possible for girth 10. It was the first (3-10)-cage discovered but it was not unique. The complete list of (3-10)-cages and the proof of minimality was given by O'Keefe and Wong in 1980. There exist three distinct (3-10)-cage graphs—the Balaban 10-cage, the Harries graph In the mathematical field of graph theory, the Harries graph or Harrie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balaban 10-cage
In the mathematical field of graph theory, the Balaban 10-cage or Balaban -cage is a 3-regular graph with 70 vertices and 105 edges named after Alexandru T. Balaban. Published in 1972, It was the first 10-cage discovered but it is not unique. The complete list of 10-cages and the proof of minimality was given by Mary R. O'Keefe and Pak Ken Wong. There exist 3 distinct -cages, the other two being the Harries graph and the Harries–Wong graph. Moreover, the Harries–Wong graph and Harries graph are cospectral graphs. The Balaban 10-cage has chromatic number 2, chromatic index 3, diameter 6, girth 10 and is hamiltonian. It is also a 3- vertex-connected graph and 3- edge-connected. The book thickness is 3 and the queue number is 2.Jessica Wolz, ''Engineering'' ''Linear Layouts with SAT''. Master Thesis, Universität Tübingen, 2018 The characteristic polynomial of the Balaban 10-cage is : (x-3) (x-2) (x-1)^8 x^2 (x+1)^8 (x+2) (x+3) \cdot :\cdot(x^2-6)^2 (x^2-5)^4 (x^2-2)^2 ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cage Graph
In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an is defined to be a graph in which each vertex has exactly neighbors, and in which the shortest cycle has length exactly . An is an with the smallest possible number of vertices, among all . A is often called a . It is known that an exists for any combination of and . It follows that all exist. If a Moore graph exists with degree and girth , it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth must have at least :1+r\sum_^(r-1)^i vertices, and any cage with even girth must have at least :2\sum_^(r-1)^i vertices. Any with exactly this many vertices is by definition a Moore graph and therefore automatically a cage. There may exist multiple cages for a given combination of and . For instance there are three nonisomorphic , each with 70 vertices: the Balaban 10-cage, the H ... [...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] |
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] |
Cubic Graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Symmetry In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with each possible ... [...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] |
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] |
