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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 proof of minimality of the number of vertices 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 (x^4-6 x^2+ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. 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 edges 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balaban 11-cage
In the mathematical field of graph theory, the Balaban 11-cage or Balaban (3,11)-cage is a 3-regular graph with 112 vertices and 168 edges named after Alexandru T. Balaban. The Balaban 11-cage is the unique (3,11)-cage. It was discovered by Balaban in 1973. The uniqueness was proved by Brendan McKay and Wendy Myrvold in 2003. The Balaban 11-cage is a Hamiltonian graph and can be constructed by excision from the Tutte 12-cage by removing a small subtree and suppressing the resulting vertices of degree two.Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008) It has independence number 52, chromatic number 3, chromatic index 3, radius 6, diameter 8 and girth 11. It is also a 3- vertex-connected graph and a 3- edge-connected graph. The characteristic polynomial of the Balaban 11-cage is: :(x-3) x^ (x^2-6)^5 (x^2-2)^ (x^3-x^2-4 x+2)^2\cdot :\cdot(x^3+x^2-6 x-2) (x^4-x^3-6 x^2+4 x+4)^4 \cdot :\cdot(x^5+x^4-8 x^3-6 x^2+12 x+4)^8. The automorphism group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Graph
In chemical graph theory and in mathematical chemistry, a molecular graph or chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory. A chemical graph is a labeled graph whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds. Its vertices are labeled with the kinds of the corresponding atoms and edges are labeled with the types of bonds. For particular purposes any of the labelings may be ignored. A hydrogen-depleted molecular graph or hydrogen-suppressed molecular graph is the molecular graph with hydrogen vertices deleted. In some important cases (topological index calculation etc.) the following classical definition is sufficient: a molecular graph is a connected, undirected graph which admits a one-to-one correspondence with the structural formula of a chemical compound in which the vertices of the graph correspond to atoms of the molecule and edges of the graph correspond to chemical ... [...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 basis (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 correspondi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Queue Number
In the mathematical field of graph theory, the queue number of a Graph (discrete mathematics), graph is a graph invariant defined analogously to book thickness, stack number (book thickness) using Queue (abstract data type), first-in first-out (queue) orderings in place of Stack (abstract data type), last-in first-out (stack) orderings. Definition A queue layout of a given graph is defined by a total ordering of the vertex (graph theory), vertices of the graph together with a partition of the edge (graph theory), 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 (abstract data type), queue data structure, by considering the vertices in ... [...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 in 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, e ... [...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 the subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is said to be ''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 minimum cut in ''G''. The edge connectivity version of Menger's theorem provides an alternative and equivalent character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected Graph (discrete mathematics), graph is said to be -vertex-connected (or -connected) if it has more than Vertex (graph theory), vertices and remains Connectivity (graph theory), 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. In complete graphs, there is no subset whose removal would disconnect the graph. Some sources modify the definition of connectivity to handle this case, by defining it as the size of the smallest subset of vertices whose deletion results in either a disconnected graph or a single vertex. For this variation, the connectivity of a complete graph K_n is n-1. An equivalent definition is that a graph with at least two vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Path
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. The computational problems of determining whether such paths and cycles exist in graphs are NP-complete; see Hamiltonian path problem for details. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Girth (graph Theory)
In graph theory, the girth of an undirected graph is the length of a shortest Cycle (graph theory), cycle contained in the graph. If the graph does not contain any cycles (that is, it is a forest (graph theory), forest), its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free graph, triangle-free. Cages A cubic graph (all vertices have degree three) of girth that is as small as possible is known as a -cage (graph theory), cage (or as a -cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance (graph Theory)
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite. In the case of a directed graph the distance between two vertices and is defined as the length of a shortest directed path from to consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, does not necessarily coincide with —so it is just a quasi-metric, and it might be the case that one is defined while the other is not. Related concepts A metric space defined over a set of points in terms of distances in a graph defined ov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
