233 research outputs found
Enumeration of Hamiltonian cycles in certain generalized Petersen graphs
AbstractThe generalized Petersen graph P(n, k) has vertex set V={u0, u1, …, un−1, v0, v1, …, vn−1} and edge set E={uiui+1, uivi, vivi+k∥ for 0≤i≤n−1 with indices taken modulo n}. The classification of the Hamiltonicity of generalized Petersen graphs was begun by Watkins, continued by Bondy and Bannai, and completed by Alspach. We now determine the precise number of Hamiltonian cycles present in each of the graphs P(n, 2). This more detailed information allows us to identify an infinite family of counterexamples to a conjecture of Greenwell and Kronk who had suggested a relation between uniquely 3-edge-colorable cubic graphs and the number of Hamiltonian cycles present
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
3nj Morphogenesis and Semiclassical Disentangling
Recoupling coefficients (3nj symbols) are unitary transformations between
binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum
operators. They have been used in a variety of applications in spectroscopy,
quantum chemistry and nuclear physics and quite recently also in quantum
gravity and quantum computing. These coefficients, naturally associated to
cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and
analytical features that make them fashinating objects to be studied on their
own. In this paper we develop a bottom--up, systematic procedure for the
generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and
algebraic methods. We provide also a novel approach to the problem of
classifying various regimes of semiclassical expansions of 3nj coefficients
(asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial,
analytical and numerical tools
Computing the Zero Forcing Number for Generalized Petersen Graphs
Let G be a simple undirected graph with each vertex colored either white or black, u be a black vertex of G, and exactly one neighbor v of u be white. Then change the color of v to black. When this rule is applied, we say u forces v, and write u ® v . A zero forcing set of a graph G is a subset Z of vertices such that if initially the vertices in Z are colored black and remaining vertices are colored white, the entire graph G may be colored black by repeatedly applying the color-change rule. The zero forcing number of G, denoted Z(G), is the minimum size of a zero forcing set.In this paper, we investigate the zero forcing number for the generalized Petersen graphs (It is denoted by P(n,k)). We obtain upper and lower bounds for the zero forcing number for P(n,k). We show that Z(P(n,2))=6 for n ³ 10, Z(P(n,3))=8 for n ³ 12 and Z(P(2k+1,k))=6 for k ³ 5
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