The purpose of the paper is to provide an answer to a long standing problem to compute the distance distribution among the nodes in a star graph, i.e., to compute the exact number of nodes at a distance k from the identity node in a star graph where k varies from 0 to the diameter of the graph. A star graph is a Cayley graph like the hypercubes; for a hypercube Qn, there are exactly ` n r ́ node...

In this note we show that the concept of incidence coloring introduced in [BM] is a special case of directed star arboricity, introduced in [AA]. A conjecture in [BM] concerning asmyptotics of the incidence coloring number is solved in the negative following an example in [AA]. We generalize Theorem 2.1 of [AMR] concerning the star arboricity of graphs to the directed case by a slight modificat...

We investigate synchronization in the Kuramoto model with noise on a star graph. By revising case of complete graph, we propose closed form self-consistency equation for conventional order parameter and generalize it Using obtained equation, demonstrate that there is crossover between abrupt at small continuous phase transition quite large noise. probe this numerically analytically.

Knowledge management is a critical issue for the next-generation web application, because the next-generation web is becoming a semantic web, a knowledge-intensive network. XML Topic Map (XTM), a new standard, is appearing in this field as one of the structures for the semantic web. It organizes information in a way that can be optimized for navigation. In this paper, a new set of hyper-graph o...

In 1996, Ota and Tokuda showed that a star-free graph with sufficiently high minimum degree admits a 2-factor. More recently it was shown that the minimum degree condition can be significantly reduced if one also requires that the graph is not only star-free but also of sufficiently high edge-connectivity. In this paper we reduce the minimum degree condition further for star-free graphs that al...

Let Sn be an n-dimensional Star graph. In this paper, we show that Sn − F is Hamiltonian laceable where F is the set of f ≤ (n − 4) pairs of adjacent faulty vertices, Sn − F is Hamiltonian where F is the set of f ≤ (n − 3) pairs of adjacent faulty vertices. We also show that Sn − F is hyper-Hamiltonian laceable where F is the set of f ≤ (n − 4) pairs of adjacent faulty vertices. Applying these ...

We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter's regular hyperbolic honeycombs (with innnite or star-shaped cells or vertex gures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m 7, star-honeycomb...

The distance $d(u,v)$ between two vertices $u$ and $v$ of a graph $G$ is equal to the length of a shortest path that connects $u$ and $v$. Define $WW(G,x) = 1/2sum_{{ a,b } subseteq V(G)}x^{d(a,b) + d^2(a,b)}$, where $d(G)$ is the greatest distance between any two vertices. In this paper the hyper-Wiener polynomials of the Cartesian product, composition, join and disjunction of graphs are compu...

We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter’s regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m ≥ 7, star-hone...