Abstract In this paper we report on the properties of matching polynomial α( G ) a graph . We present number recursion formulas for ), from which it follows that many families orthogonal polynomials arise as suitable graphs. consider relation between and characteristic graph. Finally, results provide information zeros ).

A new graph product is introduced, and the characteristic polynomial of a so–formed given as function polynomials factor graphs. class trees produced using this shown to be characterized by spectral properties.

The last decade has witnessed substantial interest in protocols for transferring information on networks of quantum mechanical objects. A variety control methods and network topologies have been proposed, the basis that transfer with perfect fidelity --- i.e. deterministic without loss is impossible through unmodulated spin chains more than a few particles. Solving original problem formulated by Bose [Phys. Rev. Lett. 91, 207901 (2003)], we determine exact number qubits (with XY Hamiltonian)...

Abstract The matching polynomial α( G, x ) of a graph G is form the generating function for number sets k independent edges . in this paper we show that if with vertex v then there tree T w such \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{\alpha (G\backslash v, x)}}{{\alpha (G, x)}} = (T\backslash w, (T, x)}}. $\end{document} This result has consequences. Here use it to prove \ , 1/ )/ certain class walks G. As an application these results establish some new properties ).

AbstractLet X be a graph on n vertices with adjacency matrix A andlet H(t) denote the matrix-valued function exp(iAt). If u and v aredistinct in X, we say perfectstatetransferfrom to occursif there is time τ such that |H(τ) u,v | = 1. Our chief problem tocharacterize cases where perfect state transfer occurs. We showthat if does occur graph, then square ofits spectral radius either an integer or lies quadratic extensionof rationals. From this deduce for any k thereonly finitely many graphs...

Cubelike graphs are the Cayley of elementary Abelian group ${\mathbb{Z}}_{2}^{n}$ (e.g., hypercube is a cubelike graph). We study perfect state transfer between two particles in quantum networks modeled by large class graphs. This generalizes results Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and Facer A (2008)].

Pretty good state transfer in networks of qubits occurs when a continuous-time quantum walk allows the transmission qubit from one node network to another, with fidelity arbitrarily close 1. We prove that Heisenberg chain n there is pretty between nodes at j-th and (n-j+1)-th position if power 2. Moreover, this condition also necessary for j=1. obtain result by applying theorem due Kronecker about Diophantine approximations, together techniques algebraic graph theory.