Quantum State Transfer in Graphs with Tails

We consider quantum state transfer on finite graphs which are attached to infinite paths. The finite graph represents an operational quantum system for performing useful quantum information tasks. In contrast, the infinite paths represent external infinite-dimensional systems which have limited (but nontrivial) interaction with the finite quantum system. We show that {\em perfect} state transfer can surprisingly still occur on the finite graph even in the presence of the infinite tails. Our techniques are based on a decoupling theorem for eventually-free Jacobi matrices, equitable partitions, and standard Lie theoretic arguments. Through these methods, we rehabilitate the notion of a dark subspace which had been so far viewed in an unflattering light.Comment: 25 pages, 7 figure

Approximate entropy of network parameters

We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We firstly define a purely structural entropy obtained by computing the approximate entropy of the so called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erd\H{o}s-R\'enyi networks. By using classical results of Pincus, we show that our entropy measure converges with network size to a certain binary Shannon entropy. On a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs permit to naturally associate to a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.Comment: 11 pages, 5 EPS figure

Spectral multiplicity functions of adjacency operators of graphs and cospectral infinite graphs

The adjacency operator of a graph has a spectrum and a class of scalar-valued spectral measures which have been systematically analyzed; it also has a spectral multiplicity function which has been less studied. The first purpose of this article is to review a small number of examples of infinite graphs $G = (V,E)$ for which the spectral multiplicity function of the adjacency operator $A_G$ of $G$ has been determined. The second purpose of this article is to show explicit examples of infinite connected graphs which are cospectral, i.e., which have unitarily equivalent adjacency operators

The rank of pseudo walk matrices : controllable and recalcitrant pairs

A pseudo walk matrixWv of a graph G having adjacency matrix A is an nXn matrix with columns (Formula presented.) whose Gram matrix has constant skew diagonals, each containing walk enumerations in G. We consider the factorization over Q of the minimal polynomial m(G, x) of A. We prove that the rank of Wv, for any walk vector v, is equal to the sum of the degrees of some, or all, of the polynomial factors of m(G, x). For some adjacency matrix A and a walk vector v, the pair (A, v) is controllable if Wv has full rank. We show that for graphs having an irreducible characteristic polynomial over Q, the pair (A, v) is controllable for any walk vector v. We obtain the number of such graphs on up to ten vertices, revealing that they appear to be commonplace. It is also shown that, for all walk vectors v, the degree of the minimal polynomial of the largest eigenvalue of A is a lower bound for the rank of Wv. If the rank of Wv attains this lower bound, then (A, v) is called a recalcitrant pair. We reveal results on recalcitrant pairs and present a graph having the property that (A, v) is neither controllable nor recalcitrant for any walk vector v.peer-reviewe