722 research outputs found
Controllability and observability of grid graphs via reduction and symmetries
In this paper we investigate the controllability and observability properties
of a family of linear dynamical systems, whose structure is induced by the
Laplacian of a grid graph. This analysis is motivated by several applications
in network control and estimation, quantum computation and discretization of
partial differential equations. Specifically, we characterize the structure of
the grid eigenvectors by means of suitable decompositions of the graph. For
each eigenvalue, based on its multiplicity and on suitable symmetries of the
corresponding eigenvectors, we provide necessary and sufficient conditions to
characterize all and only the nodes from which the induced dynamical system is
controllable (observable). We discuss the proposed criteria and show, through
suitable examples, how such criteria reduce the complexity of the
controllability (respectively observability) analysis of the grid
Spectral graph theory : from practice to theory
Graph theory is the area of mathematics that studies networks, or graphs. It arose from the need to analyse many diverse network-like structures like road networks, molecules, the Internet, social networks and electrical networks. In spectral graph theory, which is a branch of graph theory, matrices are constructed from such graphs and analysed from the point of view of their so-called eigenvalues and eigenvectors. The first practical need for studying graph eigenvalues was in quantum chemistry in the thirties, forties and fifties, specifically to describe the Hückel molecular orbital theory for unsaturated conjugated hydrocarbons. This study led to the field which nowadays is called chemical graph theory. A few years later, during the late fifties and sixties, graph eigenvalues also proved to be important in physics, particularly in the solution of the membrane vibration problem via the discrete approximation of the membrane as a graph. This paper delves into the journey of how the practical needs of quantum chemistry and vibrating membranes compelled the creation of the more abstract spectral graph theory. Important, yet basic, mathematical results stemming from spectral graph theory shall be mentioned in this paper. Later, areas of study that make full use of these mathematical results, thus benefitting greatly from spectral graph theory, shall be described. These fields of study include the P versus NP problem in the field of computational complexity, Internet search, network centrality measures and control theory.peer-reviewe
Perfect State Transfer in Laplacian Quantum Walk
For a graph and a related symmetric matrix , the continuous-time
quantum walk on relative to is defined as the unitary matrix , where varies over the reals. Perfect state transfer occurs
between vertices and at time if the -entry of
has unit magnitude. This paper studies quantum walks relative to graph
Laplacians. Some main observations include the following closure properties for
perfect state transfer:
(1) If a -vertex graph has perfect state transfer at time relative
to the Laplacian, then so does its complement if is an integer multiple
of . As a corollary, the double cone over any -vertex graph has
perfect state transfer relative to the Laplacian if and only if . This was previously known for a double cone over a clique (S. Bose,
A. Casaccino, S. Mancini, S. Severini, Int. J. Quant. Inf., 7:11, 2009).
(2) If a graph has perfect state transfer at time relative to the
normalized Laplacian, then so does the weak product if for any
normalized Laplacian eigenvalues of and of , we have
is an integer multiple of . As a corollary, a weak
product of with an even clique or an odd cube has perfect state
transfer relative to the normalized Laplacian. It was known earlier that a weak
product of a circulant with odd integer eigenvalues and an even cube or a
Cartesian power of has perfect state transfer relative to the adjacency
matrix.
As for negative results, no path with four vertices or more has antipodal
perfect state transfer relative to the normalized Laplacian. This almost
matches the state of affairs under the adjacency matrix (C. Godsil, Discrete
Math., 312:1, 2011).Comment: 26 pages, 5 figures, 1 tabl
Zero forcing sets and controllability of dynamical systems defined on graphs
In this paper, controllability of systems defined on graphs is discussed. We
consider the problem of controllability of the network for a family of matrices
carrying the structure of an underlying directed graph. A one-to-one
correspondence between the set of leaders rendering the network controllable
and zero forcing sets is established. To illustrate the proposed results,
special cases including path, cycle, and complete graphs are discussed.
Moreover, as shown for graphs with a tree structure, the proposed results of
the present paper together with the existing results on the zero forcing sets
lead to a minimal leader selection scheme in particular cases
Non-stationary quantum walks on the cycle
We consider quantum walks on the cycle in the non-stationary case where the
`coin' operation is allowed to change at each time step. We characterize, in
algebraic terms, the set of possible state transfers and prove that, as opposed
to the stationary case, it is possible to asymnptotically reach a uniform
distribution among the nodes of the associated graph.Comment: Revised version with minor change
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