Kernels of Directed Graph Laplacians

Let G denote a directed graph with adjacency matrix Q and in- degree matrix D. We consider the Kirchhoff matrix L = D − Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace--namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights

Electron Bernstein waves emission in the TJ-II Stellarator

Taking advantage of the electron Bernstein waves heating (EBWH) system of the TJ-II stellarator, an electron Bernstein emission (EBE) diagnostic was installed. Its purpose is to investigate the B-X-O radiation properties in the zone where optimum theoretical EBW coupling is predicted. An internal movable mirror shared by both systems allows us to collect the EBE radiation along the same line of sight that is used for EBW heating. The theoretical EBE has been calculated for different orientations of the internal mirror using the TRUBA code as ray tracer. A comparison with experimental data obtained in NBI discharges is carried out. The results provide a valuable information regarding the experimental O-X mode conversion window expected in the EBW heating experiments. Furthermore, the characterization of the radiation polarization shows evidence of the underlying B-X-O conversion process.Comment: 21 pages, 14 figure

The girth, odd girth, distance function, and diameter of generalized Johnson graphs

For any non-negative integers $v > k > i$, the {\em generalized Johnson graph}, $J(v,k,i)$, is the undirected simple graph whose vertices are the $k$-subsets of a $v$-set, and where any two vertices $A$ and $B$ are adjacent whenever $|A \cap B| =i$. In this article, we derive formulas for the girth, odd girth, distance function, and diameter of $J(v,k,i)$

Kernels of directed graph Laplacians

Abstract. Let G denote a directed graph with adjacency matrix Q and in-degree matrix D. We consider the Kirchhoff matrix L = D−Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was recently observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace -namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights. AMS 2002 Subject Classification: Primary 05C50. Keywords: Kirchhoff matrix. Eigenvalues of Laplacians. Graphs. Stochastic matrix. Definitions Let G denote a directed graph with vertex set V = {1, 2, ..., N } and edge set E ⊆ V ×V . To each edge uv ∈ E, we allow a positive weight ω uv to be assigned. The adjacency matrix Q is the N × N matrix whose rows and columns are indexed by the vertices, and where the ij-entry is ω ji if ji ∈ E and zero otherwise

Shift-Symmetric Configurations in Two-Dimensional Cellular Automata: Irreversibility, Insolvability, and Enumeration

The search for symmetry as an unusual yet profoundly appealing phenomenon, and the origin of regular, repeating configuration patterns have long been a central focus of complexity science and physics. To better grasp and understand symmetry of configurations in decentralized toroidal architectures, we employ group-theoretic methods, which allow us to identify and enumerate these inputs, and argue about irreversible system behaviors with undesired effects on many computational problems. The concept of so-called configuration shift-symmetry is applied to two-dimensional cellular automata as an ideal model of computation. Regardless of the transition function, the results show the universal insolvability of crucial distributed tasks, such as leader election, pattern recognition, hashing, and encryption. By using compact enumeration formulas and bounding the number of shift-symmetric configurations for a given lattice size, we efficiently calculate the probability of a configuration being shift-symmetric for a uniform or density-uniform distribution. Further, we devise an algorithm detecting the presence of shift-symmetry in a configuration. Given the resource constraints, the enumeration and probability formulas can directly help to lower the minimal expected error and provide recommendations for system's size and initialization. Besides cellular automata, the shift-symmetry analysis can be used to study the non-linear behavior in various synchronous rule-based systems that include inference engines, Boolean networks, neural networks, and systolic arrays.Comment: 22 pages, 9 figures, 2 appendice

Te (R,t) Measurements using Electron Bernstein Wave Thermal Emission on NSTX

The National Spherical Torus Experiment (NSTX) routinely studies overdense plasmas with ne of (1–5) X 1019 m-3 and total magnetic field of <0.6 T, so that the first several electron cyclotron harmonics are overdense. The electrostatic electron Bernstein wave (EBW) can propagate in overdense plasmas, exhibits strong absorption, and is thermally emitted at electron cyclotron harmonics. These properties allow thermal EBW emission to be used for local Te measurement. A significant upgrade to the previous NSTX EBW emission diagnostic to measure thermal EBW emission via the oblique B-X-O mode conversion process has been completed. The new EBW diagnostic consists of two remotely steerable, quad-ridged horn antennas, each of which is coupled to a dual channel radiometer. Fundamental (8–18 GHz) and second and third harmonic (18–40 GHz) thermal EBW emission and polarization measurements can be obtained simultaneously

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page