Distinguishing Number for some Circulant Graphs

Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial automorphism of $G$. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs depending on $n$. In this paper, we study circulant graphs of order $n$ where the adjacency is defined using a symmetric subset $A$ of $\mathbb{Z}_n$, called generator. We give a construction of a family of circulant graphs of order $n$ and we show that this class has distinct distinguishing numbers and these lasters are not depending on $n$

A New Game Invariant of Graphs: the Game Distinguishing Number

The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A distinguishing $d$-coloring is a coloring $c:V(G)\rightarrow\{1,...,d\}$ invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing number. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph $G$ with a set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $G$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then the Gentle wins if the coloring is distinguishing and the Rascal wins otherwise. This game leads to define two new invariants for a graph $G$, which are the minimum numbers of colors needed to ensure that the Gentle has a winning strategy, depending on who starts. These invariants could be infinite, thus we start by giving sufficient conditions to have infinite game distinguishing numbers. We also show that for graphs with cyclic automorphisms group of prime odd order, both game invariants are finite. After that, we define a class of graphs, the involutive graphs, for which the game distinguishing number can be quadratically bounded above by the classical distinguishing number. The definition of this class is closely related to imprimitive actions whose blocks have size $2$. Then, we apply results on involutive graphs to compute the exact value of these invariants for hypercubes and even cycles. Finally, we study odd cycles, for which we are able to compute the exact value when their order is not prime. In the prime order case, we give an upper bound of $3$

A gravimetric quasi-geoid evaluation in the Northern region of Algeria using EGM96 and GPS/Levelling.

The use of GPS for the estimation of orthometric heights in a given region, with the help of existing levelling data requires the determination of a local geoid and the bias between the local levelling and the one implicitly defined when the geoid is calculated which is generally based on the gravity anomalies data. The heights of new data can be collected swiftly without using the orthometric heights from levelling; it is what one calls commonly levelling by GPS. In this framework, the Least Squares Collocation method (LSC) has been used to evaluate the quality of the available GPS-Levelling data, to determine a gravimetric geoid in the North region of Algeria and to estimate the constant datum bias. The data used in the setting of this study are: The geopotential model EGM96, a total number of 2534 gravity anomalies, as well as 43 GPS points connected to the geodetic network levelling present on the whole North part of Algerian

Event-based controller synthesis by bounding methods

International audienceTwo event-triggered algorithms for digital implementation of a continuous-time stabilizing controller are proposed in this work. The first algorithm updates the control value in order to keep the time evolution of a given Lyapunov-like function framed between two auxiliary functions; whereas the second one actualizes the control value so that the state trajectory of the system stays enclosed between two a priori defined templates. In both cases, a natural hybrid formulation of the event-based stabilizing control problem is used to prove the main results of this work. Furthermore, the existence of a minimum inter-event time greater than zero is proved. Numerical simulations are provided to illustrate the digital implementation of the event-sampling algorithms for nonlinear systems

Distinguishing numbers and distinguishing indices of oriented graphs

A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected graph G is a mapping λ from the set of vertices (resp. the set of edges) of G to the set of labels {1,. .. , r} such that no non-trivial automorphism of G preserves all the vertex (resp. edge) labels. The distinguishing number D(G) and the distinguishing index D (G) of G are then the smallest r for which G admits a distinguishing r-vertex-labelling or r-edge-labelling, respectively. The distinguishing chromatic number D χ (G) and the distinguishing chromatic index D χ (G) are defined similarly, with the additional requirement that the corresponding labelling must be a proper colouring. These notions readily extend to oriented graphs, by considering arcs instead of edges. In this paper, we study the four corresponding parameters for oriented graphs whose underlying graph is a path, a cycle, a complete graph or a bipartite complete graph. In each case, we determine their minimum and maximum value, taken over all possible orientations of the corresponding underlying graph, except for the minimum values for unbalanced complete bipartite graphs K m,n with m = 2, 3 or 4 and n > 3, 6 or 13, respectively, or m ≥ 5 and n > 2 m − m 2 , for which we only provide upper bounds

Event-Based Sampling Algorithm for Setpoint Tracking Using a State-Feedback Controller

International audienceEvent-based control techniques are investigated for output reference tracking in the case of linear time-invariant systems. In event-based control, the controller remains at rest if the system is behaving according to some predefined conditions, the feedback loop being closed only when the system states violate these conditions. In this work a reference system, which consists in the continuously-controlled version of the system under study, is employed. Based on the difference between the state of the event-triggered system and that of the reference system, we define a Lyapunov-like function, and show that if we can keep this function confined to a certain region, the tracking error would also be bounded. The trespassing of this function outside of the desired region is used as an event-triggering condition

A Simple ADMM Solution To Sparse-Modeling-Based Detectors For Massive MIMO Systems

International audienceWe give a simple yet efficient Alternating Direction Method of Multipliers algorithm for solving sparse-modeling-based detectors [7, 9] for massive MIMO systems. Our solution relies on a special reformulation of the associated optimization problem by describing the constraints as a Cartesian power of the probability simplex. Simulation results show that the proposed algorithm is as accurate as the best known solvers (interior point methods), while its complexity remains linear with respect to the size of the system

Uncertainty and Quality rating in Analytical Vulnerability Assessment

Fragility curves represent a major component of seismic risk and vulnerability assessment of buildings and infrastructure facilities. A recently conducted extensive literature review under the framework of developing the “GEM Guide for Selecting of Existing Analytical Fragility Curves and Compilation of the Database”, shows that there is a wealth of existing analytically derived fragility curves that can be used for future applications. However, the main challenge in using these curves is how to identify and, if necessary, combine suitable fragility curves from a pool of curves with different characteristics and, often unknown, reliability. The present article introduces a rating system that has been developed following detail review and critique of the various methodologies on the derivation of analytical fragility curves that have been generated in the past two decades. The main scope is to provide guidance, either in choosing suitable and robust existing fragility curves or in generating new fragility curves. The quality rating system rates the quality of a curve according to the effect that various parameters, simulation procedures and assumptions on the reliability of fragility curve. It also assists and steers potential analysts towards a better identification and quantification of expected uncertainties throughout the process