The Deformed Consensus Protocol

International audienceThis paper studies a generalization of the standard continuous-time consensus protocol, obtained by replacing the Laplacian matrix of the communication graph with the so-called deformed Laplacian. The deformed Laplacian is a second-degree matrix polynomial in the real variable s which reduces to the standard Laplacian for s equal to unity. The stability properties of the ensuing deformed consensus protocol are studied in terms of parameter s for some special families of undirected and directed graphs, and for arbitrary graph topologies by leveraging the spectral theory of quadratic eigenvalue problems. Examples and simulation results are provided to illustrate our theoretical findings

The von Neumann Theil index : characterizing graph centralization using the von Neumann index

We show that the von Neumann entropy (from herein referred to as the von Neumann index) of a graph’s trace normalized combinatorial Laplacian provides structural information about the level of centralization across a graph. This is done by considering the Theil index, which is an established statistical measure used to determine levels of inequality across a system of ‘agents’, for example, income levels across a population. Here, we establish a Theil index for graphs, which provides us with a macroscopic measure of graph centralization. Concretely, we show that the von Neumann index can be used to bound the graph’s Theil index, and thus we provide a direct characterization of graph centralization via the von Neumann index. Because of the algebraic similarities between the bound and the Theil index, we call the bound the von Neumann Theil index. We elucidate our ideas by providing examples and a discussion of different n=7 vertex graphs. We also discuss how the von Neumann Theil index provides a more comprehensive measure of centralization when compared to traditional centralization measures, and when compared to the graph’s classical Theil index. This is because it more accurately accounts for macro-structural changes that occur from micro-structural changes in the graph (e.g. the removal of a vertex). Finally, we provide future direction, showing that the von Neumann Theil index can be generalized by considering the Rényi entropy. We then show that this generalization can be used to bound the negative logarithm of the graph’s Jain fairness index

Bounds for the Generalized Distance Eigenvalues of a Graph

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D(G) be the distance matrix, DL(G) be the distance Laplacian, DQ(G) be the distance signless Laplacian, and Tr(G) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by Dα(G) the generalized distance matrix, i.e., Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue

International Journal of Mathematical Combinatorics, Vol.6

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010

Peer Reviewe