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Graph theoretic methods in multiagent networks
This accessible book provides an introduction to the analysis and design of dynamic multiagent networks. Such networks are of great interest in a wide range of areas in science and engineering, including: mobile sensor networks, distributed robotics such as formation flying and swarming, quantum networks, networked economics, biological synchronization, and social networks. Focusing on graph theoretic methods for the analysis and synthesis of dynamic multiagent networks, the book presents a powerful new formalism and set of tools for networked systems.
The book's three sections look at foundations, multiagent networks, and networks as systems. The authors give an overview of important ideas from graph theory, followed by a detailed account of the agreement protocol and its various extensions, including the behavior of the protocol over undirected, directed, switching, and random networks. They cover topics such as formation control, coverage, distributed estimation, social networks, and games over networks. And they explore intriguing aspects of viewing networks as systems, by making these networks amenable to control-theoretic analysis and automatic synthesis, by monitoring their dynamic evolution, and by examining higher-order interaction models in terms of simplicial complexes and their applications.
The book will interest graduate students working in systems and control, as well as in computer science and robotics. It will be a standard reference for researchers seeking a self-contained account of system-theoretic aspects of multiagent networks and their wide-ranging applications.
This book has been adopted as a textbook at the following universities:
University of Stuttgart, GermanyRoyal Institute of Technology, SwedenJohannes Kepler University, AustriaGeorgia Tech, USAUniversity of Washington, USAOhio University, USA
eBook
Expander families and Cayley graphs : a beginner's guide
\"The theory of expander graphs is a rapidly developing topic in mathematics and computer science, with applications to communication networks, error-correcting codes, cryptography, complexity theory, and much more. Expander Families and Cayley Graphs: A Beginner's Guide is a comprehensive introduction to expander graphs, designed to act as a bridge between classroom study and active research in the field of expanders. It equips those with little or no prior knowledge with the skills necessary to both comprehend current research articles and begin their own research. Central to this book are four invariants that measure the quality of a Cayley graph as a communications network-the isoperimetric constant, the second-largest eigenvalue, the diameter, and the Kazhdan constant. The book poses and answers three core questions: How do these invariants relate to one another? How do they relate to subgroups and quotients? What are their optimal values/growth rates? Chapters cover topics such as: ℗ʺ Graph spectra ℗ʺ A Cheeger-Buser-type inequality for regular graphs ℗ʺ Group quotients and graph coverings ℗ʺ Subgroups and Schreier generators ℗ʺ Ramanujan graphs and the Alon-Boppana theorem ℗ʺ The zig-zag product and its relation to semidirect products of groups ℗ʺ Representation theory and eigenvalues of Cayley graphs ℗ʺ Kazhdan constants The only introductory text on this topic suitable for both undergraduate and graduate students, Expander Families and Cayley Graphs requires only one course in linear algebra and one in group theory. No background in graph theory or representation theory is assumed. Examples and practice problems with varying complexity are included, along with detailed notes on research articles that have appeared in the literature. Many chapters end with suggested research topics that are ideal for student projects\"-- Provided by publisher.
Book
On an uncountable family of graphs whose spectrum is a Cantor set
For each p≥1, the star automaton group GSp is an automaton group which can be defined starting from a star graph on p+1 vertices. We study Schreier graphs associated with the action of the group GSp on the regular rooted tree Tp+1 of degree p+1 and on its boundary ∂Tp+1. With the transitive action on the n-th level of Tp+1 is associated a finite Schreier graph Γnp, whereas there exist uncountably many orbits of the action on the boundary, represented by infinite Schreier graphs which are obtained as the limits of the sequence {Γnp}n≥1 in the Gromov–Hausdorff topology. We obtain an explicit description of the spectrum of the graphs {Γnp}n≥1. Then, by using amenability of GSp, we prove that the spectrum of each infinite Schreier graph is the union of a Cantor set of zero Lebesgue measure, which is the Julia set of the quadratic map fp(z)=z2−2(p−1)z−2p, and a countable collection of isolated points supporting the Kesten–Neumann–Serre spectral measure. We also give a complete classification of the infinite Schreier graphs up to isomorphism of unrooted graphs, showing that they may have 1, 2 or 2p ends, and that the case of 1 end is generic with respect to the uniform measure on ∂Tp+1.
Journal Article
NOWHERE-ZERO \\(3\\)-FLOWS IN CAYLEY GRAPHS OF ORDER \\(8p\\)
It is proved that Tutte’s \\(3\\)-flow conjecture is true for Cayley graphs on groups of order \\(8p\\) where p is an odd prime.
Journal Article
Bavard duality for the relative Gromov seminorm
The relative Gromov seminorm is a finer invariant than stable commutator length where a relative homology class is fixed. We show a duality result between bounded cohomology and the relative Gromov seminorm, analogously to Bavard duality for scl. We give an application to computations of scl in graphs of groups. We also explain how our duality result can be given a purely algebraic interpretation via a relative version of the Hopf formula. Moreover, we show that this leads to a natural generalisation of a result of Calegari on a connection between scl and the rotation quasimorphism.
Journal Article
Slab theorem and halfspace theorem for constant mean curvature surfaces in H2×R
We prove that a properly embedded annular end of a surface in H2×R with constant mean curvature 0
Journal Article
Effective Edge Domination in iterated Jump Graph of Intuitionistic Fuzzy Graph
Intuitionistic Fuzzy Graphs (InFGs) serve as a sophisticated framework for modeling complex and uncertain phenomena across diverse domains, such as decision-making, economics, medicine, computer science, and engineering. In this research, we develop and analyse the properties of jump graphs in the context of InFGs. The vertex set of the jump graph J(G) of a graph G is defined as the edge set of G, with adjacency between vertices in J(G) established if and only if the corresponding edges in G are non-incident. We systematically construct sequences of jump graphs for InFGs through iterative processes and investigate the structural characteristics of these sequences. Moreover, we introduce the concept of an effective edge dominating set for jump graphs of InFGs and rigorously determine the effective edge domination number for certain classes of graphs. These contributions enhance the theoretical foundation of InFGs and extendtheir applicability to solving real-world problems characterized by uncertainty and complexity
Journal Article
Weakly Modular Graphs and Nonpositive Curvature
This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of
cell complexes derived from them. The unifying themes of our investigation are various “nonpositive curvature\" and “local-to-global”
properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a
far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs,
bridged graphs, and dual polar graphs occurring under different disguises (
We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple
connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit
characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the
clique-Helly graphs with simply connected clique complexes. With
eBook
Graphs that embedded in any fixed surfaces with sufficiently large maximum degree Δ is total-(Δ+1)-colorable
Let G be a graph that can be embedded in a surface Σ of Euler characteristic c′(Σ) . In this paper, we proved that there exists an integer Δ0=4⋅(5+√49−24c′)⋅[2−c′+12(5+√49−24c′)] such that the total chromatic number of G is Δ(G)+1 if Δ(G)≥Δ0.
Journal Article