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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
Strong secure domination in graphs
In a graph G , subset T ⊆ V(G) is defined as a dominating set of G if, for every vertex not in T , a neighbor in T exists. The subset T ⊆ V(G) is a secure dominating set of G when T functions as a dominating set of G and, for each vertex x ∈ V(G) \\ T , there is a vertex y ∈ T satisfying x , y are adjacent and ( T \\ y ) ∪ x is a dominating set of G . Furthermore, if for every vertex x ∈ V(G) \\ T , there is a vertex y ∈ T satisfying x , y are adjacent and d G ( y ) ≥ d G ( x ), then a secure dominating set T of G is also a strong secure dominating set of G . In this paper, we prove that the decision problem corresponding to the strong secure domination problem is NP-complete on split and bipartite graphs, and determine the exact value of the strong secure domination number for the middle graph of the path.
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
Classification of higher grade ℓ graphs for U(N)2×O(D) multi-matrix models
The authors studied in [Ann. Inst. Henri Poincaré D 9 (2022), 367–433], a complex multi-matrix model with U(N)2×O(D) symmetry, and whose double scaling limit where simultaneously the large-N and large-D limits were taken while keeping the ratio N/D =M finite and fixed. In this double scaling limit, the complete recursive characterization of the Feynman graphs of arbitrary genus for the leading order grade ℓ=0 was achieved. In this current study, we classify the higher order graphs in ℓ. More specifically, ℓ=1 and ℓ=2 with arbitrary genus, in addition to a specific class of two-particle-irreducible (2PI) graphs for higher ℓ⩾3 but with genus zero. Furthermore, we demonstrate that each 2PI graph with a single O(D)-loop with an arbitrary ℓ corresponds to a reduced alternating knot diagram with ℓ crossings as listed in the Rolfsen knot table, or a resulting alternating knot diagram obtained after performing the Tait flyping moves. We generalize to 2PR by considering the connected sum and the Reidemeister move I.
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