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Algebraic Signed Graphs: A Review
Published in N. P. Shrimali, Nita H. Shah, Recent Advancements in Graph Theory, 2020
A marked signed graph is an ordered pair Σμ = (Σ, μ), where Σ = (G, σ) is a signed graph and μ: V(Σ) → { + , − } is a function from the vertex set V(Σ) into the set { + , − }, called the marking of Σ.
Bipartite consensus of nonlinear multi-agent systems based on delayed output signals
Published in International Journal of Systems Science, 2023
Wenjie Zhang, Xianfu Zhang, Yanjie Chang, Yanan Qi
The research results mentioned above, however, merely involved cooperative interactions among agents in spite of agents may also show competitive interactions with their neighbour agents. To represent cooperative and competitive interactions among agents, the signed graph is adopted, where the positive and negative weights stand for cooperation and competition between neighbour agents, respectively. With signed graphs in hand, Altafini (2013) proposed the concept of bipartite consensus firstly, where all agents converged to values with identical modulus. Based on the work of Altafini (2013), plenty of results on the bipartite consensus were discussed (see, e.g. Fiore et al., 2017; Qin et al., 2017; Valcher & Misra, 2014). Particularly, by virtue of signed graphs, K. Li et al. (2021) designed an output-feedback-based predefined-time bipartite consensus protocol for uncertain nonlinear multi-agent systems. More recently, Qi et al. (2023) utilised dynamic event-triggered and self-triggered controls to achieve the bipartite consensus for a class of high-order nonlinear multi-agent systems.
Bipartite consensus of multi-agent systems with reduced-order observer-based distributed control protocols
Published in International Journal of Systems Science, 2021
A weighted signed digraph , where is the set of nodes, is the set of edges, is used to model the communication network of a multi-agent system. In the network, an edge from node to is denoted by , which means that node can get information from node . The weight associated with edge is denoted by , and is positive/negative if the edge is positive/negative. If there is no edge from node to node , then . Furthermore, we assume that . The topology of a graph can be described by its adjacency matrix . Let explicitly denotes a graph, whose adjacency matrix is . A graph is undirected if , and directed otherwise. A graph, in which all the edges are positive, can be represented by a non-negative graph. A signed graph is a graph that have both positive and negative edges. A path from node to node exists if there are a sequence of edges . If there exists at least one node, i.e. the so called root, having a directed path to any other nodes, then the digraph has a directed spanning tree. Let , and it is called signature matrices set. The column vector with all entries are 1 is denoted by .