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Conditions for Unique Localizability in Cooperative Localization of Wireless ad hoc and Sensor Networks
Published in Chao Gao, Guorong Zhao, Hassen Fourati, Cooperative Localization and Navigation, 2019
Redundant rigidity is a stronger type of rigidity than (plain) rigidity. A rigid graph G = (V,E) is redundantly rigid if G' = (V,E'), where E' = E\{e} remains rigid for every edge e in E. For example, the graph in Figure 2.4 is not redundantly rigid, because if we remove the edge v2v3, it becomes flexible. On the other hand, the graph in Figure 2.5 is redundantly rigid; that is, if we remove any one of the edges in this graph, it still remains rigid. We note that a redundantly rigid graph does not need to have all the possible edges between the set of given vertices. For example, the edge v3v5 is missing in the graph in Figure 2.5. A graph that has edges between every pair of vertices is called a complete graph. An example of a complete graph is shown in Figure 2.6. We note that a complete graph G = (V,E) with |V| > 3 is always a redundantly rigid graph.
Variational Methods and Functional Legendre Transforms
Published in A.N. Vasiliev, Patricia A. Millard, Functional Methods in Quantum Field Theory and Statistical Physics, 2019
where sym denotes complete symmetrization in the arguments 1–4. The first term on the right-hand side, the vertex γ4, is generated by the graph the first term in the curly brackets is generated by the first of the 4-irreducible graphs (6.97), and the other terms are generated by the 4-reducible graphs (6.109). We note that in an even theory the kernel (6.99) coincides with γ4. The terms following the curly brackets in (6.113) represent the contribution of the 4-irreducible graphs of Γ above sixth order in the number of lines. In an even theory the first such contribution is the “envelope” in (6.113), which in Γ corresponds to a complete graph with five vertices. We recall that a complete graph is one in which any pair of vertices is connected by a single line. The symmetry group of a complete graph contains all permutations of its n vertices, and the symmetry number is n!.
Applications of Graph Theory
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
It is usual to restate the problem in a slightly different graph-theoretic form. The graph described above is replaced by a complete graph with one vertex for every town. (Recall that a complete graph is one in which there is a unique edge joining every pair of distinct vertices.) An edge is given weight equal to the shortest distance between the corresponding towns using the roads of the network. These shortest distances can be found by applying algorithm 11.7 to the original graph.
A network-based approach to examine the impact of within-city industry agglomeration on total factor productivity
Published in Quality Technology & Quantitative Management, 2020
Zhigang Cai, Ying Li, Yu Yvette Zhang
If one connects each pair of nodes with an arc in a network, the resulting network graph is a complete graph. However, the network-based measurements of interest to us are not meaningful on a complete graph. Hence, for the purpose of our research, we construct incomplete network graphs. That is, in each of our networks, not all pairs of nodes are connected. The key to the construction of an incomplete network graph is to specify the criterion which instructs whether two nodes are connected with an arc. Our construction relies on the intensity of mutual influence between two firms, which we define as Rij. If and only if the intensity of mutual influence between two firms is beyond a predetermined threshold, we connect the two nodes (firms) with an arc.
Robust formation control of thrust-propelled vehicles under deterministic and stochastic topology
Published in International Journal of Systems Science, 2019
M. Kabiri, H. Atrianfar, M. B. Menhaj
To represent the interconnection topology, we make a use of undirected graph with a node set , edge set and weighted adjacency matrix which is defined such that and if . For the undirected graph, we assume for all . To describe the time-varying topologies, we use a piecewise right-continuous switching function , where N is the total number of all possible communication graphs. denotes the communication graph at time t. A path is a sequence of adjacent edges of the form where . An undirected graph is called connected if every two distinct nodes are connected by a path. The union of a collection of graphs ,…, with the node set is a graph denoted by with the same node set whose edge set is the union of edge sets of all graphs. A switching topology is said to be jointly connected if there exists an infinite sequence of nonempty, bounded and contiguous time intervals , , with and for some constant such that the union of undirected graphs across each time interval is connected. We assume that a complete graph is a graph in which each pair of graph vertices is connected by an edge. It is also assumed that there is a sequence of non-overlapping subintervals where , and , for some integer and given constant . The communication topology is supposed to be fixed on each time subinterval and switches at .