Graph completely connected

Fig. 12.1 Graphs contain nodes connected by edges. A completely connected graph (right) has an edge between all pairs of nodes.

The traveling salesman problem (TSP) (Lawler et al. 1985), the classic problem in which a mythical traveler must find a minimum-length cycle through a set of nodes in a completely connected graph. 

Graph Gc is obtained from Gb by making nodes vy,..., Vjo complete. The subgraph G c 1 = 1 is still completely connected with respect to j = A/j, 1 but this... 

In this section we present several results concerning the eigenvalues and eigenvectors of Jg for graphs with complete nodes or completely connected subgraphs. See [305] for proofs of the following theorems. 

Theorem 13.16 Consider a network of n reactors represented by a graph Q and assume there exists Mq defined by (13.113), such that Mq is a proper subset of Mg. Then there exist subgraphs Gi, i = 0,..., 1, such that Mg = M t/j, i = 1,..., are completely connected with respect to Mq ... 

As a second example of the nonlinear dynamics of Lengyel-Epstein networks, we consider the graph Qi, in Fig. 13.3b. As discussed above, this graph has three completely connected subgraphs, gI i, 2> b,3- Por Gb,iy Theorem 13.14 and... 

Molecular graphs, complete or incomplete, can be conveniently represented by connectivity matrices as can be seen in the examples in Fig. 3.1 and in Refs. [1-9]. A complete graph where connectedness is indicated by 1 and disjointedness by 0 will have a non-zero entry for every non-diagonal element of the matrix while an incomplete graph has finite entries only for connected vertices and zero elsewhere in the matrix (Fig. 3.1). 

Another common limitation of aU known connectivity graphs—complete or incomplete—or of their matrix surrogates is their inherent insensitivity to optical isomerism. This limitation is circumvented if the experimental dataset includes the active isomers. 

Conventional CA models are defined on particular lattice-networks, the sites of which are populated with discrete-valued dynamic elements evolving under certain local transition functions. Such a network with N sites is simply a general (undirected) graph G of size N and is completely defined by the NxN) connectivity matrix... 

Figure 1.9 shows that equation fi contains (is connected to) variables v and v2. The complete graph for the set of equations is shown in Figure 1.10. 

In system C, many stacks are connected in series. Very small currents are generated at still higher voltages. As the number of stacks in series is increased, the maximum achievable power quickly approaches the power which a reversible system would generate, i.e. complete conversion of the available free energy. (A reversible system is reversible at every point in each stack, not just at the stack outlets.) The shaded area in the graph nearly fills the entire area under the curve - the reversible power. 

Figure 5.1 Schematic presentation of town Konigsberg with its seven bridges and connections. The plan of the town (a) is converted into graphs (b and c), which are completely equivalent representation of the object in (5.1a) and between themselves in Graph Theory.

---