Disconnectivity graphs

Conversely, a disconnected graph (null graph) coniains... 

Figure 6 The topological disconnectivity graph of alanine hexapeptide. (Adapted from Ref. 67.)...

If B— [bij] is an N xN matrix in which bu equals the degree of vertex i, bij = —1 if vertices i and j are adjacent and bij = 0 otherwise, then the number of spanning tree of G is equal to the determinant of any principal minor of B [hararybO]. The extremes occur for totally disconnected graphs that have no spanning trees and thus a complexity of zero, and for complete graphs of order N that contain the maximum possible number of distinct trees on N vertices. ... 

The union GiUG2 is defined as the graph which vertex set is V,UV2 and edge set is E[UE2. A disconnected graph is a graph which can be expressed as the union of two graphs [54a]... 

Graph in which for each pair of vertices i,j e V((t) at least one path exists. Otherwise Q is called a disconnected graph. The simplest disconnected graph is a graph with an isolated vertex and the vertices not joined by a path belong to different components of the graph. The number of components of a graph is denoted lc(Q). 

Note that information about density of states associated with either the local minima or the saddles are not included in the disconnectivity graph, and the graph generally depends on the choice of the spacing between the energies at which the superbasin analysis is performed. There does not exist any meaning about one dimensional horizontal axis. 

Figure 5. Disconnectivity graphs for (a) BLN model in terms of a sample of 500 minima and 636 saddles, and for (b) Go-like BLN model a sample of 500 minima and 805 saddles. The energy is in the units of e. [Reprinted with permission from M. Miller and D. J. Wales, J. Chem. Phys. Ill, 6610 (1999). Copyright 1999, American Institute of Physics.]...

An edge is a cut edge if its removal produces a disconnected graph it cannot be a part of a cyde similarly, a vertex is a cut vertex if its removal produces a disconnected graph. Of course, each vertex inddent to a cut edge is a cut vertex, but a cut vertex can also be a part of a cycle. 

The subscript c indicates that only connected diagrams have to be included because the contributions due to disconnected graphs can be eliminated using the factorization property (Goldstone 1957)... 

We can then follow the previous development to replace e-bonds by (1 -I-/(-bonds. This gives a sum of connected and disconnected (product) graphs of /-bonds and Zi-circles. The disconnected graphs, whose factors contain no white Zj-circles, sum to cancel the factor S in the denominator, so that... 

Note that for A > 2 this graphical sum contains disconnected graphs that become connected if a/2(l,2) bond is added. For example the expansion for P2(1>2) contains... 

Notice that P2(/2) contains disconnected graphs, but those which are disconnected can be connected by an /-bond between 1 and 2. All of the disconnected graphs arise from the product term Mi(1)ui(2). 

The disconnectivity graph is constructed by performing the superbasin analysis at a series of energies, plotted on a vertical scale. At each energy, a... 

Figure 1.3. Schematic examples of potential energy surfaces (potential energy as a function of some generalized coordinate) and the corresponding disconnectivity graphs. In each case, the dotted lines indicate the energy levels at which the superbasin analysis has been made, (a) A gently sloping funnel with high barriers ( willow tree ) (b) a steeper funnel with lower barriers ( palm tree ) and (c) a rough landscape ( banyan tree ).

In this section we present disconnectivity graphs for a selection of LJ clusters to illustrate the effects of size on the PES [211]. These effects are both general—the number of stationary points on the PES increases exponentially with size—and specific—the clusters we have chosen illustrate a number of interesting features. 

We used the methods described in Section III.B to obtain samples of stationary points for these clusters. Only LJ13 is small enough for an exhaustive search of all minima and transition states to be possible. For the other clusters, the searches were terminated once we were confident that we had obtained an accurate representation of the low-energy regions of the PES. The disconnectivity graphs constructed from these samples are shown in Figure 1.7. 

Figure 1.7. Disconnectivity graphs for (a) LJ,3. (b) LJ19, (c) LJ, (d) LJ, 8, (e) LJ55, and (f) LJ75. In (a) all the minima are represented. In the other parts only the branches leading to the (b) 250, (c) 200, (d) 150, (e) 900, and (f) 250 lowest-energy minima are shown. The numbers adjacent to the nodes indicate the number of minima that the nodes represent. The branches associated with the minima depicted in Figure 1.4 and Figure 1.6 are labeled by the letters that indicate their energetic rank. Energy is in units of e.

Here, we use monotonic sequence basin (MSB s) to produce a more coarse-grained picture of the PES. Disconnectivity graphs that only include the minimum at the bottom of each MSB can be produced by excluding all the minima directly connected to a lower-energy minimum. In the resulting... 

For LJ55 the MSB analysis leads to a remarkable simplification of the disconnectivity graph (Fig. 1.9b). The single-defect minima produce just one MSB, and the fine structure of the two-defect minima collapses onto the band of MSB s that branches off at -273e and -272e. The remaining... 

Figure 1.9. Disconnectivity graphs for (a) LJ38 and (b) LJ55. Only the branches corresponding to monotonic sequence basin bottoms are shown. Energy is in units of e.

Figure 1.15. Disconnectivity graphs for Mu with p = 4 and p = 6 plotted on the same energy scale (in units of the pair well depth, e).

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