Rooted labeled graphs

As the objects we are interested in this chapter are connected, let Ck be the number of connected labeled graphs of k vertices. kC rooted connected labeled graphs exist because k ways exist of choosing a root. The number of rooted, labeled graphs of n vertices in which the root is in a connected component containing k vertices is feQ ( This expression, summed from... 

To define this concept more precisely, we need to define a labeled graph. A labeled graph has exactly the same definition as that given above for a graph, except that the field points (if any) as well as the root points have labels, usually numbers, and no two points have the same label. It is easy to define topological equivalence for labeled graphs. Two labeled graphs, which have the same number of root points, the same set of labels on the root points, the same... 

To overcome this limitation, it is sufficient to restrict root labels to positive values. By this modification, the size of the resulting graph is doubled at most. Multiplying a positive BMD (p BMD) with a positive number is still a const ant-time operation, but multiplication with negative numbers additionally requires to copy the graph and to negate the terminal values. 

Normalise the scores of die first two PCs obtained in question 4 by dividing by die square root of the sum of squares at each pH. Plot the graph of the normalised scores of PC2 vs PCI, labelling each point as in question 4, and comment. 

The integral contains a factor for the density point function for each field point and a factor for the appropriate bond function for each bond. The arguments of these functions are the coordinates corresponding to the labels on the graph. Moreover, if more than one bond of a type connects a pair of molecules, there is a numerical factor [N/(i,y) ] There is also a factor for the reciprocal of the symmetry number. Note that if the graph contains no root points, its value is just a number since all the x coordinates are integrated over. If the graph has m root points, where m 0, the value is a function of Xi,..., x ,. 

F hb( i 2) = sum of all topologically different irreducible graphs that have two root points (labeled 1 and 2), any number of field points, fo and fnaYp bonds, at most one bond between any pair of points, an fuBYp bond between the roots, no pair of overlapping points that remain at least doubly connected when all fo bonds between overlapping pairs are removed, and no pair of points connected by an /hb bond that has a residual that has no /hb attached to either root (115)... 

Figure 9. (a) labelling of different sites on the 3-simplex (b) definition of weights for rooted graphs... 

---