Extended Schlafli or vertex symbols

The extended Schlafli notation has been put forward by O Keeffe [5,9] and is, as the name indicates, an expansion of the Schlafli notation. This notation is also called vertex symbol or long notation. Now we will not only note the size of the smallest rings of a link pair, but we will also count how many of these there are. 

all possible ways to unite a link pair using smallest rings are counted and then added as a subscript index to the ring size as M i-Ny2 0y. The connectivity p is now deduced from the number of link pairs, each of these having a Mji symbol assigned to it. Thus summing the number of link pairs is equal to / (/ -1 )/2 (equation 4.1). For example for a three-connected net there will be three link pairs (Myi-Nyj-Oys) and for a four-connected net there will be six (My -Ny2-Oy3 Py4-Qy5-Ry j). The subscript is used to avoid confusion with the Schlafli notation. 

The order of the link pairs are the same as for the short (Schlafli) notation when p-3 and p 4. However, for nets where the nodes have p-4 a different order is used. The link pairs are paired opposite to each other and ordered with the lowest numbers first. In these pairs, each link is used once. Designating the links as a, b, c and d, the order w ill be (ab,cd)(ac,bd)(ad,bc). 

Although there seems to be a theoretical possibility of having nets with the same set of vertex symbol and CIO value, for practical purposes this it is an 

This enumeration may seem forbidding to work out by hand , especially if we consider nets with more than one type of node, but fortunately, there is now both free and commercial software that will do the ring counting more or less automatically (depending on your type of net-connection) [11-14]. A brief introduction will be given in Chapter ll 

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