***** To join INSNA, visit http://www.insna.org ***** Juno, what you're asking essentially is the following: given n nodes, find the number of all regular graphs with diameter 2. Of course, the complete graph is one of these graphs but I guess you want to exclude it, right? I'm not sure if this (as a combinatorical problem) is solvable but I'll give a try to it (when I find time) and in case I reach to a formula I'll send it to you. Formulating this problem in terms of sociomatrices, it becomes: How many symmetric nxn matrices with entries 0 or 1 exist in such a way every row (or column) has the same number (sum) of 1s, which are either isolated (having 0 to their right and left) or there are at most two consecutive 1s (surrounded by 0s). Incidentally in your example (n = 6) there is a third graph that you need to consider and this is the following n 1 2 3 4 5 6 1 1 1 1 2 1 1 1 3 1 1 1 4 1 1 1 5 1 1 1 6 1 1 1 --Moses On Mon, 16 Jan 2006, Juno Blaauw wrote: > ***** To join INSNA, visit http://www.insna.org ***** > > Hello SOCNETTERS, > > My name is Juno Blaauw and I am a political research student at the > University of Amsterdam. I am writing this e-mail on behalf of Meindert > Fennema, Jean Tillie and myself. > > Meindert Fennema and Jean Tillie are studying ethnic civic communities. > Lately they have been interested (among other things) in introducing a > normative definition of a civic community and in measuring the difference > between a given empirical civic community and this normative definition. I > have been working with them on this (thesis). We have been able to formulate > a, in our eyes, satisfactory definition of a normative civic community, > which we have called an 'old boys civic community', and a measure for the > above mentioned difference. The only problem is that our normative > definition does not lead to one unique social network. In other words: on a > given number of nodes many social networks qualify as old boys civic > communities. > > Before getting to the specific question we would like to ask you all, > I willfirst give you our definitions. First of all our definition of a > civic > community: "a civic community [...] consists of many voluntary associations > that are related to each other by way of overlapping membership and > interlocking directorates' (Fennema, 2004, p. 433.)" The voluntary > associations are formal organisations and the interlocking directorates are > people that are board members of two of these organisations. Thus, a civic > community can be defined as the set of formal organisations and the ties, in > the form of interlocking directorates, between them. A civic community that is > defined like this can be depicted as a graph were the nodes > representorganisations and the lines interlocking directorates. > > Secondly, our definition of an old boys civic community: A civic community > is an old boys civic community if and only if (a) any two organisations are > either directly linked by an interlocking directorate or have a maximum of > two interlocking directorates between them and (b) each of the > civiccommunity's organisations is adjacent to the same number of > otherorganisations. > > Now, the question we would like to ask all of you is this: how can we find > out how many and which old boys civic communities there are on a > givennumber of nodes? Note that we are only interested in minimal old > boys civic > communities, that is in old boys civic communities with a minimal number of > interlocking directorates (ties). > > To illustrate our problem: Here are two minimal old boys civic communities > on six nodes. Are these the only two? If so, how can we proof that? If not, > how can we find the other ones? > > n 1 2 3 4 5 6 > 1 1 1 1 > 2 1 1 1 > 3 1 1 1 > 4 1 1 1 > 5 1 1 1 > 6 1 1 1 > > and > > n 1 2 3 4 5 6 > 1 1 1 1 > 2 1 1 1 > 3 1 1 1 > 4 1 1 1 > 5 1 1 1 > 6 1 1 1 > > We have no idea whether or not this is actually an easy problem that has > already been solved by somebody else. We are just hoping that you can inform > us of anything you seem fit, given what we have told you so far. > > Please let me know if you need any additional information. Thank you in > advance! > > Juno Blaauw. > > _____________________________________________________________________ > SOCNET is a service of INSNA, the professional association for social > network researchers (http://www.insna.org). To unsubscribe, send > an email message to [log in to unmask] containing the line > UNSUBSCRIBE SOCNET in the body of the message. > _____________________________________________________________________ SOCNET is a service of INSNA, the professional association for social network researchers (http://www.insna.org). To unsubscribe, send an email message to [log in to unmask] containing the line UNSUBSCRIBE SOCNET in the body of the message.