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One of the many interesting conjectures proposed by S. M. Ulam in (5) can be stated as follows:
If G and H are two graphs with p points vi and ui respectively (p ⩾ 3) such that for all i, G — vi is isomorphic with H — ui then G and H are themselves isomorphic.
P. J. Kelly (3) has shown this to be true for trees. The conjecture is, of course, not true for p = 2, but Kelly has verified by exhaustion that it holds for all of the other graphs with at most six points. Harary and Palmer (2) found the same to be true of the seven-point graphs.
In (1) Harary reformulated the conjecture as a problem of reconstructing G from its subgraphs G — vi and derived several of the invariants of G from the collection G — vi.
The purpose of this paper is to integrate the theorems on enumerating subgraphs and supergraphs in (2) and (3) respectively by generalizing to a result which includes both of these as special cases. In this process we again utilize the powerful enumeration method of Pólya (4).
In the theory of probability, the conditional can be treated by an operation analogous to division. Many properties of the conditional can best be studied by means of the corresponding multiplication (called the cross-product). An implicative Boolean ring is defined [2] in terms of a cross-product and the usual Boolean operations. The cross-product is the only device yet known in which the events corresponding to conditional probabilities are themselves elements of the Boolean ring. The fact that such advice was not introduced by Boole is probably the reason why Boolean algebra has been very little used in the theory of probability, although probability was one of the principal applications which Boole had in mind.
A graph G consists of a finite set of p points and q lines joining pairs of these points. Each line joins two distinct points and no pair of points is joined by more than one line. A subgraph of G is a graph whose points and lines are also in G. If every pair of points of a graph is joined by a line, the graph is called complete and is denoted by Kp. A planar graph can be embedded in the plane, that is, drawn in the plane in such a way that none of its lines intersect.
The definition of the genus γ(G) of a graph G is very well known (König
2): it is the minimum genus among all orientable surfaces in which G can be
drawn without intersections of its edges. But there are very few graphs whose
genus is known. The purpose of this note is to answer this question for one
family of graphs by determining the genus of the n-cube.
The graph Qn called the n-cube has 2n vertices each of which is a binary
sequence a1a2. . . an of length n, where ai = 0 or 1.
Among the unsolved problems in graphical enumeration listed in (4) is included the determination of the number of graphs and digraphs with a given partition. Parthasarathy (9) has developed a formulation for counting graphs with a given partition by making a suitable modification of the method given in (2) for the enumeration of graphs. We present here an analogous modification that leads to a formula for the number of digraphs with a given partition. Not surprisingly, the main combinatorial device for this purpose is provided by the classical theorem due to Pólya.
A set of points M of a graph G is a point cover if each line of G is incident with at least one point of M. A minimum cover (abbreviated m.c.) for G is a point cover with a minimum number of points. The point covering number α(G) is the number of points in any minimum cover of G. Let [V1, V2, … , Vr], r > 1 be a partition of V(G), the set of points of G. Let Gi be the subgraph of G spanned by Vi for i = 1, 2, … , r.
In a previous paper (2), one of us has derived a formula for the counting series for bicoloured graphs.2 These are graphs each of whose points has been coloured with exactly one of two colours in such a way that every two adjacent points have different colours.
In this paper we first enumerate bicoloured graphs without isolated points and connected bicoloured graphs. This leads us to corresponding problems for bicolourable graphs. Such a graph has the property that its points can be coloured with two colours so as to obtain a bicoloured graph.
From a mathematical perspective many problems in anthropology concerned with the analysis of structures, patterns, and configurations are combinatorial in nature. There are three types of combinatorial problems:
The existence problem asks, “Is there a structure of a certain type?”
The counting problem asks, “How many such structures are there?”
The optimization problem asks, “Which is the best structure according to some criterion?” (Roberts 1984).
The minimum spanning tree problem (MSTP) is an optimization problem, well known in many fields; its history is detailed in Graham and Hell (1985). The problem has applications to the design of all kinds of networks, including communication, computer, transportation, and other flow networks. It also has applications to problems of network reliability and classification, among many others. Our purpose here is to describe some applications of the MSTP to anthropology – in particular to problems of size, clustering, and simulation in networks of various kinds. We proceed by presenting in a unified format the three standard MST algorithms of Kruskal (1956), Prim (1957), and Boruvka (1926a,b), describing the advantages and some of the applications of each one.
We first illustrate the MSTP intuitively, as follows. A large corporation with offices in many cities, v1, …, vn, wishes to determine the monthly telephone charge. All the distances d(vi, vj) are known and are distinct.
In the course of transforming verbal propositions into images many things are made explicit that were previously implicit and hidden.
Herbert A. Simon, Models of My Life
Oceanists have increasingly come to recognize the limitations of the laboratory analogy that treats island societies as isolated experiments in adaptive radiation. Reconstructions of regional exchange systems (Hage and Harary 1991), archaeological evidence of sustained inter-island contacts (Kirch 1988a), firsthand accounts of traditional voyaging techniques (Lewis 1972), and the evident contradiction between neoevolutionist assumptions and the facts of Oceanic ethnography and prehistory (Friedman 1981) conduce to a network perspective that views island societies as elements of communication systems. Most islands in the Pacific are, in fact, distributed in groups, and most island societies are, or once were, connected to other island societies – as colonists, trade partners, tributaries, allies, wife-givers, and in various other ways. In acknowledging the importance of these connections many researchers, including anthropologists, archaeologists, and linguists, are now using or recommending the application of network concepts to answer a range of fundamental questions concerning
the settlement of island groups (Levison, Ward, and Webb 1973; Ward, Webb, and Levison 1976; Green 1979; Kirch 1988a; Irwin 1992);
the location of trade centers (Irwin 1974, 1978, 1983; Kirch 1988b; Hunt 1988);
the development of social stratification and social complexity (Reid 1977; Friedman 1981; Kirch 1984a; Lilley 1985; Graves and Hunt 1990);
the differentiation of cultural complexes (Green 1978);
the diversification of dialects and languages (Pawley and Green 1984; Marck 1986);
the distribution of physical and cultural traits (Terrell 1986);
the selection of subsistence practices (Harris 1979);
the evolution of kinship structures (Epling, Kirk, and Boyd 1973; Marshall 1984).
This book is the third work in a comprehensive program of research on applications of graph theory to anthropology. Graph theory is an explosively developing branch of pure mathematics with increasingly important applications to many fields, including architecture, biology, chemistry, computer science, cognitive science, economics, geography, and operations research. It is our belief that anthropology belongs with this company of subjects. Our aims are (1) to solve certain theoretical and methodological problems in anthropology by using the concepts, theorems, and techniques of graph theory; (2) to provide a common framework for structural analysis by demonstrating the applicability of graph theory to a wide spectrum of social and cultural phenomena; (3) to promote connections between various areas of anthropology and between anthropology and other disciplines in which graph theoretic modeling has proven useful; (4) to preserve continuity with the historical tradition of structural analysis in anthropology; and (5) to make graph theoretic models accessible to all structurally minded anthropologists and other social scientists.
In our first book, Structural Models in Anthropology (Hage and Harary 1983), we presented graph theory as a family of models for the analysis of social, symbolic, and cognitive relations. We used graphs, digraphs, and networks, together with their associated matrices, to study such diverse topics as mediation and power in exchange systems, reachability in social networks, efficiency in cognitive schemata, and productivity in subsistence modes. We exploited duality laws for graphs and the interaction between graphs and groups to analyze transformations and permutations in myths and symbolic systems.
Every network N, with underlying graph G, has one or more dominating sets. Historically, this concept originated with von Neumann in his pioneering work with Morgenstern (1944) on the theory of games. In game theory, a given game may have several strategies deciding which move to make in any given game situation. A strategy is said to dominate another one if the person using the first strategy defeats his opponent using the second one in a two-person game. This was formalized to domination in digraphs by Richardson (1953) and studied by Harary and Richardson (1959).
Ore (1962) generalized the concept of domination in digraphs to graphs G. This is entirely analogous to the domination of the 64 squares of a conventional chessboard by Queens. This Queen domination problem was mentioned in Chapter 1. In particular, the placing of eight Queens on a chessboard so that no Queen threatens (dominates) any other Queen was completely solved by Euler in the eighteenth century.
Ore defined a node v in G as dominating itself and all nodes adjacent to it, that is, v dominates its closed neighborhood N[v]. Domination in graphs is now the most active area of research in graph theory (Laskar and Walikar 1981; Hedetniemi and Laskar 1990).
For our present purposes, every island network has some dominating set of islands. We now use the combinatorial model of domination in graphs to describe local political hierarchies in the Caroline Islands in Micronesia, alliance structures in the Tuamotu Islands in Polynesia, and pottery monopolies in two trade networks in Melanesia.
Contrary to common perception and belief, most island societies of the Pacific were not isolated, but were connected to other island societies by relations of kinship and marriage, trade and tribute, language and history. Using network models from graph theory, the authors analyse the formation of island empires, the social basis of dialect groups, the emergence of economic and political centres, the evolution and devolution of social stratification and the evolution of kinship terminologies, marriage systems and descent groups from common historical prototypes. The book is at once a unique and important contribution to Oceania studies, anthropology and social network analysis.
It goes without saying that … merely formal studies can never be an end in themselves. But there is, on the other hand, always the danger that in historical or functional studies of kinship problems this formal aspect may be unduly neglected.
Paul Kirchhoff, “Kinship Organization”
There have been two major attempts to construct formal evolutionary models of kinship organization in Oceania: Murdock's (1949) derivation of Malayo-Polynesian societies from a Hawaiian prototype, and Marshall's (1984) derivation of Island Oceanic sibling terminologies from a distributional prototype. Murdock's model, known more generally as the “bilateral hypothesis,” is part of a universal theory of social evolution representing the culmination of a lifetime of cross-cultural research on kinship organization, while Marshall's model is the most recent contribution to a theoretical discussion of sibling classification and social organization which began with the ethnographic researches of Codrington (1891) a century ago. Both models are controversial, primarily because of arguments from historical linguistics (Blust 1980, 1984; Bender 1984; Clark 1984). But there are problems of interpretation as well. Our purpose is to examine the graph theoretic foundation of these models. We will find that Murdock's model provides no valid reason for inferring that kinship organization in Proto-Malayo-Polynesian (PMP) society was Hawaiian in type. If anything, it was Iroquois or Nankanse, neither of which is a bilateral type of social organization. We will see that Marshall's model can be replaced by the one implicit in Milke's (1938) historical reconstruction of Proto-Oceanic (POC) sibling terms.
The contention between older and younger brothers is a celebrated condition of Hawaiian – indeed Polynesian – myth and practice.
Marshall Sahlins, Historical Metaphors and Mythical Realities
Children of brothers love one another, children of sisters fear one another.
Marshallese proverb, August Erdland, The Marshall Islanders
In the conclusion of his monograph on Tonga, Gifford remarked that “the parallels in the social organization of Tonga and the remainder of Polynesia and Micronesia are obvious” (1929:350). Unfortunately he did not elaborate, choosing instead to discuss parallels and possible genetic connections between Oceania and Japan. It is clear, however, that the sexually dual, matrilineal variant of the conical clan is found in the Marshall Islands in eastern Micronesia. It also appears that the Micronesian and Polynesian variants are genetically related, having a common origin in Proto-Oceanic society. Like its Tongan counterpart, the Marshallese conical clan was a socially encompassing, politically expansive structure, associated with asymmetric marriage alliance and implicated in the formation of island empires. Linguistic evidence suggests that the Marshallese variant represents the ancestral form of all the differently permuted forms of social organization in Nuclear Micronesia.
The Marshallese conical clan
Descent
The Marshallese clan, called jowi in the Ralik chain and jou in the Ratak chain, consisted of a group of lineages called bwij, which traced descent through females from a common ancestress (Krämer 1906; Krämer and Nevermann 1938; Erdland 1914; Mason 1947, 1954; Spoehr 1949a, b; Kiste 1974).