This IMA Volume in Mathematics and its Applications Applications of Combinatorics and Graph Theory to the Biological and Social Sciences is based on the proceedings of a workshop which was an integral part of the 1987-88 IMA program on APPLIED COMBINATORICS. We are grateful to the Scientific Committee: Victor Klee (Chairman), Daniel Kleitman, Dijen Ray-Chaudhuri and Dennis Stanton for planning and implementing an exciting and stimulating year­ long program. We especially thank the Workshop Organizers, Joel Cohen and Fred Roberts, for organizing a workshop which brought together many of the major figures in a variety of research fields connected with the application of combinatorial ideas to the social and biological sciences. A vner Friedman Willard Miller APPLICATIONS OF COMBINATORICS AND GRAPH THEORY TO THE BIOLOGICAL AND SOCIAL SCIENCES: SEVEN FUNDAMENTAL IDEAS FRED S. RoBERTS* Abstract. To set the stage for the other papers in this volume, seven fundamental concepts which arise in the applications of combinatorics and graph theory in the biological and social sciences are described. These ideas are: RNA chains as "words" in a 4 letter alphabet; interval graphs; competition graphs or niche overlap graphs; qualitative stability; balanced signed graphs; social welfare functions; and semiorders. For each idea, some basic results are presented, some recent results are given, and some open problems are mentioned.

Applications of Combinatorics and Graph Theory to the Biological and Social Sciences: Seven Fundamental Ideas.- Social Welfare and Aggregation Procedures: Combinatorial and Algorithmic Aspects.- Consecutive One's Properties for Matrices and Graphs Including Variable Diagonal Entries.- Probabilistic Knowledge Spaces: A Review.- Uniqueness in Finite Measurement.- Conceptual Scaling.- The Micro-Macro Connection: Exact Structure and Process.- Sign-Patterns and Stability.- Food Webs, Competition Graphs, Competition-Common Enemy Graphs and Niche Graphs.- Qualitatively Stable Matrices and Convergent Matrices.- Tree Structures in Immunology.- Meaningless Statements, Matching Experiments, and Colored Digraphs (Applications of Graph Theory and Combinatorics to the Theory of Measurement.- Combinatorial Aspects of Enzyme Kinetics.- Spatial Models of Power and Voting Outcomes.- Some Mathematics for DNA Restriction Mapping.