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Báo cáo toán học: "Glossary of Signed and Gain Graphs and Allied Areas"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Glossary of Signed and Gain Graphs and Allied Areas. | Glossary of Signed and Gain Graphs and Allied Areas by Thomas Zaslavsky Department of Mathematical Sciences Binghamton University Binghamton New York U.S.A. 13902-6000 E-mail zaslav@math .binghamton.edu 1998 July 21 Second Edition 1998 September 16 Typeset by Ụ S-T . X 1 THE ELECTRONIC JOURNAL OF COMBINATORICS DS9 2 Table of Contents 1. Basics .p. 3 Notation 3 Partitions 3 2. Graphs .4 Graph Elements 4 Kinds of Graphs 5 Graph Structures 7 Graph Operations 9 Switching Subgraphs Graph Relations 10 Graph Invariants Matrices 11 Graph Problems 11 3. Signed Gain and Biased Graphs .12 Basic Concepts of Signed Graphs 12 Aspects of balance Clusterability Additional Basic Concepts for Gain and Biased Graphs 15 Structures 17 Orientation 18 Vertex Labels States 19 Examples 20 Particular General Operations 22 Switching Negation Subgraphs and contractions Subdivision and splitting Relations 26 Line Graphs 27 Covering or Derived Graphs 28 Matrices 29 Matroids 29 Topology of signed graphs 31 Coloring 32 Flows 33 Invariants 33 Chromatic invariants Problems 35 4. Applications .36 Chemistry 36 Physics Spin Glasses 36 Vector models Ising mo dels Gauge models Social Science 40 Operations Research 41 NOTES Key. a term usually one rarely used for which there is a preferred variant. Citations. Citations are to A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas Electronic Journal of Combinatorics 1998 Dynamic Surveys in Combinatorics DS8. THE ELECTRONIC JOURNAL OF COMBINATORICS DS9 3 BASICS Notation To simplify descriptions I adopt some standard notation. I generally call a graph r a signed graph a gain graph and when indicated by the context a permutation gain graph and a biased graph r B . The sign function of is Ơ the gain function of is that is I use upper and lower case consistently for the graph and its edge labelling. The gain group of is 0. Partitions partition of a set Unordered class of pairwise disjoint nonempty subsets whose union is the whole set. The empty