New York: Addison-Wesley, 1990. — 352 p.Graph Theory has developed into a very active area of mathematical research. Whereas twenty years ago many mathematics departments had no graph theorists, it is now not uncommon to find several in a single department. A major impetus for this growth has certainly been the wide applicability of graph theory, especially in computer science. Besides the dozen or so extant introductory texts on Graph Theory, monographs have been written in recent years covering such specialized areas as connectivity, colorability, extremal graphs, random graphs, ramsey theory, and groups and surfaces. A number of recent introductory texts have used algorithms as the common thread, a reasonable approach because of the interest of computer scientists in graph theory. One concept that pervades all of graph theory is that of distance. Distance is used in isomorphism testing, graph operations, hamiltonicity problems, extremal problems on connectivity and diameter, and convexity in graphs. Distance is the basis of many concepts of symmetry in graphs. The important application of facility location on networks is based on various types of graphical centrality, all of which are defined using distance. Many graph algorithms depend on the idea of finding collections of long paths within a graph or network. Since distance is such a pervasive notion in graph theory, the time has come for a text focusing on distance in graphs. Distance in Graphs is based on the classic Graph Theory by F.H. and brings Graph Theory up to date on the topics covered. This text can be used by advanced undergraduates and beginning graduate students in mathematics and computer science and also as a comprehensive reference work for researchers in graph theory, communication networks, and the many fields using graph theory in applications. The basic mathematical background required here is "mathematical sophistication," although a good one-year course in discrete mathematics (or its equivalent) would be helpful. We have made every effort to see that concepts are carefully explained and motivated. Accompanying figures are used to elucidate and illustrate concepts throughout the text. Clear, illuminating proofs for all major theorems are given. Exercises are included to further clarify the material from each section. Many of the exercises contain results extending the concepts discussed in the section and cite the original source to provide a direction for further reading.
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