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Discrete mathematics and graph theory: a concise study companion and guide
Erciyes K., Springer International Publishing, New York, NY, 2021. 352 pp.  Type: Book (978-3-030611-14-9)
 Date Reviewed: Jul 12 2022
 For today’s students of computer science (CS), engineering, and mathematics, discrete mathematics is considered a fundamental course demanding in-depth study. The author employs an algorithmic approach, quite rightfully, to write this reference book, giving a thorough review of the basic concepts of discrete mathematics with an emphasis on graph theory. Given that graphs are important data structures that find several applications in bioinformatics, the Internet, representing networks (including social networks), games, chemical structures, and so on, the author’s attempt is certainly commendable. The book has two parts, with Part 1 (“Fundamentals of Discrete Mathematics”) comprising ten chapters and Part 2 (“Graph Theory”) comprising five. Here are some of the book’s salient features: Concepts of discrete mathematics are reviewed thoroughly; Strong emphasis on graph theory, which is generally not found in most texts on the subject; Major algorithmic techniques are well covered; Problems of discrete mathematics and graph theory are tackled using an algorithmic approach; and Chapter summaries, review questions, several exercises and examples, and a helpful bibliography for further reading. On the negative side, at times the author either oversimplifies or unnecessarily complicates the concepts. For example, in the opening line of the preface, the author’s description of discrete mathematics is clearly an oversimplification: “a branch of mathematics that studies structures which take distinct values.” (“Countable values” would have been mathematically more correct.) Likewise, proving the sum of two even numbers to be even using the method of contradiction (pages 31 and 32) is an unnecessary complication where one could simply have 2m+2n=2(m+n), with m and n being integers. Proofs by contradiction are provided only when a direct proof is not available. The second example of proving the irrationality of the square root of two by contradiction (page 32) is a better example, but here again the author needs to be reminded that proofs by contradiction are more like verifying the conjecture without supplying a proof (while the conjecture does get verified, it is also revealed that you don’t know the proof; for if you knew it, you ought to have given it directly). Furthermore, a direct proof is in fact available by establishing that the continued fraction expansion (CFE) of the square root of two is aperiodic and infinite (only irrationals have this property). Nevertheless, this accessible reference book should be well received by undergraduate-level CS, engineering, and mathematics students. Reviewer:  Soubhik Chakraborty Review #: CR147470
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