Here are several sources for getting more background or detail on discrete math topics, as well as many applications and examples. Many of the problems, although intended for the college level, have been used successfully at lower levels.
Discrete Mathematics and its Applications (Book, College)
Kenneth Rosen, McGraw Hill, 2nd Ed., 1991; $70.
Moshe Vardi, a computer science professor at Rice University, asked college/university faculty which book they used to teach their introductory course on discrete mathematics. Over a dozen faculty responded: Rosen's book was the most popular text in the survey. The book has a companion volume, with further applications, written by various authors: Applications of Discrete Mathematics, J. Michaels and K. Rosen, Editors (McGraw Hill). Other texts mentioned several times in Vardi's survey were Discrete Mathematics, by Ken Ross and Charles Wright, Prentice-Hall, 3rd Ed., 1992; and Discrete Mathematics with Applications, by Susanna Epp, Wadsworth, 1990. (In , the author discusses her experience teaching basic logic in the course at DePaul University that led to this text.)
Applied Combinatorics (Book, College-Graduate)
Fred S. Roberts, Prentice-Hall, 1984; $70.
This is a good source of applications of discrete mathematics. The book also describes many classic algorithms, such as those for finding shortest paths, minimum spanning trees, Eulerian tours, and maximum flow. I (Franzblau) have used the text several times for an intermediate-level undergraduate course; students find it somewhat difficult, so I prefer using it as a resource for myself.
Graph Theory Applications (Book, College)
L.R. Foulds, Springer-Verlag, 1992; paper, $49.
This text was recommended by Susan Picker LP `90 (private communication):
The distinctive thing about this text is the variety of applications, including social sciences, economics, physics, biology, chemistry, civil engineering, operations research, circuit design, matrices, algorthms, architecture, and industrial engineering.
Graphs: An Introductory Approach (Book, College)
Robin Wilson and John Watkins, Wiley, NY, 1990. One of the referees of this article strongly recommended this text, as a ``great introduction to graphs.''
Graphs, Models, and Finite Mathematics (Book, College)
Joseph Malkevitch and Walter Meyer, Prentice-Hall, 1974. This book is the source of some of the material in the HiMAP module Problem Solving Using Graphs (described above in Section 3).