Enumeration of Independent Sets in Graphs

An independent set is one of the most natural structures in a graph to focus on, from both a pure and applied perspective. In the realm of graph theory, and any concept it can represent, an independent set is the mathematical way of capturing a set of objects, none of which are related to each other. As graph theory grows, many questions about independent sets are being asked and answered, many of which are concerned with the enumeration of independent sets in graphs. We provide a detailed introduction to general graph theory for those who are not familiar with the subject, and then develop the basic language and notation of independent set theory before cataloging some of the history and major results of the field. We focus particularly on the enumeration of independent sets in various classes of graphs, with the heaviest focus on those defined by maximum and minimum degree restrictions. We provide a brief, specific history of this topic, and present some original results in this area. We then speak about some questions which remain open, and end the work with a conjecture for which we provide strong, original evidence. In the appendices, we cover all other necessary prerequisites for those without a mathematical background

2011 IMSAloquium, Student Investigation Showcase

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Teaching Time Savers: Some Advice on Giving Advice

There are always a lot of questions that need to be answered at the beginning of a course. When are office hours? What are the grading policies? How many exams will there be? Will late homework be accepted? We have all seen the answers to these sorts of questions form the bulk of a standard course syllabus, and most of us feel an obligation (and rightly so) to provide such information

College algebra students' understanding and algebraic thinking and reasoning with functions

The purpose of study was to investigate college algebra students' understanding of function concepts. In addition, their solution strategies and algebraic thinking and reasoning were explored. Twenty-four volunteer students from one college algebra recitation class participated in the study to access their understanding of functions. Five out of 24 volunteer students were selected to participate in problem-solving interview sessions to provide a rich description of their understandings of functions and their algebraic thinking and reasoning. A Function Understanding Questionnaire was administered to gather these college students' understandings of functions after they completed the college algebra course. The questionnaire consisted of four questions asking students to identify their understanding of: (1) the definition of function, (2) multiple representations of functions, (3) the use of functions in doing mathematics, and (4) the use of functions in the real-world situations. Formal interviews prior to, during, and after instruction on functions with the five students were conducted, and their work on homework problems, quizzes and tests were explored to clarify these college students' understanding of functions and to explore their solution strategies and algebraic thinking and reasoning while solving problems. Overall, instruction supported students' understanding of functions. The students' definitions of a function improved toward a more formal definition. In addition, students had a better understanding of multiple representations, function transformations, and the application of functions to new mathematical situations and to real-world situations after completing the course. Algebraic reasoning includes the ability to approach and solve mathematical problems in multiple ways. The students in this study were able to use different methods to solve mathematical problems when they were encouraged to do so. However, the instruction did not encourage this activity. Perhaps for this reason, their algebraic thinking and reasoning abilities did not seem to progress as much. In concert with the recommendation of the several mathematics education organizations, more research needs to deal with the development of algebraic thinking and reasoning. In addition, research involving the communication of mathematical ideas and connection of mathematical understanding, thinking, and reasoning to other mathematics disciplines, to different subject areas, and to real-world situations are recommended