Discrete Mathematics and Combinatorics Commons^™

Conflict Dynamics In Scale-Free Networks With Degree Correlations And Hierarchical Structure, Eduardo Jacobo-Villegas, Bibiana Obregón-Quintana, Lev Guzmán-Vargas, Larry S. Liebovitch

Publications and Research

We present a study of the dynamic interactions between actors located on complex networks with scale-free and hierarchical scale-free topologies with assortative mixing, that is, correlations between the degree distributions of the actors. The actor’s state evolves according to a model that considers its previous state, the inertia to change, and the influence of its neighborhood. We show that the time evolution of the system depends on the percentage of cooperative or competitive
interactions. For scale-free networks, we find that the dispersion between actors is higher when all interactions are either cooperative or competitive, while a balanced presence of interactions …

Using Magic To Teach Computer Programming, Dale F. Reed, Ronald I. Greenberg

Computer Science: Faculty Publications and Other Works

Magic can be used in project-based instruction to motivate students and provide a meaningful context for learning computer programming. This work describes several magic programs of the “Choose a Number” and “Pick a Card” varieties, making connections to underlying computing concepts.

Magic tricks presented as demonstrations and programming assignments elicit wonder and captivate students’ attention, so that students want to understand and replicate the work to show it to friends and family members. Capturing student interest and curiosity motivates them to learn the underlying programming concepts.

Two “Choose a Number” programs are shown where the computer is able to identify …

Relaxed Wythoff Has All Beatty Solutions, Jon Kay, Geremías Polanco

Mathematics Sciences: Faculty Publications

We find conditions under which the P-positions of three subtraction games arise as pairs of complementary Beatty sequences. The first game is due to Fraenkel and the second is an extension of the first game to non-monotone settings. We show that the P-positions of the second game can be inferred from the recurrence of Fraenkel's paper if a certain inequality is satisfied. This inequality is shown to be necessary if the P-positions are known to be pairs of complementary Beatty sequences, and the family of irrationals for which this inequality holds is explicitly given. We highlight several games in the …

Structure Of Number Theoretic Graphs, Lee Trent

Mathematical Sciences Technical Reports (MSTR)

The tools of graph theory can be used to investigate the structure
imposed on the integers by various relations. Here we investigate two
kinds of graphs. The first, a square product graph, takes for its vertices
the integers 1 through n, and draws edges between numbers whose product
is a square. The second, a square product graph, has the same vertex set,
and draws edges between numbers whose sum is a square.
We investigate the structure of these graphs. For square product
graphs, we provide a rather complete characterization of their structure as
a union of disjoint complete graphs. For …

Extremal Problems In Graph Saturation And Covering, Adam Volk

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation considers several problems in extremal graph theory with the aim of finding the maximum or minimum number of certain subgraph counts given local conditions. The local conditions of interest to us are saturation and covering. Given graphs F and H, a graph G is said to be F-saturated if it does not contain any copy of F, but the addition of any missing edge in G creates at least one copy of F. We say that G is H-covered if every vertex of G is contained in at least one copy of H. In the former setting, we …

Combinatorial Optimization With Photonics-Inspired Clock Models, Mostafa Honari-Latifpour, Matthew S. Mills, Mohammad-Ali Miri

Publications and Research

NP-hard combinatorial optimization problems are in general hard problems that their computational complexity grows faster than polynomial scaling with the size of the problem. Thus, over the years there has been a great interest in developing unconventional methods and algorithms for solving such problems. Here, inspired by the nonlinear optical process of q-photon down-conversion, in which a photon is converted into q degenerate lower energy photons, we introduce a nonlinear dynamical model that builds on coupled single-variable phase oscillators and allows for efficiently approximating the ground state of the classical q-state planar Potts Hamiltonian. This reduces the exhaustive search in …