# Discrete Math Seminar (DMS)

The **Discrete Math Seminar (DMS)** is a research seminar intended for Kennesaw State faculty working in the various
fields of algebra, number theory, and discrete math. A main goal of the seminar is
to encourage collaborative work between KSU and neighboring institutions. Seminars
often involved advanced mathematical knowledge. However, the seminars are open to
anyone who is interested in attending.

**SPRING 2017**

**Wednesday, February 22, 2017 **- 3:30-4:30pm in D-250 (Marietta Campus)

**Ariel Keller**, Emory University*"On Disjoint Cycles and Degree Conditions"*- ABSTRACT: We consider degree sum conditions sufficient to imply the existence of k vertex disjoint cycles in a graph. In particular, sigma_t(G) is the minimum degree sum over all sets in G of t independent vertices. We prove that if a graph G has order at least 7k+1 and sigma_4(G) >= 8k-3, with k >= 2, then G contains k disjoint cycles.

**Wednesday, March 1, 2017 **- 3:30-4:30pm in D-237 (Marietta Campus)

**Dr. Stuart Borrett**, Department of Biology and Marine Biology & Center for Marine Science, University of North Carolina Wilmington; Duke Network Analysis Center, Duke University*"Network Ecology: Using Math to Understand Ecosystems"*- ABSTRACT: Living systems are linked through multiple networks of energy, matter, and informational exchanges. Patterns in these exchange networks reveal information about the structure, function, and behavior of these complex systems as well as the processes that create them. Ecological Network Analysis (ENA) is a method to investigate the energy and matter exchange networks in ecological systems. In this presentation I review the formal features of the ENA model, and introduce an organizational skeleton for the multitude of techniques. I then illustrate applications of ENA to investigate ecological problems of both a theoretical and applied nature. To conclude, I will characterize a number of open mathematical and statistical challenges for ENA including: (1) construction of useful null models, (2) benchmarking network metrics, (3) violation of series convergence criteria, and (4) an uncertainty analyses that enables stronger inference. ENA has provided novel insights into food web organization, ecosystem functioning, estuarine biogeochemistry, and the sustainability of urban and industrial systems. With further development, it may be a useful decision tool for ecosystem management and sustainable development.

**Wednesday, March 22, 2017** - 3:30-4:30pm in D-250 (Marietta Campus)

**Dr. Babak Moazzez**, Assistant Professor of Mathematics, Kennesaw State University*"Integer Programming Approach to Static Monopolies in Graphs"*- ABSTRACT: A subset
*M*of vertices of a graph is called a*static monopoly*, if any vertex*v*outside*M*has at least [½ deg (*v*)] neighbors in*M.*The minimum static monopoly problem has been extensively studied in graph theoretical context. We look at this problem from an integer programming point of view for the first time and give a linear formulation for it. We study the facial structure of the corresponding polytope, classify facet defining inequalities of the integer programming formulation and introduce some families of valid inequalities. We show that in the presence of a vertex cut or an edge cut in the graph, the problem can be solved more efficiently by adding some strong valid inequalities. An algorithm is given that solves the minimum monopoly problem in trees and cactus graphs in linear time. We test our methods by performing several experiments on randomly generated graphs. A software package is introduced that solves the minimum monopoly problem using open source integer linear programming solvers.

**FALL 2016**

**Tuesday, August 30, 2016** - 3:30-4:30pm in D-225 (Marietta Campus)

**Jess Fuller**, Emory University*"Saturation and Constructing (K_t-e)-Saturated Graphs"*

**Tuesday, September 20, 2016** - 3:30-4:30pm in D-225 (Marietta Campus)

**Linh Le**, Kennesaw State University*"Utilizing Item Graphs in Detecting Purchase Patterns"*

**Tuesday, October 25, 2016** - 3:30-4:30pm in D-225 (Marietta Campus)

**Steve Edwards**, Department of Mathematics, Kennesaw State University*"When is Zero not Zero? A Discrete Interloper Reveals Nothing (or: From Fibonacci to Cross Polytopes and Beyond via Alternating Binomial Sums)"*- ABSTRACT: A Fibonacci identity leads us to an infinite family of alternating binomial sums, each of which equals zero. We generalize to a second family of sums that equal a binomial coefficient. We show that each sum generates a family of doubly-recursive sequences, the only known one being the Cross-Polytope numbers. The two families turn out to be related. Within the doubly-recursive sequences are the sequences of numbers for which 2n - k is a perfect square.

**Tuesday, November 8, 2016** - 3:30-4:30pm in D-225 (Marietta Campus)

**Joel Fowler**, Department of Mathematics, Kennesaw State University*"Counting Random Strings That Don't Look Random"*- ABSTRACT: Long random strings of characters often don't look entirely random because of the likely appearance of substrings with identifiable patterns, such as runs of repeated or alternating characters. We look at the enumeration of strings that are free of regular substrings, when those substrings are generated by the repeated action of permutations on the set of characters. The recurrence relations obtained can be thought of as an extension of the Fibonacci number enumeration of binary strings with no two ones side by side.