A decomposition of the complete graph on v vertices, Kv, into copies of K3 with a pendant edge is called a "lollipop" system of order v, denoted LS(v). We give necessary and sufficient conditions for the existence of a LS(v) admitting an automorphism consisting of two disjoint cycles. We also give a brief proof that the previously known sufficient conditions for the existence of a cyclic LS(v) are in fact necessary.

A vertex that is contained in some minimum dominating set of a graph G is a good vertex, otherwise it is bad. Let g(G) (respectively, b(G)) denote the number of good (respectively, bad) vertices in a graph G. We determine for which triples (x, y, z) there exists a graph G such that γ(G) = x, g(G) = y, and b(G) = z. Then we give graphs realizing these triples. Also, we show that no graph has g(G) = b(G) = γ(G) and characterize the graphs G for which g(G) = b(G) = γ(G) + 1.

The classical Eneström-Kakeya Theorem states that if p(z)=∑v=0navzv is a polynomial satisfying 0≤a0≤a1 ≤...≤an, then all of the zeros of p(z) lie in the region z≤1 in the complex plane. Many generalizations of the Eneström-Kakeya theorem exist which put various conditions on the coefficients of the polynomial (such as monotonicity of the moduli of the coefficients). We will introduce several results which put conditions on the coefficients of even powers of z and on the coefficients of odd powers of z. As a consequence, our results will be applicable to some polynomials to which these related results are not applicable.

Denote the total domination number of a graph G by γt(G). A graph G is said to be total domination edge critical, or simply γt-critical, if γt(G + e) < γt(G) for each edge e ∈ E(Ḡ). For 3 t-critical graphs G, that is, γt-critical graphs with γt(G) = 3, the diameter of G is either 2 or 3. We study the 3t-critical graphs G with diam G = 2.

A defensive alliance in a graph G = (V,E) is a set of vertices S ⊆ V satisfying the condition that for every vertex v ∈ S, the number of neighbors v has in S plus one (counting v) is at least as large as the number of neighbors it has in V - S. Because of such an alliance, the vertices in S, agreeing to mutually support each other, have the strength of numbers to be able to defend themselves from the vertices in V - S. A defensive alliance S is called global if it e ects every vertex in V - S, that is, every vertex in V - S is adjacent to at least one member of the alliance S. Note that a global defensive alliance is a dominating set. We study global defensive alliances in graphs.

A worm ω is a continuous rectifiable arc of unit length in the Cartesian plane. Let W denote the class of all worms. A planar region C is called a cover for W if it contains a copy of every worm in W. That is, C will cover or contain any member ω of W after an appropriate translation and/or rotation of ω is completed (no reflections). The open problem of determining a cover C of smallest area is attributed to Leo Moser [7], [8]. This paper reduces the smallest known upper bound for this area from 0.275237 [10] to 0.260437.

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. We study graphs having the same total domination number as their complements. In particular, we characterize the cubic graphs having this property. Also we characterize such graphs with total domination numbers equal to two or three, and we determine properties of the ones with larger total domination numbers.

Let Sn be the set of all permutations on [n]:= {1, 2,..., n}. We denote by κn the smallest cardinality of a subset A of Sn+1 that covers Sn, in the sense that each π ∈ Sn may be found as an order-isomorphic subsequence of some π′ in A. What are general upper bounds on κn? If we randomly select νn elements of Sn+1, when does the probability that they cover Sn transition from 0 to 1? Can we provide a fine-magnification analysis that provides the "probability of coverage" when νn is around the level given by the phase transition? In this paper we answer these questions and raise others.

We discuss methods for a priori selection of parameters to be estimated in inverse problem formulations (such as Maximum Likelihood, Ordinary and Generalized Least Squares) for dynamical systems with numerous state variables and an even larger number of parameters. We illustrate the ideas with an in-host model for HIV dynamics which has been successfully validated with clinical data and used for prediction and a model for the reaction of the cardiovascular system to an ergometric workload.

The nonlinear Schrödinger (NLS) equation is a classical field equation that describes weakly nonlinear wave-packets in one-dimensional physical systems. It is in a class of nonlinear partial differential equations (PDEs) that pertain to several physical and biological systems. In this project we apply a pseudo-spectral solution-estimation method to a modified version of the NLS equation as a means of searching for solutions that are solitons, where a soliton is a self-reinforcing solitary wave that maintains its shape over time. We use the pseudo-spectral method to determine whether cardiac action potential states, which are perturbed solutions to the Fitzhugh-Nagumo nonlinear PDE, create soliton-like solutions. We then use symmetry group properties of the NLS equation to explore these solutions and find new ones.