Given a connected graph with domination (or total domination) number γ ≥ 2, we ask for the maximum number mγ and m T,γof dominating and total dominating sets of size γ. An exact answer is provided for γ = 2 and lower bounds are given for m γ and mT,γ;γ≥3.

A subset S ⊆ V in a graph G = (V,E) is a [1, k]-set for a positive integer k if for every vertex v ∈ V \ S, 1 ≤ |N(v) ∩ S|; ≤ k, that is, every vertex v ∈ V \ S is adjacent to at least one but not more than k vertices in S. We consider [1, k]-sets that are also independent, and note that not every graph has an independent [1, k]-set. For graphs having an independent [1, k]-set, we define the lower and upper [1, k]-independence numbers and determine upper bounds for these values. In addition, the trees having an independent [1, k]-set are characterized. Also, we show that the related decision problem is NP-complete.

A set A ⊆ [n] ∪ {0} is said to be a 2-additive basis for [n] if each j ∈ [n] can be written as j = x + y, x, y ∈ A, x ≤ y. If we pick each integer in [n] ∪ {0} independently with probability p = pn → 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is [(1 - α)n, (1 + α)n] (or equivalently [αn, (2 - α)n]) for some 0 < α < 1. We use a delicate application of Janson's correlation inequalities in conjuction with the Stein-Chen method of Poisson approximation to tease out a very sharp threshold for the emergence of a 2-additive basis. Generalizations to k-additive bases are then given.

A graph G is a (t, r)-regular graph if every collection of t independent vertices is collectively adjacent to exactly r vertices. Let p, s, and m be positive integers, where m ≥ 2 and let G be a (2,r)-regular graph. If n is sufficiently large, then G is isomorphic to G = Ks + mKp, where 2(p- 1) + s = r. A nested (2, r)-regular graph is constructed by replacing selected cliques in a (2,r)-regular graph with a (2, r')-regular graph and joining the vertices of the peripheral cliques. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs.

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In this paper, we consider decompositions of the complete graph on υv vertices into 4-cycles with a pendant edge which admit automorphisms consisting of: (1) a single cycle of length υ, (2) f fixed points and a cycle of length υ - f, or (3) two disjoint cycles.

A decomposition D of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. The intersection graph I(D) of the decomposition D has a vertex for each part of the partition and two parts A and B are adjacent iff they share a common node in H. If I(D) ≅ H, then D is an automorphic decomposition of H. If n(G) = x(H) as well, then we say that D is a fully automorphic decomposition. In this paper, we examine the question of whether a fully automorphic host will have an even degree of regularity. We also give several examples of fully automorphic decompositions as well as necessary conditions for their existence.

In a recent work by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. In this paper, we extend this study by considering the windmill and the double star. Simple necessary and sufficient conditions are given for the solvability of each graph. We also discuss an open problem concerning the range of values for which a graph has a terminal state with k pegs.

Because quantitative biology requires skills and concepts from a disparate collection of different disciplines, the scientists of the near future will increasingly need to rely on collaborations to produce results. Correspondingly, students in disciplines impacted by quantitative biology will need to be taught how to create and engage in such collaborations. In response to this important curricular need, East Tennessee State University and Georgia Technological University/Emory University cooperated in an unprecedented curricular experiment in which theoretically oriented students at East Tennessee State designed biophysical models that were implemented and tested experimentally by biomedical engineers at the Wallace H. Coulter Department of Biomedical Engineering at Georgia Technological University and Emory University. Implementing the collaborations between two institutions allowed an assessment of the student collaborations from before the groups of students had met for the first time until after they had finished their projects, thus providing insight about the formation and conduct of such collaborations that could not have been obtained otherwise.