A set S of vertices in a graph G = (V, E) is a locating-dominating set if S is a dominating set of G, and every pair of distinct vertices {u, v} in V - S is located with respect to S, that is, if the set of neighbors of u that are in S is not equal to the set of neighbors of v that are in S. We give a construction of trees that have unique minimum locating-dominating sets.

Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The primary purpose of this paper is to use generating functions to prove other vector partition identities that parallel results of integer partitions.

Previous work by Flaxman (2004) and Biers-Ariel et al. (2018) focused on the number of distinct words embedded in a string of words of length n. In this thesis, we will extend this work to permutations, focusing on the maximum number of distinct permutations contained in a permutation on [n] = {1,2,...,n} and on the expected number of distinct permutations contained in a random permutation on [n]. We further considered the problem where repetition of subsequences are as a result of the occurrence of (Type A and/or Type B) replications. Our method of enumerating the Type A replications causes double counting and as a result causes the count of the number of distinct sequences to go down.

There has often been a connection between music and mathematics. The world of musical composition is full of combinations of orderings of different musical notes, each of which has different sound quality, length, and em phasis. One of the more intricate composition styles is twelve-tone music, where twelve unique notes (up to octave isomorphism) must be used before they can be repeated. In this thesis, we aim to show multiple ways in which mathematics can be used directly to compose twelve-tone musical scores.

We present SetPad, a new application prototype that lets computer science students explore discrete math problems by sketching set expressions using pen-based input. Students can manipulate the expressions interactively with the tool via pen or multi-touch interface. Likewise, discrete mathematics instructors can use SetPad to display and work through set problems via a projector to better demonstrate the solutions to the students. We discuss the implementation and feature set of the application, as well as results from both an informal perceived usefulness evaluation for students taking a computer science foundation exam in addition to a formal user study measuring the effectiveness of the tool when solving set proof problems. The results indicate that SetPad was well received, allows for efficient solutions to proof problems, and has the potential to have a positive impact when used as as an individual student application or an instructional tool.

The quaternions are an extension of the complex numbers which were first described by Sir William Rowan Hamilton in 1843. In his description, he gave the equation of the multiplication of the imaginary component similar to that of complex numbers. Many mathematicians have studied the zeros of quaternionic polynomials. Prominent of these, Ivan Niven pioneered a root-finding algorithm in 1941, Gentili and Struppa proved the Fundamental Theorem of Algebra (FTA) for quaternions in 2007. This thesis finds the zeros of quaternionic polynomials using the Fundamental Theorem of Algebra. There are isolated zeros and spheres of zeros. In this thesis, we also find the automorphisms of the zeros of the polynomials and the automorphism group.

We provide a combinatorial interpretation of the frequency of any irreducible representation of Sn in representations of Sn arising from group actions on words. Recognizing that representations arising from group actions naturally split across orbits yields combinatorial interpretations of the irreducible decompositions of representations from similar group actions. The generalization from group actions on words to group actions on matrices gives rise to representations that prove to be much less transparent. We share the progress made thus far on the open problem of determining the irreducible decomposition of certain representations of Sm × Sn arising from group actions on matrices.

We study the local list number of graphs introduced by Lennerstad and Eriksson. A labelling of a graph on n vertices is a bijection from vertex set to the set {1,…, n}. Given such a labelling c a vertex u is distance consistent if for all vertices v and w |c(u)-c(v)|=|c(u)-c(w)|+1 implies d(u,w)≤ d(u,v). A graph G is k-distance consistent if there is a labelling with k distance-consistent vertices. The local list number of a graph G is the largest k such that G is  k-distance consistent. We determine the local list number of cycles, complete bipartite graphs and some trees as well as prove bounds for some families of trees. We show that the local list number of even cycles is two, and of odd cycles is three. We also show that, if  k, l≥ 3 , the complete bipartite graph  Kk,l has local list number one, the star graph Sn=K1,n has local list number 3, and K2,k  has local list number 2. Finally, we show that for each n≥3 and each k such that 3≤k≤n there is a tree with local list number k. / Vi studerar det lokala listtalet introducerat av Lennerstad och Eriksson. En märkning av en graf på n hörn är en bijektion från hörnmängden till mängden {1, . . . , n}. Givet en sådan märkning c är ett hörn u avståndskonsistent om för alla hörn v och w |c(u) − c(v)| = |c(u) − c(w)| = 1 implicerar d(u, w) ≤d(u, v). En graf G är k-avståndskonsistent om det nns en märkning med k avståndskonsistenta hörn. Det lokala listtalet av en graf G är det största k sådan att G är k -avståndskonsistent. Vi bestämmer den lokala listtalet av cykler, kompletta bipartita grafer och vissa träd och visar som gränser för några familjer av träd. Vi visar att det lokla listtalet av jämna cykler är två, och av udda cykler är tre. Vi visar också att, om k, l ≥ 3, den kompletta bipartita grafen Kk,l har lokalt listtal ett, stjärngrafen Sn = K1,n har lokalt listtal 3, och K2,k har lokalt listtal 2. Slutligen, visar vi att för varje n≥3 och varje k sådant att 3 ≤ k≤n finns ett träd med lokalt listtal k.

Dating back to the 1930's, Ramsey theory still intrigues many who study combinatorics. Roughly put, it makes the profound assertion that complete disorder is impossible. One view of this problem is in edge-colorings of complete graphs. For forbidden graphs H1,...,Hk and a graph G, we write G "arrows" (H1,...,Hk) if every k-edge-coloring of G contains a monochromatic copy of Hi in color i for some i=1,2,...,k. If c is a (red, blue)-edge-coloring of G, we say c is a bad coloring if G contains no red K3or blue K1,t under c. A graph G is (H1,...,Hk)-Ramsey-minimal if G arrows (H1,...,Hk) but no proper subgraph of G has this property. Given a family F of graphs, we say that a graph G is F-saturated if no member of F is a subgraph of G, but for any edge xy not in E(G), G + xy contains a member of F as a subgraph. Letting Rmin(K3, K1,t) be the family of (K3,K1,t)-Ramsey minimal graphs, we study the saturation number, denoted sat(n,Rmin(K3,K1,t)), which is the minimum number of edges among all Rmin(K3,K1,t)-saturated graphs on n vertices. We believe the methods and constructions developed in this thesis will be useful in studying the saturation numbers of (K4,K1,t)-saturated graphs.

<p>Whether the growth series of a group is a rational function is investigated in this paper.Parry showed certain torus bundle groups of even trace exhibits rational growth, and thisresult has been extended by the author, Turbo Ho and Mark Pengitore. In this paper, bothresults are combined into a single proof used in [1], and the result is pushed further into thenegative case not covered in earlier works</p>