Discrete Mathematics and Combinatorics Commons™

All Articles in Discrete Mathematics and Combinatorics

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Bounding The Number Of Compatible Simplices In Higher Dimensional Tournaments, 2019 University of Kentucky

Bounding The Number Of Compatible Simplices In Higher Dimensional Tournaments, Karthik Chandrasekhar

Theses and Dissertations--Mathematics

A tournament graph G is a vertex set V of size n, together with a directed edge set EV × V such that (i, j) ∈ E if and only if (j, i) ∉ E for all distinct i, jV and (i, i) ∉ E for all iV. We explore the following generalization: For a fixed k we orient every k-subset of V by assigning it an orientation. That is, every facet of the (k − 1)-skeleton of the (n − 1)-dimensional simplex on V is given an orientation. In this dissertation we bound the number of compatible k-simplices ...

Visualizing Random Walks On Supercharacter Theories For Uppertriangular Unipotent Matrices, Justin Willson

The conjugacy classes of some finite groups are provably wild in the sense that there is no general solution to finding their conjugacy classes other than brute force. One solution to this problem is to create a supercharacter theory for a group which has the effect of gluing together conjugacy classes into more manageable chunks. In this paper we apply these methods to the group of uppertriangular unipotent matrices and analyze the combinatorics of this added structure.

Induction Of Nontrivial Supercharacter Theories For Finite Groups, Jesse Franklin

This study focuses on the partitions of a group that arise from: action by conjugation, a two sided multiplicative generalization of conjugation, and inclusion of a subgroup into the group. Since conjugacy classes correspond to irreducible characters, studying the partitions in a group compatible with conjugacy classes in the subgroup, and by analogy, studying the partition of a group compatible with superclasses in a subgroup, invariances in the group can be derived from the subgroup's simpler structure. The fusion of conjugacy classes, and superclasses, has some effects on the calculation of an induced and superinduced function. However, these effects ...

2019 Bucknell University

Positivity Among P-Partition Generating Functions Of Partially Ordered Sets, Nate Lesnevich

Honors Theses

We find necessary and separate sufficient conditions for the difference between two labeled partially ordered set's (poset) partition generating functions to be positive in the fundamental basis. We define the notion of a jump sequence for a poset and show how different conditions on the jump sequences of two posets are necessary for those posets to have an order relation in the fundamental basis. Our sufficient conditions are of two types. First, we show how manipulating a poset's Hasse diagram produces a poset that is greater according to the fundamental basis. Secondly, we also provide tools to explain ...

Ordering Cacti With Signless Laplacian Spread, 2018 China University of Mining and Technology

Ordering Cacti With Signless Laplacian Spread, Zhen Lin, Shu-Guang Guo

Electronic Journal of Linear Algebra

A cactus is a connected graph in which any two cycles have at most one vertex in common. The signless Laplacian spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated signless Laplacian matrix. In this paper, all cacti of order n with signless Laplacian spread greater than or equal to n − 1/2 are determined.

Two Linear Preserver Problems On Graphs, 2018 Hunan University

Two Linear Preserver Problems On Graphs, Yanan Hu, Zhenhua Lyu

Electronic Journal of Linear Algebra

Let n, t, k be integers such that 3 ≤ t,k ≤ n. Denote by G_n the set of graphs with vertex set {1,2,...,n}. In this paper, the complete linear transformations on G_n mapping K_t-free graphs to K_t-free graphs are characterized. The complete linear transformations on G_n mapping C_k-free graphs to C_k-free graphs are also characterized when n ≥ 6.

Subsets Of Vertices Of The Same Size And The Same Maximum Distance, 2018 Karlsruhe Institute of Technology

Subsets Of Vertices Of The Same Size And The Same Maximum Distance, Maria Axenovich, Dominik Duerrschnabel

Theory and Applications of Graphs

For a simple connected graph $G=(V,E)$ and a subset $X$ of its vertices, let $$d^*(X) = \max\{{\rm dist}_G(x,y): x,y\in X\}$$ and let

$h^*(G)$ be the largest $k$ such that there are disjoint vertex subsets $A$ and $B$ of $G$, each of size $k$ such that $d^*(A) = d^*(B).$

Let $h^*(n) = \min \{h^*(G): |V(G)|=n\}$. We prove that $h^*(n) = \lfloor (n+1)/3 \rfloor,$ for $n\geq 6.$ This solves the homometric set problem restricted to the largest distance exactly. In addition we compare $h^*(G)$ with ...

Pentagons In Triangle-Free Graphs, 2018 Iowa State University

Pentagons In Triangle-Free Graphs, Bernard Lidicky, Florian Pfender

Mathematics Publications

For all n≥9, we show that the only triangle-free graphs on n vertices maximizing the number 5-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by Erd\H{o}s, and extends results by Grzesik and Hatami, Hladky, Kral, Norin and Razborov, where they independently showed this same result for large n and for all n divisible by 5.

Italian Domination On Ladders And Related Products, 2018 East Tennessee State University

Electronic Theses and Dissertations

An Italian dominating function on a graph $G = (V,E)$ is a function such that $f : V \to \{0,1,2\}$, and for each vertex $v \in V$ for which $f(v) = 0$, we have $\sum_{u\in N(v)}f(u) \geq 2$. The weight of an Italian dominating function is $f(V) = \sum_{v\in V(G)}f(v)$. The minimum weight of all such functions on a graph $G$ is called the Italian domination number of $G$. In this thesis, we will consider Italian domination in various types of products of a graph $G$ with the complete ...

Finite Simple Graphs And Their Associated Graph Lattices, 2018 Middle Tennessee State University

Finite Simple Graphs And Their Associated Graph Lattices, James B. Hart, Brian Frazier

Theory and Applications of Graphs

In his 2005 dissertation, Antoine Vella explored combinatorical aspects of finite graphs utilizing a topological space whose open sets are intimately tied to the structure of the graph. In this paper, we go a step further and examine some aspects of the open set lattices induced by these topological spaces. In particular, we will characterize all lattices that constitute the opens for finite simple graphs endowed with this topology, explore the structure of these lattices, and show that these lattices contain information necessary to reconstruct the graph and its complement in several ways.

Inducibility Of Directed Paths, 2018 Hankuk University of Foreign Studies

Inducibility Of Directed Paths, Ilkyoo Choi, Bernard Lidicky, Florian Pfender

Mathematics Publications

A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles.

Erasure Coding For Distributed Matrix Multiplication For Matrices With Bounded Entries, 2018 Iowa State University

Erasure Coding For Distributed Matrix Multiplication For Matrices With Bounded Entries, Li Tang, Konstantinos Konstantinidis, Aditya Ramamoorthy

Electrical and Computer Engineering Publications

Distributed matrix multiplication is widely used in several scientific domains. It is well recognized that computation times on distributed clusters are often dominated by the slowest workers (called stragglers). Recent work has demonstrated that straggler mitigation can be viewed as a problem of designing erasure codes. For matrices A and B, the technique essentially maps the computation of ATB into the multiplication of smaller (coded) submatrices. The stragglers are treated as erasures in this process. The computation can be completed as long as a certain number of workers (called the recovery threshold) complete their assigned tasks. We present a novel ...

Polychromatic Colorings On The Integers, 2018 Karlsruhe Institute of Technology

Polychromatic Colorings On The Integers, Maria Axenovich, John Goldwasser, Bernard Lidicky, Ryan R. Martin, David Offner, John Talbot, Michael Young

Bernard Lidický

We show that for any set S ⊆ Z, |S| = 4 there exists a 3-coloring of Z in which every translate of S receives all three colors. This implies that S has a codensity of at most 1/3, proving a conjecture of Newman. We also consider related questions in Zd, d ≥ 2.

2018 Rose-Hulman Institute of Technology

A Proof Of The "Magicness" Of The Siam Construction Of A Magic Square, Joshua Arroyo

A magic square is an n x n array filled with n2 distinct positive integers 1, 2, ..., n2 such that the sum of the n integers in each row, column, and each of the main diagonals are the same. A Latin square is an n x n array consisting of n distinct symbols such that each symbol appears exactly once in each row and column of the square. Many articles dealing with the construction of magic squares introduce the Siam method as a "simple'' construction for magic squares. Rarely, however, does the article actually prove that the construction ...

On Orders Of Elliptic Curves Over Finite Fields, 2018 Columbia University

On Orders Of Elliptic Curves Over Finite Fields, Yujin H. Kim, Jackson Bahr, Eric Neyman, Gregory Taylor

In this work, we completely characterize by $j$-invariant the number of orders of elliptic curves over all finite fields $F_{p^r}$ using combinatorial arguments and elementary number theory. Whenever possible, we state and prove exactly which orders can be taken on.

On The Largest Distance (Signless Laplacian) Eigenvalue Of Non-Transmission-Regular Graphs, 2018 East China Normal University

On The Largest Distance (Signless Laplacian) Eigenvalue Of Non-Transmission-Regular Graphs, Shuting Liu, Jinlong Shu, Jie Xue

Electronic Journal of Linear Algebra

Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by \$W ...

Spectral Bounds For The Connectivity Of Regular Graphs With Given Order, 2018 Maastricht University

Spectral Bounds For The Connectivity Of Regular Graphs With Given Order, Aida Abiad, Boris Brimkov, Xavier Martinez-Rivera, Suil O, Jingmei Zhang

Electronic Journal of Linear Algebra

The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to connectivity attributes such as the vertex- and edge-connectivity, isoperimetric number, and characteristic path length. In this paper, two upper bounds are presented for the second-largest eigenvalues of regular graphs and multigraphs of a given order which guarantee a desired vertex- or edge-connectivity. The given bounds are in terms of the order and degree of the graphs, and hold with equality for infinite families of graphs. These results ...

Transformations On Double Occurrence Words Motivated By Dna Rearrangement, 2018 University of South Florida

Transformations On Double Occurrence Words Motivated By Dna Rearrangement, Daniel Cruz, Margherita Maria Ferrari, Lukas Nabergall, Natasa Jonoska, Masahico Saito

Annual Symposium on Biomathematics and Ecology: Education and Research

No abstract provided.

2018 Illinois State University

The Influence Of Canalization On The Robustness Of Finite Dynamical Systems, Claus Kadelka

Annual Symposium on Biomathematics and Ecology: Education and Research

No abstract provided.

Combinatorial Geometry Of Threshold-Linear Networks, 2018 Illinois State University

Combinatorial Geometry Of Threshold-Linear Networks, Christopher Langdon

Annual Symposium on Biomathematics and Ecology: Education and Research

No abstract provided.