# Discrete Mathematics and Combinatorics Commons™

## All Articles in Discrete Mathematics and Combinatorics

1,216 full-text articles. Page 1 of 49.

2024 Sheffield Hallam University

#### Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett

##### Journal of Humanistic Mathematics

Mathematics is often presented as a neatly polished finished product, yet its development is messy and often full of mis-steps that could have been avoided with hindsight. An experience with a puzzle illustrates this conflict. The puzzle asks for the probability that a group of four and a group of two are seated adjacently within a hundred seats, and is solved using combinatorics techniques.

Recent Studies On The Super Edge-Magic Deficiency Of Graphs, 2024 Kokushikan University

#### Recent Studies On The Super Edge-Magic Deficiency Of Graphs, Rikio Ichishima, Susana C. Lopez, Francesc Muntaner, Yukio Takahashi

##### Theory and Applications of Graphs

A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left(G\right) \right\vert \right\}$ such that $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant for each $uv\in E\left( G\right)$. Also, $G$ is called super edge-magic if $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$. Furthermore, the super edge-magic deficiency $\mu_{s}\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G \cup nK_{1}$ is super edge-magic or $+ \infty$ if there exists no such …

A Survey Of Maximal K-Degenerate Graphs And K-Trees, 2024 Purdue University

#### A Survey Of Maximal K-Degenerate Graphs And K-Trees, Allan Bickle

##### Theory and Applications of Graphs

This article surveys results on maximal $k$-degenerate graphs, $k$-trees,

and related classes including simple $k$-trees, $k$-paths, maximal

outerplanar graphs, and Apollonian networks. These graphs are important

in many problems in graph theory and computer science. Types of results

surveyed include structural characterizations, enumeration, degree

sets and sequences, chromatic polynomials, algorithms, and related

extremal problems.

On The Singular Pebbling Number Of A Graph, 2024 University of Waterloo

#### On The Singular Pebbling Number Of A Graph, Harmony R. Morris

In this paper, we define a new parameter of a connected graph as a spin-off of the pebbling number (which is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble). This new parameter is the singular pebbling number, the smallest t such that a player can be given any configuration of at least t pebbles and any target vertex and can successfully move pebbles so that exactly one pebble ends on the target vertex. We also prove that the singular pebbling number of any graph on 3 or more vertices is equal …

Reducing Food Scarcity: The Benefits Of Urban Farming, 2023 Brigham Young University

#### Reducing Food Scarcity: The Benefits Of Urban Farming, S.A. Claudell, Emilio Mejia

##### Journal of Nonprofit Innovation

Urban farming can enhance the lives of communities and help reduce food scarcity. This paper presents a conceptual prototype of an efficient urban farming community that can be scaled for a single apartment building or an entire community across all global geoeconomics regions, including densely populated cities and rural, developing towns and communities. When deployed in coordination with smart crop choices, local farm support, and efficient transportation then the result isn’t just sustainability, but also increasing fresh produce accessibility, optimizing nutritional value, eliminating the use of ‘forever chemicals’, reducing transportation costs, and fostering global environmental benefits.

Imagine Doris, who is …

Difference Of Facial Achromatic Numbers Between Two Triangular Embeddings Of A Graph, 2023 Seikei University

#### Difference Of Facial Achromatic Numbers Between Two Triangular Embeddings Of A Graph, Kengo Enami, Yumiko Ohno

##### Theory and Applications of Graphs

A facial $3$-complete $k$-coloring of a triangulation $G$ on a surface is a vertex $k$-coloring such that every triple of $k$-colors appears on the boundary of some face of $G$. The facial $3$-achromatic number $\psi_3(G)$ of $G$ is the maximum integer $k$ such that $G$ has a facial $3$-complete $k$-coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge.

For two triangulations $G$ and $G'$ on a surface, $\psi_3(G)$ may not be equal to $\psi_3(G')$ even if $G$ …

The Ricci Curvature On Simplicial Complexes, 2023 Shimane University

#### The Ricci Curvature On Simplicial Complexes, Taiki Yamada

##### Theory and Applications of Graphs

We define the Ricci curvature on simplicial complexes modifying the definition of the Ricci curvature on graphs, and prove upper and lower bounds of the Ricci curvature. These properties are generalizations of previous studies. Moreover, we obtain an estimate of the eigenvalues of the Laplacian on simplicial complexes by the Ricci curvature.

Toughness Of Recursively Partitionable Graphs, 2023 University of Vermont

#### Toughness Of Recursively Partitionable Graphs, Calum Buchanan, Brandon Du Preez, K. E. Perry, Puck Rombach

##### Theory and Applications of Graphs

A simple graph G = (V,E) on n vertices is said to be recursively partitionable (RP) if G ≃ K1, or if G is connected and satisfies the following recursive property: for every integer partition a1, a2, . . . , ak of n, there is a partition {A1,A2, . . . ,Ak} of V such that each |Ai| = ai, and each induced subgraph G[Ai] is RP (1 ≤ i ≤ k). We show that if S is a …

Wiener Index In Graphs Given Girth, Minimum, And Maximum Degrees, 2023 Alabama State University, USA

#### Wiener Index In Graphs Given Girth, Minimum, And Maximum Degrees, Fadekemi J. Osaye, Liliek Susilowati, Alex S. Alochukwu, Cadavious Jones

##### Theory and Applications of Graphs

Let $G$ be a connected graph of order $n$. The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. The well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ by Kouider and Winkler \cite{Kouider} was improved significantly by Alochukwu and Dankelmann \cite{Alex} for graphs containing a vertex of large degree $\Delta$ to $W(G) \leq {n-\Delta+\delta \choose 2} \big( \frac{n+2\Delta}{\delta+1}+4 \big)$. In this paper, we give upper bounds on the Wiener index of $G$ in terms of order …

2023 Indian Institute of Technology Hyderabad

#### On The Hardness Of The Balanced Connected Subgraph Problem For Families Of Regular Graphs, Harsharaj Pathak

##### Theory and Applications of Graphs

The Balanced Connected Subgraph problem (BCS) was introduced by Bhore et al. In the BCS problem we are given a vertex-colored graph G = (V, E) where each vertex is colored “red” or “blue”. The goal is to find a maximum cardinality induced connected subgraph H of G such that H contains an equal number of red and blue vertices. This problem is known to be NP-hard for general graphs as well as many special classes of graphs. In this work we explore the time complexity of the BCS problem in case of regular graphs. We prove that the BCS …

2023 William & Mary

#### The Vulnerabilities To The Rsa Algorithm And Future Alternative Algorithms To Improve Security, James Johnson

The RSA encryption algorithm has secured many large systems, including bank systems, data encryption in emails, several online transactions, etc. Benefiting from the use of asymmetric cryptography and properties of number theory, RSA was widely regarded as one of most difficult algorithms to decrypt without a key, especially since by brute force, breaking the algorithm would take thousands of years. However, in recent times, research has shown that RSA is getting closer to being efficiently decrypted classically, using algebraic methods, (fully cracked through limited bits) in which elliptic-curve cryptography has been thought of as the alternative that is stronger than …

2023 East Tennessee State University

#### A Bridge Between Graph Neural Networks And Transformers: Positional Encodings As Node Embeddings, Bright Kwaku Manu

##### Electronic Theses and Dissertations

Graph Neural Networks and Transformers are very powerful frameworks for learning machine learning tasks. While they were evolved separately in diverse fields, current research has revealed some similarities and links between them. This work focuses on bridging the gap between GNNs and Transformers by offering a uniform framework that highlights their similarities and distinctions. We perform positional encodings and identify key properties that make the positional encodings node embeddings. We found that the properties of expressiveness, efficiency and interpretability were achieved in the process. We saw that it is possible to use positional encodings as node embeddings, which can be …

2023 Clemson University

#### Zero-Knowledge Reductions And Confidential Arithmetic, Marvin Jones

##### All Dissertations

The changes in computing paradigms to shift computations to third parties have resulted in the necessity of these computations to be provable. Zero-knowledge arguments are probabilistic arguments that are used to to verify computations without secret data being leaked to the verifying party.

In this dissertation, we study zero-knowledge arguments with specific focus on reductions. Our main contributions are:

1. Provide a thorough survey in a variety of zero-knowledge techniques and protocols.
2. Prove various results of reductions that can be used to study interactive protocols in terms of subroutines. Additionally, we identify an issue in the analogous definition of zero-knowledge for …

Foundations Of Memory Capacity In Models Of Neural Cognition, 2023 California Polytechnic State University, San Luis Obispo

#### Foundations Of Memory Capacity In Models Of Neural Cognition, Chandradeep Chowdhury

##### Master's Theses

A central problem in neuroscience is to understand how memories are formed as a result of the activities of neurons. Valiant’s neuroidal model attempted to address this question by modeling the brain as a random graph and memories as subgraphs within that graph. However the question of memory capacity within that model has not been explored: how many memories can the brain hold? Valiant introduced the concept of interference between memories as the defining factor for capacity; excessive interference signals the model has reached capacity. Since then, exploration of capacity has been limited, but recent investigations have delved into the …

Generalized Vulnerability Measures Of Graphs, 2023 Clemson University

#### Generalized Vulnerability Measures Of Graphs, Julia Vanlandingham

##### All Theses

Several measures of vulnerability of a graph look at how easy it is to disrupt the network by removing/disabling vertices. As graph-theoretical parameters, they treat all vertices alike: each vertex is equally important. For example, the integrity parameter considers the number of vertices removed and the maximum number of vertices in a component that remains. We consider the generalization of these measures of vulnerability to weighted vertices in order to better model real-world applications. In particular, we investigate bounds on the weighted versions of connectivity and integrity, when polynomial algorithms for computation exist, and other characteristics of the generalized measures.

Jumping Frogs On Cyclic Graphs, 2023 Murray State University

#### Jumping Frogs On Cyclic Graphs, Jake Mitchell

##### Honors College Theses

From the traditional game of Solitaire to modern video games like Candy Crush and Five Nights at Freddy’s, single-player games have captivated audiences for gener- ations. We investigate a lesser-known single-player game, the Jumping Frogs problem, on various classes of simple graphs, a graph with no multiple edges or looped ver- tices. We determine whether frogs can be stacked together on one vertex of a given graph. In a graph with k vertices and one frog on each vertex, the frogs must make legal jumps to form a stack of k frogs. The problem is known to be solvable on …

Combinatorially Orthogonal Paths, 2023 Texas A&M University Texarkana

#### Combinatorially Orthogonal Paths, Sean A. Bailey, David E. Brown, Leroy Beaseley

##### Communications on Number Theory and Combinatorial Theory

Vectors x=(x1,x2,...,xn)T and y=(y1,y2,...,yn)T are combinatorially orthogonal if |{i:xiyi≠0}|≠1. An undirected graph G=(V,E) is a combinatorially orthogonal graph if there exists f:V→ℝn such that for any u,vV, uvE iff f(u) and f(v) are combinatorially orthogonal. We will show that every graph has a combinatorially orthogonal representation. We will show …

2023 University of Massachusetts Amherst

#### Intersection Cohomology Of Rank One Local Systems For Arrangement Schubert Varieties, Shuo Lin

##### Doctoral Dissertations

In this thesis we study the intersection cohomology of arrangement Schubert varieties with coefficients in a rank one local system on a hyperplane arrangement complement. We prove that the intersection cohomology can be computed recursively in terms of certain polynomials, if a local system has only $\pm 1$ monodromies. In the case where the hyperplane arrangement is generic central or equivalently the associated matroid is uniform and the local system has only $\pm 1$ monodromies, we prove that the intersection cohomology is a combinatorial invariant. In particular when the hyperplane arrangement is associated to the uniform matroid of rank $n-1$ …

Facets Of The Union-Closed Polytope, 2023 University of Massachusetts Amherst

#### Facets Of The Union-Closed Polytope, Daniel Gallagher

##### Doctoral Dissertations

In the haze of the 1970s, a conjecture was born to unknown parentage...the union-closed sets conjecture. Given a family of sets $\FF$, we say that $\FF$ is union-closed if for every two sets $S, T \in \FF$, we have $S \cup T \in \FF$. The union-closed sets conjecture states that there is an element in at least half of the sets of any (non-empty) union-closed family. In 2016, Pulaj, Raymond, and Theis reinterpreted the conjecture as an optimization problem that could be formulated as an integer program. This thesis is concerned with the study of the polytope formed by taking …

Msis-Kadelka: On The Uniqueness Of Network Identification, 2023 University of Dayton

#### Msis-Kadelka: On The Uniqueness Of Network Identification, Alan Veliz-Cuba, Elena Dimitrova

##### Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.