# Discrete Mathematics and Combinatorics Commons™

Articles 1 - 30 of 1115

## Full-Text Articles in Discrete Mathematics and Combinatorics

Counting Power Domination Sets In Complete M-Ary Trees, Hays Whitlatch, Katharine Shultis, Olivia Ramirez, Michele Ortiz, Sviatlana Kniahnitskaya Jan 2023

#### Counting Power Domination Sets In Complete M-Ary Trees, Hays Whitlatch, Katharine Shultis, Olivia Ramirez, Michele Ortiz, Sviatlana Kniahnitskaya

##### Theory and Applications of Graphs

Motivated by the question of computing the probability of successful power domination by placing k monitors uniformly at random, in this paper we give a recursive formula to count the number of power domination sets of size k in a labeled complete m-ary tree. As a corollary we show that the desired probability can be computed in exponential with linear exponent time.

Hs-Integral And Eisenstein Integral Mixed Circulant Graphs, Monu Kadyan, Bikash Bhattacharjya Jan 2023

#### Hs-Integral And Eisenstein Integral Mixed Circulant Graphs, Monu Kadyan, Bikash Bhattacharjya

##### Theory and Applications of Graphs

A mixed graph is called \emph{second kind hermitian integral} (\emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of the second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set $S$ for which a mixed circulant graph $\text{Circ}(\mathbb{Z}_n, S)$ is HS-integral. We also show that a mixed circulant graph is Eisenstein integral if and only if it is HS-integral. Further, we express the eigenvalues and the HS-eigenvalues of unitary oriented circulant graphs in terms of generalized M$\ddot{\text{o}}$bius function.

Algorithmic Methods For Covering Arrays Of Higher Index, Ryan Dougherty, Kristoffer Kleine, Michael Wagner, Charles J. Colbourn, Dimitris E. Simos Dec 2022

#### Algorithmic Methods For Covering Arrays Of Higher Index, Ryan Dougherty, Kristoffer Kleine, Michael Wagner, Charles J. Colbourn, Dimitris E. Simos

##### West Point Research Papers

Covering arrays are combinatorial objects used in testing large-scale systems to increase confidence in their correctness. To do so, each interaction of at most a specified number t of factors is represented in at least one test; that is, the covering array has strength t and index 1. For certain systems, the outcome of running a test may be altered by variability of the interaction effect or by measurement error of the test result. To improve the efficacy of testing, one can ensure that each interaction of t or fewer factors is represented in at least λ tests. When λ …

Meertens Number And Its Variations, Chai Wah Wu Dec 2022

#### Meertens Number And Its Variations, Chai Wah Wu

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

In 1998, Bird introduced Meertens numbers as numbers that are invariant under a map similar to the Gödel encoding. In base 10, the only known Meertens number is 81312000. We look at some properties of Meertens numbers and consider variations of this concept. In particular, we consider variations of Meertens numbers where there is a finite time algorithm to decide whether such numbers exist, exhibit infinite families of these variations and provide bounds on parameters needed for their existence.

(R1979) Permanent Of Toeplitz-Hessenberg Matrices With Generalized Fibonacci And Lucas Entries, Hacène Belbachir, Amine Belkhir, Ihab-Eddine Djellas Dec 2022

#### (R1979) Permanent Of Toeplitz-Hessenberg Matrices With Generalized Fibonacci And Lucas Entries, Hacène Belbachir, Amine Belkhir, Ihab-Eddine Djellas

##### Applications and Applied Mathematics: An International Journal (AAM)

In the present paper, we evaluate the permanent and determinant of some Toeplitz-Hessenberg matrices with generalized Fibonacci and generalized Lucas numbers as entries.We develop identities involving sums of products of generalized Fibonacci numbers and generalized Lucas numbers with multinomial coefficients using the matrix structure, and then we present an application of the determinant of such matrices.

Nov 2022

#### On Rainbow Cycles And Proper Edge Colorings Of Generalized Polygons, Matt Noble

##### Theory and Applications of Graphs

An edge coloring of a simple graph G is said to be proper rainbow-cycle-forbidding (PRCF, for short) if no two incident edges receive the same color and for any cycle in G, at least two edges of that cycle receive the same color. A graph G is defined to be PRCF-good if it admits a PRCF edge coloring, and G is deemed PRCF-bad otherwise. In recent work, Hoffman, et al. study PRCF edge colorings and find many examples of PRCF-bad graphs having girth less than or equal to 4. They then ask whether such graphs exist having girth greater than …

On The Uniqueness Of Continuation Of A Partially Defined Metric, Evgeniy Petrov Nov 2022

#### On The Uniqueness Of Continuation Of A Partially Defined Metric, Evgeniy Petrov

##### Theory and Applications of Graphs

The problem of continuation of a partially defined metric can be efficiently studied using graph theory. Let $G=G(V,E)$ be an undirected graph with the set of vertices $V$ and the set of edges $E$. A necessary and sufficient condition under which the weight $w\colon E\to\mathbb R^+$ on the graph $G$ has a unique continuation to a metric $d\colon V\times V\to\mathbb R^+$ is found.

Extension Of Fundamental Transversals And Euler’S Polyhedron Theorem, Joy Marie D'Andrea Nov 2022

#### Extension Of Fundamental Transversals And Euler’S Polyhedron Theorem, Joy Marie D'Andrea

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

No abstract provided.

Modularity And Boolean Network Decomposition, Matthew Wheeler Nov 2022

#### Modularity And Boolean Network Decomposition, Matthew Wheeler

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

No abstract provided.

Path-Stick Solitaire On Graphs, Jan-Hendrik De Wiljes, Martin Kreh Oct 2022

#### Path-Stick Solitaire On Graphs, Jan-Hendrik De Wiljes, Martin Kreh

##### Theory and Applications of Graphs

In 2011, Beeler and Hoilman generalised the game of peg solitaire to arbitrary connected graphs. Since then, peg solitaire and related games have been considered on many graph classes. In this paper, we introduce a variant of the game peg solitaire, called path-stick solitaire, which is played with sticks in edges instead of pegs in vertices. We prove several analogues to peg solitaire results for that game, mainly regarding different graph classes. Furthermore, we characterise, with very few exceptions, path-stick-solvable joins of graphs and provide some possible future research questions.

Combinatorial Algorithms For Graph Discovery And Experimental Design, Raghavendra K. Addanki Oct 2022

#### Combinatorial Algorithms For Graph Discovery And Experimental Design, Raghavendra K. Addanki

##### Doctoral Dissertations

In this thesis, we study the design and analysis of algorithms for discovering the structure and properties of an unknown graph, with applications in two different domains: causal inference and sublinear graph algorithms. In both these domains, graph discovery is possible using restricted forms of experiments, and our objective is to design low-cost experiments.

First, we describe efficient experimental approaches to the causal discovery problem, which in its simplest form, asks us to identify the causal relations (edges of the unknown graph) between variables (vertices of the unknown graph) of a given system. For causal discovery, we study algorithms …

Alpha Labeling Of Amalgamated Cycles, Christian Barrientos Oct 2022

#### Alpha Labeling Of Amalgamated Cycles, Christian Barrientos

##### Theory and Applications of Graphs

A graceful labeling of a bipartite graph is an \a-labeling if it has the property that the labels assigned to the vertices of one stable set of the graph are smaller than the labels assigned to the vertices of the other stable set. A concatenation of cycles is a connected graph formed by a collection of cycles, where each cycle shares at most either two vertices or two edges with other cycles in the collection. In this work we investigate the existence of \a-labelings for this kind of graphs, exploring the concepts of vertex amalgamation to produce a family of …

(Si10-054) Nonsplit Edge Geodetic Domination Number Of A Graph, P. Arul Paul Sudhahar, J. Jeba Lisa Oct 2022

#### (Si10-054) Nonsplit Edge Geodetic Domination Number Of A Graph, P. Arul Paul Sudhahar, J. Jeba Lisa

##### Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we have defined an inventive parameter called the nonsplit edge geodetic domination number of a graph, and some of its general properties are studied. The nonsplit edge geodetic domination number of some standard graph is obtained. In this work, we also determine the realization results of the nonsplit edge geodetic domination number and the edge geodetic number of a graph.

(Si10-089) Integer Cordial Labeling Of Alternate Snake Graph And Irregular Snake Graph, Pratik Shah, Dharamvirsinh Parmar Oct 2022

#### (Si10-089) Integer Cordial Labeling Of Alternate Snake Graph And Irregular Snake Graph, Pratik Shah, Dharamvirsinh Parmar

##### Applications and Applied Mathematics: An International Journal (AAM)

If a graph G admits integer cordial labeling, it is called an integer cordial graph. In this paper we prove that Alternate m-triangular Snake graph, Quadrilateral Snake graph, Alternate m- quadrilateral Snake graph, Pentagonal Snake graph, Alternate m-pentagonal Snake graph, Irregular triangular Snake graph, Irregular quadrilateral Snake graph, and Irregular pentagonal Snake graphs are integer cordial graphs.

Conflict Dynamics In Scale-Free Networks With Degree Correlations And Hierarchical Structure, Eduardo Jacobo-Villegas, Bibiana Obregón-Quintana, Lev Guzmán-Vargas, Larry S. Liebovitch Oct 2022

#### Conflict Dynamics In Scale-Free Networks With Degree Correlations And Hierarchical Structure, Eduardo Jacobo-Villegas, Bibiana Obregón-Quintana, Lev Guzmán-Vargas, Larry S. Liebovitch

##### Publications and Research

We present a study of the dynamic interactions between actors located on complex networks with scale-free and hierarchical scale-free topologies with assortative mixing, that is, correlations between the degree distributions of the actors. The actor’s state evolves according to a model that considers its previous state, the inertia to change, and the influence of its neighborhood. We show that the time evolution of the system depends on the percentage of cooperative or competitive
interactions. For scale-free networks, we find that the dispersion between actors is higher when all interactions are either cooperative or competitive, while a balanced presence of interactions …

(Si10-130) On Regular Inverse Eccentric Fuzzy Graphs, N. Meenal, J. Jeromi Jovita Oct 2022

#### (Si10-130) On Regular Inverse Eccentric Fuzzy Graphs, N. Meenal, J. Jeromi Jovita

##### Applications and Applied Mathematics: An International Journal (AAM)

Two new concepts of regular inverse eccentric fuzzy graphs and totally regular inverse eccentric fuzzy graphs are established in this article. By illustrations, these two graphs are compared and the results are derived. Equivalent condition for the existence of these two graphs are found. The exact values of Order and Size for some standard inverse eccentric graphs are also derived.

Asymptotic Classes, Pseudofinite Cardinality And Dimension, Alexander Van Abel Sep 2022

#### Asymptotic Classes, Pseudofinite Cardinality And Dimension, Alexander Van Abel

##### Dissertations, Theses, and Capstone Projects

We explore the consequences of various model-theoretic tameness conditions upon the behavior of pseudofinite cardinality and dimension. We show that for pseudofinite theories which are either Morley Rank 1 or uncountably categorical, pseudofinite cardinality in ultraproducts satisfying such theories is highly well-behaved. On the other hand, it has been shown that pseudofinite dimension is not necessarily well-behaved in all ultraproducts of theories which are simple or supersimple; we extend such an observation by constructing simple and supersimple theories in which pseudofinite dimension is necessarily ill-behaved in all such ultraproducts. Additionally, we have novel results connecting various forms of asymptotic classes …

The On-Line Width Of Various Classes Of Posets., Israel R. Curbelo Aug 2022

#### The On-Line Width Of Various Classes Of Posets., Israel R. Curbelo

##### Electronic Theses and Dissertations

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\'edi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In the survey paper by Bosek et al., variants of the problem were studied where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size $d$. We prove …

Aug 2022

#### Minimal Differential Graded Algebra Resolutions Related To Certain Stanley-Reisner Rings, Todd Anthony Morra

##### All Dissertations

We investigate algebra structures on resolutions of a special class of Cohen-Macaulay simplicial complexes. Given a simplicial complex, we define a pure simplicial complex called the purification. These complexes arise as a generalization of certain independence complexes and the resultant Stanley-Reisner rings have numerous desirable properties, e.g., they are Cohen-Macaulay. By realizing the purification in the context of work of D'alì, et al., we obtain a multi-graded, minimal free resolution of the Alexander dual ideal of the Stanley-Reisner ideal. We augment this in a standard way to obtain a resolution of the quotient ring, which is likewise minimal and multi-graded. …

Verifying Sudoku Puzzles, Chelsea Schweer Aug 2022

#### Verifying Sudoku Puzzles, Chelsea Schweer

##### Electronic Theses, Projects, and Dissertations

Sudoku puzzles, created by Meki Kaji around 1983, consist of a square 9 by 9 grid made up of 9 rows, 9 columns, and nine 3 by 3 square sub-grids called blocks. The goal of the puzzle is to be able to place the numbers 1 through 9 in every row, column, and block where no number is repeated in each row, column, and block. Imagine being given a completed Sudoku puzzle and having to check that it was solved correctly. You could just check all the rows columns and blocks (27 items), but is there a smaller number of …

Characteristic Sets Of Matroids, Dony Varghese Aug 2022

#### Characteristic Sets Of Matroids, Dony Varghese

##### Doctoral Dissertations

Matroids are combinatorial structures that generalize the properties of linear independence. But not all matroids have linear representations. Furthermore, the existence of linear representations depends on the characteristic of the fields, and the linear characteristic set is the set of characteristics of fields over which a matroid has a linear representation. The algebraic independence in a field extension also defines a matroid, and also depends on the characteristic of the fields. The algebraic characteristic set is defined in the similar way as the linear characteristic set.

The linear representations and characteristic sets are well studied. But the algebraic representations and …

Radio Number Of Hamming Graphs Of Diameter 3, Jason Devito, Amanda Niedzialomski, Jennifer Warren Jul 2022

#### Radio Number Of Hamming Graphs Of Diameter 3, Jason Devito, Amanda Niedzialomski, Jennifer Warren

##### Theory and Applications of Graphs

For $G$ a simple, connected graph, a vertex labeling $f:V(G)\to \Z_+$ is called a \emph{radio labeling of $G$} if it satisfies $|f(u)-f(v)|\geq\diam(G)+1-d(u,v)$ for all distinct vertices $u,v\in V(G)$. The \emph{radio number of $G$} is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto $\{1,2,\dots,|V(G)|\}$ exists, $G$ is called a \emph{radio graceful} graph. We determine the radio number of all diameter 3 Hamming graphs and show that an infinite subset of them is radio graceful.

On The Integer-Antimagic Spectra Of Non-Hamiltonian Graphs, Wai Chee Shiu, Richard M. Low Jul 2022

#### On The Integer-Antimagic Spectra Of Non-Hamiltonian Graphs, Wai Chee Shiu, Richard M. Low

##### Theory and Applications of Graphs

Let $A$ be a nontrivial abelian group. A connected simple graph $G = (V, E)$ is $A$-\textbf{antimagic} if there exists an edge labeling $f: E(G) \to A \setminus \{0\}$ such that the induced vertex labeling $f^+: V(G) \to A$, defined by $f^+(v) = \Sigma$ $\{f(u,v): (u, v) \in E(G) \}$, is a one-to-one map. In this paper, we analyze the group-antimagic property for Cartesian products, hexagonal nets and theta graphs.

Restrained Reinforcement Number In Graphs, Kazhal Haghparast, Jafar Amjadi, Mustapha Chellali, Seyed Mahmoud Sheikholeslami Jul 2022

#### Restrained Reinforcement Number In Graphs, Kazhal Haghparast, Jafar Amjadi, Mustapha Chellali, Seyed Mahmoud Sheikholeslami

##### Theory and Applications of Graphs

There are three types of affine regular polygons in AG(2, q): ellipse, hyperbola and parabola. The first two cases have been investigated in previous papers. In this note, a particular class of geometric one-factorizations of the complete graph Kn arising from parabolas is constructed and described in full detail. With the support of computer aided investigation, it is also conjectured that up to isomorphisms this is the only one-factorization where each one-factor is either represented by a line or a parabola.