A Stronger Strong Schottky Lemma For Euclidean Buildings, 2023 The Graduate Center, City University of New York

#### A Stronger Strong Schottky Lemma For Euclidean Buildings, Michael E. Ferguson

*Dissertations, Theses, and Capstone Projects*

We provide a criterion for two hyperbolic isometries of a Euclidean building to generate a free group of rank two. In particular, we extend the application of a Strong Schottky Lemma to buildings given by Alperin, Farb and Noskov. We then use this extension to obtain an infinite family of matrices that generate a free group of rank two. In doing so, we also introduce an algorithm that terminates in finite time if the lemma is applicable for pairs of certain kinds of matrices acting on the Euclidean building for the special linear group over certain discretely valued fields.

Counting Power Domination Sets In Complete M-Ary Trees, 2023 Gonzaga University

#### 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, 2023 Indian Institute of Technology Guwahati

#### 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.

On The Uniqueness Of Continuation Of A Partially Defined Metric, 2023 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine

#### 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.

On Rainbow Cycles And Proper Edge Colorings Of Generalized Polygons, 2023 Middle Georgia State University

#### 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 …

Algorithmic Methods For Covering Arrays Of Higher Index, 2022 United States Military Academy

#### 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, 2022 International Business Machines

#### 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, 2022 RECITS Laboratory

#### (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.

Extension Of Fundamental Transversals And Euler’S Polyhedron Theorem, 2022 University of South Florida

#### 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, 2022 University of Florida

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

*Annual Symposium on Biomathematics and Ecology Education and Research*

No abstract provided.

Path-Stick Solitaire On Graphs, 2022 Freie Universität Berlin

#### 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, 2022 University of Massachusetts Amherst

#### 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, 2022 Valencia College

#### 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, 2022 Rani Anna Government College for Women

#### (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, 2022 C. U. Shah University

#### (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.

(Si10-130) On Regular Inverse Eccentric Fuzzy Graphs, 2022 J.J. College of Arts and Science (Autonomous)

#### (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.

Conflict Dynamics In Scale-Free Networks With Degree Correlations And Hierarchical Structure, 2022 Universidad Nacional Autonoma de Mexico

#### 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 …

Asymptotic Classes, Pseudofinite Cardinality And Dimension, 2022 The Graduate Center, City University of New York

#### 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., 2022 University of Louisville

#### 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 …

Characteristic Sets Of Matroids, 2022 University of Tennessee, Knoxville

#### 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 …