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.

The Influence Of Canalization On The Robustness Of Finite Dynamical Systems, 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.

Reducing The Maximum Degree Of A Graph By Deleting Vertices: The Extremal Cases, 2018 University of Malta

#### Reducing The Maximum Degree Of A Graph By Deleting Vertices: The Extremal Cases, Peter Borg, Kurt Fenech

*Theory and Applications of Graphs*

Let $\lambda(G)$ denote the smallest number of vertices that can be removed from a non-empty graph $G$ so that the resulting graph has a smaller maximum degree. In a recent paper, we proved that if $n$ is the number of vertices of $G$, $k$ is the maximum degree of $G$, and $t$ is the number of vertices of degree $k$, then $\lambda (G) \leq \frac{n+(k-1)t}{2k}$. We also showed that $\lambda (G) \leq \frac{n}{k+1}$ if $G$ is a tree. In this paper, we provide a new proof of the first bound and use ...

Using Magic In Computing Education And Outreach, 2018 Loyola University Chicago

#### Using Magic In Computing Education And Outreach, Ronald I. Greenberg, Dale F. Reed

*Computer Science: Faculty Publications and Other Works*

This special session explores the use of magic tricks based on computer science ideas; magic tricks help grab students' attention and can motivate them to invest more deeply in underlying CS concepts. Error detection ideas long used by computer scientists provide a particularly rich basis for working such "magic'', with a CS Unplugged parity check activity being a notable example. Prior work has shown that one can perform much more sophisticated tricks than the relatively well-known CS Unplugged activity, and these tricks can motivate analyses across a wide variety of computer science concepts and are relevant to learning objectives across ...

Bounds On The Sum Of Minimum Semidefinite Rank Of A Graph And Its Complement, 2018 Central Michigan University

#### Bounds On The Sum Of Minimum Semidefinite Rank Of A Graph And Its Complement, Sivaram Narayan, Yousra Sharawi

*Electronic Journal of Linear Algebra*

The minimum semi-definite rank (msr) of a graph is the minimum rank among all positive semi-definite matrices associated to the graph. The graph complement conjecture gives an upper bound for the sum of the msr of a graph and the msr of its complement. It is shown that when the msr of a graph is equal to its independence number, the graph complement conjecture holds with a better upper bound. Several sufficient conditions are provided for the msr of different classes of graphs to equal to its independence number.

Microstructure Design Using Graphs, 2018 Iowa State University

#### Microstructure Design Using Graphs, Pengfei Du, Adrian Zebrowski, Jaroslaw Zola, Baskar Ganapathysubramanian, Olga Wodo

*Mechanical Engineering Publications*

Thin films with tailored microstructures are an emerging class of materials with applications such as battery electrodes, organic electronics, and biosensors. Such thin film devices typically exhibit a multi-phase microstructure that is confined, and show large anisotropy. Current approaches to microstructure design focus on optimizing bulk properties, by tuning features that are statistically averaged over a representative volume. Here, we report a tool for morphogenesis posed as a graph-based optimization problem that evolves microstructures recognizing confinement and anisotropy constraints. We illustrate the approach by designing optimized morphologies for photovoltaic applications, and evolve an initial morphology into an optimized morphology exhibiting ...

Stranded Cellular Automaton And Weaving Products, 2018 Rose-Hulman Institute of Technology

#### Stranded Cellular Automaton And Weaving Products, Hao Yang

*Mathematical Sciences Technical Reports (MSTR)*

In order to analyze weaving products mathematically and find out valid weaving products, it is natural to relate them to Cellular Automaton. They are both generated based on specific rules and some initial conditions. Holden and Holden have created a Stranded Cellular Automaton that can represent common weaving and braiding products. Based on their previous findings, we were able to construct a Java program and analyze various aspects of the automaton they created. This paper will discuss the complexity of the Stranded Cellular Automaton, how to determine whether a weaving product holds together or not based on the automaton and ...

Ideals, Big Varieties, And Dynamic Networks, 2018 Portland State University

#### Ideals, Big Varieties, And Dynamic Networks, Ian H. Dinwoodie

*Mathematics and Statistics Faculty Publications and Presentations*

The advantage of using algebraic geometry over enumeration for describing sets related to attractors in large dynamic networks from biology is advocated. Examples illustrate the gains.

Tutte-Equivalent Matroids, 2018 California State University - San Bernardino

#### Tutte-Equivalent Matroids, Maria Margarita Rocha

*Electronic Theses, Projects, and Dissertations*

We begin by introducing matroids in the context of finite collections of vectors from a vector space over a specified field, where the notion of independence is linear independence. Then we will introduce the concept of a matroid invariant. Specifically, we will look at the Tutte polynomial, which is a well-defined two-variable invariant that can be used to determine differences and similarities between a collection of given matroids. The Tutte polynomial can tell us certain properties of a given matroid (such as the number of bases, independent sets, etc.) without the need to manually solve for them. Although the Tutte ...

Identifying Combinatorially Symmetric Hidden Markov Models, 2018 Aberystwyth University

#### Identifying Combinatorially Symmetric Hidden Markov Models, Daniel Burgarth

*Electronic Journal of Linear Algebra*

A sufficient criterion for the unique parameter identification of combinatorially symmetric Hidden Markov Models, based on the structure of their transition matrix, is provided. If the observed states of the chain form a zero forcing set of the graph of the Markov model, then it is uniquely identifiable and an explicit reconstruction method is given.

The Largest Eigenvalue And Some Hamiltonian Properties Of Graphs, 2018 University of South Carolina Aiken

#### The Largest Eigenvalue And Some Hamiltonian Properties Of Graphs, Rao Li

*Electronic Journal of Linear Algebra*

In this note, sufficient conditions, based on the largest eigenvalue, are presented for some Hamiltonian properties of graphs.

Proof Of A Conjecture Of Graham And Lovasz Concerning Unimodality Of Coefficients Of The Distance Characteristic Polynomial Of A Tree, 2018 Iowa State University

#### Proof Of A Conjecture Of Graham And Lovasz Concerning Unimodality Of Coefficients Of The Distance Characteristic Polynomial Of A Tree, Ghodratollah Aalipour, Aida Abiad, Zhanar Berikkyzy, Leslie Hogben, Franklin H.J. Kenter, Jephian C.-H. Lin, Michael Tait

*Electronic Journal of Linear Algebra*

The conjecture of Graham and Lov ́asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established.

Extremal Octagonal Chains With Respect To The Spectral Radius, 2018 Anhui University of Science and Technology

#### Extremal Octagonal Chains With Respect To The Spectral Radius, Xianya Geng, Shuchao Li, Wei Wei

*Electronic Journal of Linear Algebra*

Octagonal systems are tree-like graphs comprised of octagons that represent a class of polycyclic conjugated hydrocarbons. In this paper, a roll-attaching operation for the calculation of the characteristic polynomials of octagonal chain graphs is proposed. Based on these characteristic polynomials, the extremal octagonal chains with n octagons having the maximum and minimum spectral radii are identified.

Topological Recursion And Random Finite Noncommutative Geometries, 2018 The University of Western Ontario

#### Topological Recursion And Random Finite Noncommutative Geometries, Shahab Azarfar

*Electronic Thesis and Dissertation Repository*

In this thesis, we investigate a model for quantum gravity on finite noncommutative spaces using the topological recursion method originated from random matrix theory. More precisely, we consider a particular type of finite noncommutative geometries, in the sense of Connes, called spectral triples of type ${(1,0)} \,$, introduced by Barrett. A random spectral triple of type ${(1,0)}$ has a fixed fermion space, and the moduli space of its Dirac operator ${D=\{ H , \cdot \} \, ,}$ ${H \in {\mathcal{H}_N}}$, encoding all the possible geometries over the fermion space, is the space of Hermitian matrices ${\mathcal{H}_N}$. A distribution of ...

Minimal Graphs With A Specified Code Map Image, 2018 University of Texas of the Permian Basin

#### Minimal Graphs With A Specified Code Map Image, Paul Feit

*Theory and Applications of Graphs*

Let $G$ be a graph and $e_1,\cdots ,e_n$ be $n$ distinct vertices. Let $\rho$ be the metric on $G$. The code map on vertices, corresponding to this list, is $c(x)=(\rho (x,e_1),\cdots ,\rho (x,e_n))$. This paper introduces a variation: begin with $V\subseteq\bbz^n$ for some $n$, and consider assignments of edges $E$ such that the identity function on $V$ is a code map for $G=(V,E)$. Refer to such a set $E$ as a {\em code-match.}

An earlier paper classified subsets of $V$ for which at least one code-match exists. We prove ...

Inertia Sets Allowed By Matrix Patterns, 2018 Saint Olaf College

#### Inertia Sets Allowed By Matrix Patterns, Adam H. Berliner, Dale D. Olesky, Pauline Van Den Driessche

*Electronic Journal of Linear Algebra*

Motivated by the possible onset of instability in dynamical systems associated with a zero eigenvalue, sets of inertias $\sn_n$ and $\SN{n}$ for sign and zero-nonzero patterns, respectively, are introduced. For an $n\times n$ sign pattern $\mc{A}$ that allows inertia $(0,n-1,1)$, a sufficient condition is given for $\mc{A}$ and every superpattern of $\mc{A}$ to allow $\sn_n$, and a family of such irreducible sign patterns for all $n\geq 3$ is specified. All zero-nonzero patterns (up to equivalence) that allow $\SN{3}$ and $\SN{4}$ are determined, and are described by their associated digraphs.

Fractional Difference Operators And Related Boundary Value Problems, 2018 University of Nebraska-Lincoln

#### Fractional Difference Operators And Related Boundary Value Problems, Scott C. Gensler

*Dissertations, Theses, and Student Research Papers in Mathematics*

In this dissertation we develop a fractional difference calculus for functions on a discrete domain. We start by showing that the Taylor monomials, which play a role analagous to that of the power functions in ordinary differential calculus, can be expressed in terms of a family of polynomials which I will refer to as the Pochhammer polynomials. These important functions, the Taylor monomials, were previously described by other scholars primarily in terms of the gamma function. With only this description it is challenging to understand their properties. Describing the Taylor monomials in terms of the Pochhammer polynomials has made it ...

The Expected Number Of Patterns In A Random Generated Permutation On [N] = {1,2,...,N}, 2018 East Tennessee State University

#### The Expected Number Of Patterns In A Random Generated Permutation On [N] = {1,2,...,N}, Evelyn Fokuoh

*Electronic Theses and Dissertations*

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 ...