We show that the cop number of every generalized Petersen graph is at most 4. The strategy is to play a modified game of cops and robbers on an infinite cyclic covering space where the objective is to capture the robber or force the robber towards an end of the infinite graph. We prove that finite isometric subtrees are 1-guardable and apply this to determine the exact cop number of some families of generalized Petersen graphs. We also extend these ideas to prove that the cop number of any connected I-graph is at most 5. 1 arXiv:1509.04696v1 [math.CO]

Background Small molecule metabolites produced by the microbiome are known to be neuroactive and are capable of directly impacting the brain and central nervous system, yet there is little data on the contribution of these metabolites to the earliest stages of neural development and neural gene expression. Here, we explore the impact of deriving zebrafish embryos in the absence of microbes on early neural development as well as investigate whether any potential changes can be rescued with treatment of metabolites derived from the zebrafish gut microbiota. Results Overall, we did not observe any gross morphological changes between treatments but did observe a significant decrease in neural gene expression in embryos raised germ-free, which was rescued with the addition of zebrafish metabolites. Specifically, we identified 354 genes significantly downregulated in germ-free embryos compared to conventionally raised embryos via RNA-Seq analysis. Of these, 42 were rescued with a single treatment of zebrafish gut-derived metabolites to germ-free embryos. Gene ontology analysis revealed that these genes are involved in prominent neurodevelopmental pathways including transcriptional regulation and Wnt signaling. Consistent with the ontology analysis, we found alterations in the development of Wnt dependent events which was rescued in the germ-free embryos treated with metabolites. Conclusions These findings demonstrate that gut-derived metabolites are in part responsible for regulating critical signaling pathways in the brain, especially during neural development.

In this Macaulay2 package we implement a type of object called a LinearCode. We implement functions that compute basic parameters and objects associated with a linear code, such as generator and parity check matrices, the dual code, length, dimension, and minimum distance, among others. We implement a type of object called an EvaluationCode, a construction which allows users to study linear codes using tools of algebraic geometry and commutative algebra. We implement functions to generate important families of linear codes, such as Hamming codes, cyclic codes, Reed-Solomon codes, Reed-Muller codes, Cartesian codes, monomial-Cartesian codes, and toric codes. In addition, we implement functions for the syndrome decoding algorithm and locally recoverable code construction, which are important tools in applications of linear codes.1. INTRODUCTION. Coding theory has been extensively studied since 1948, when Claude Shannon [1948] proved in his seminal paper that linear codes can be used to reliably transmit information from a single source to a single receiver through a noisy channel. Since then, coding theory has found many important engineering applications. For example, coding theory has been used in designing reliable data storage systems, radio communication protocols, and in the emerging field of quantum computers. Coding theory has close ties with many areas in mathematics including linear algebra, commutative algebra, algebraic geometry, and combinatorics.In this note we introduce the new [Macaulay2] package called CodingTheory. The goal of this package is to provide a range of functions for constructing linear and evaluation codes over finite fields, and for computing some of their main properties. To this aim, we implement two types of objects, LinearCode and EvaluationCode. The package also includes implementations of functions for generating important families of linear codes like Hamming codes, cyclic codes, Reed-Solomon codes, Reed-Muller codes, Cartesian codes, monomial-Cartesian codes and toric codes. It also has functions for the syndrome decoding algorithm and locally recoverable codes.The organization of this note is as follows. In Section 2 we describe various ways to construct a linear code over a finite field using the CodingTheory package. In Section 3 we show how to compute the main parameters of a linear code: length, dimension, and minimum distance. We also illustrate how to

Let G be a graph G whose largest independent set has size m. A permutation π of {1, . . . , m} is an independent set permutation of G ifwhere a k (G) is the number of independent sets of size k in G. In 1987 Alavi, Malde, Schwenk and Erdős proved that every permutation of {1, . . . , m} is an independent set permutation of some graph. They raised the question of determining, for each m, the smallest number f (m) such that every permutation of {1, . . . , m} is an independent set permutation of some graph with at most f (m) vertices, and they gave an upper bound on f (m) of roughly m 2m . Here we settle the question, determining f (m) = m m .Alavi et al. also considered matching permutations, defined analogously to independent set permutations. They observed that not every permutation is a matching permutation of some graph, putting an upper bound of 2 m−1 on the number of matching permutations of {1, . . . , m}. Confirming their speculation that this upper bound is not tight, we improve it to O(2 m / √ m).We also consider an extension of independent set permutations to weak orders, and extend Alavi et al.'s main result to show that every weak order on {1, . . . , m} can be realized by the independent set sequence of some graph with largest independent set size m, and with at most m m+2 vertices.

Let G be a graph G whose largest independent set has size m. A permutation π of m {1, …, } is an independent