Staphylococcus pseudintermedius: Population Genetics and Antimicrobial Resistance

Staphylococcus pseudintermedius is a Gram-positive coagulase-negative coccus. It is a normal inhabitant of the skin of dogs. However, clinical disease can be observed in animals that are immunossuppressed or if the skin barrier is altered. This bacterium is recognized as the main cause of canine pyoderma and has also been associated with other conditions such as infection of the urinary tract, the ears, and surgical wounds. Methicillin resistance and resistance to other antimicrobials regularly used by veterinarians is common among S. pseudintermedius which can complicate treatment. The first report of mecA, gene responsible for methicillin resistance, in S. pseudintermedius is from 1999. Since then, resistance to methicillin and to other antimicrobials has become increasingly more common, making this bacterium a possible reservoir for antimicrobial resistance genes. The reason for the increase in the presence of antimicrobial resistance among S. pseudintermedius is still not well understood. This research focuses on characterization of S. pseudintermedius isolates from the United States in order to determine their genetic diversity, antimicrobial susceptibility profiles, and possible relationships among the two. A description of the genetically related populations that are present in the country may help in the understanding of the mechanisms of expansion of this microorganism. Also, the availability of more current information on the susceptibility to antimicrobials should help in the reestablishment of the consequences of misusage of antimicrobials and highlights the need for the development of novel treatment alternatives

Generalized Paley graphs equienergetic with their complements

We consider generalized Paley graphs $\Gamma(k,q)$, generalized Paley sum graphs $\Gamma^+(k,q)$, and their corresponding complements $\bar \Gamma(k,q)$ and $\bar \Gamma^+(k,q)$, for $k=3,4$. Denote by $\Gamma = \Gamma^*(k,q)$ either $\Gamma(k,q)$ or $\Gamma^+(k,q)$. We compute the spectra of $\Gamma(3,q)$ and $\Gamma(4,q)$ and from them we obtain the spectra of $\Gamma^+(3,q)$ and $\Gamma^+(4,q)$ also. Then we show that, in the non-semiprimitive case, the spectrum of $\Gamma(3,p^{3\ell})$ and $\Gamma(4,p^{4\ell})$ with $p$ prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs $\Gamma(3,p)$ and $\Gamma(4,p)$ for any $\ell \in \mathbb{N}$, respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of $\Gamma^*(k,q)$ such that $\Gamma^*(k,q)$ and $\bar \Gamma^*(k,q)$ are equienergetic for $k=3,4$. In a previous work we have classified all bipartite regular graphs $\Gamma_{bip}$ and all strongly regular graphs $\Gamma_{srg}$ which are complementary equienergetic, i.e.\@ $\{\Gamma_{bip}, \bar{\Gamma}_{bip}\}$ and $\{\Gamma_{srg}, \bar{\Gamma}_{srg}\}$ are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs $\{\Gamma, \bar \Gamma\}$ which are neither bipartite nor strongly regular.Comment: 22 page

The spectra of generalized Paley graphs of qℓ+1q^\ell+1 powers and applications

We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set $\mathbb{F}_{q^m}$ and connection set the nonzero $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}$, as well as their complements. We explicitly compute the spectrum of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular (of Latin square type in half of the cases). By using the spectral information we compute several invariants of these graphs. We exhibit infinite families of pairs of equienergetic non-isospectral graphs. As applications, on the one hand we solve Waring's problem over $\mathbb{F}_q^m$ for the exponents $q^\ell+1$, for each $q$ and for infinite values of $\ell$ and $m$. We obtain that the Waring's number $g(q^\ell+1,q^m)=1$ or $2$, depending on $m$ and $\ell$, thus tackling some open cases. On the other hand, we construct infinite towers of Ramanujan graphs in all characteristics.Comment: 27 pages, 3 tables. A little modification of the title. Corollary 4.8 removed. Added Section 6 on "Energy". Minor typos corrected. Ihara zeta functions at the end correcte

The Waring's problem over finite fields through generalized Paley graphs

We show that the Waring's number over a finite field $\mathbb{F}_q$, denoted $g(k,q)$, when exists, coincides with the diameter of the generalized Paley graph $\Gamma(k,q)=Cay(\mathbb{F}_{q},R_k)$ with $R_k=\{x^k : x\in \mathbb{F}_q^*\}$. We find infinite new families of exact values of $g(k,q)$ from a characterization of graphs $\Gamma(k,q)$ which are also Hamming graphs previously proved by Lim and Praeger in 2009. Then, we show that every positive integer is the Waring number for some pair $(k,q)$ with $q$ not a prime. Finally, we find a lower bound for $g(k,p)$ with $p$ prime by using that $\Gamma(k,p)$ is a circulant graph in this case.Comment: 16 pages. Small additions and typos corrected. We added. at the end, a small subsection comparing our lower bound for Waring numbers with the other 3 lower bounds know

The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs

We use known characterizations of generalized Paley graphs which are cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields and to the computation of Gaussian periods.Comment: 20 pages, 2 tables. Added information on Cartesian products of graphs. A general reviwe was made, with small additions improving readability. Some typos corrected. arXiv admin note: text overlap with arXiv:1908.0809

Spectral properties of generalized Paley graphs

We study the spectrum of generalized Paley graphs $\Gamma(k,q)=Cay(\mathbb{F}_q,R_k)$, undirected or not, with $R_k=\{x^k:x\in \mathbb{F}_q^*\}$ where $q=p^m$ with $p$ prime and $k\mid q-1$. We first show that the eigenvalues of $\Gamma(k,q)$ are given by the Gaussian periods $\eta_{i}^{(k,q)}$ with $0\le i\le k-1$. Then, we explicitly compute the spectrum of $\Gamma(k,q)$ with $1\le k \le 4$ and of $\Gamma(5,q)$ for $p\equiv 1\pmod 5$ and $5\mid m$. Also, we characterize those GP-graphs having integral spectrum, showing that $\Gamma(k,q)$ is integral if and only if $p$ divides $(q-1)/(p-1)$. Next, we focus on the family of semiprimitive GP-graphs. We show that they are integral strongly regular graphs (of pseudo-Latin square type). Finally, we characterize all integral Ramanujan graphs $\Gamma(k,q)$ with $1\le k \le 4$ or where $(k,q)$ is a semiprimitive pair.Comment: 29 pages, 2 tables. The old manuscript arXiv:1908.08097 has grown and we divided it into two different manuscripts with different names, this is the first half, and the other one is in progres

Weight distribution of cyclic codes defined by quadratic forms and related curves

We consider cyclic codes CL associated to quadratic trace forms inm variables (Formula Presented) determined by a family L of q-linearized polynomials R over Fqm, and three related codes CL,0, CL,1, and CL,2. We describe the spectra for all these codes when L is an even rank family, in terms of the distribution of ranks of the forms QR in the family L, and we also computethe complete weight enumerator for CL. In particular, considering the family L = ‹xql›, with l fixed in N, we give the weight distribution of four parametrized families of cyclic codes Cl, Cl,0,Cl,1, and Cl,2 over Fq with zeros(Formula Presented) respectively,where q = ps with p prime, α is a generator of F*qm, and m/(m,l)is even. Finally, we give simple necessary and sufficient conditions for Artin–Schreier curves yp−y = xR(x)+βx, p prime, associated to polynomials R ∈ L to be optimal. We then obtain several maximal and minimal such curves inthe case (Formula Presented).Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin