In the first part, we assign to each positive integer $n$ a digraph $\Gamma (n,5),$ whose set of vertices consists of elements of the ring $\mathbb {Z}_n=\{0,1,\cdots ,n-1\}$ with the addition and the multiplication operations modulo $n,$ and for which there is a directed edge from $a$ to $b$ if and only if $a^5\equiv b\pmod n$. Associated with $\Gamma (n,5)$ are two disjoint subdigraphs: $\Gamma _1(n,5)$ and $\Gamma _2(n,5)$ whose union is $\Gamma (n,5).$ The vertices of $\Gamma _1(n,5)$ are coprime to $n,$ and the vertices of $\Gamma _2(n,5)$ are not coprime to $n.$ In this part, we study the structure of $\Gamma (n,5)$ in detail. \endgraf In the second part, we investigate the zero-divisor graph $G(\mathbb {Z}_n)$ of the ring $\mathbb {Z}_n.$ Its vertex- and edge-connectivity are discussed.
In this paper we derive the Integration-by-Parts Formula using the generalized Riemann approach to stochastic integrals, which is called the Itô-Kurzweil-Henstock integral.
In recent papers, S. N. Begum and A. S. A. Noor have studied join partial semilattices (JP-semilattices) defined as meet semilattices with an additional partial operation (join) satisfying certain axioms. We show why their axiom system is too weak to be a satisfactory basis for the authors’ constructions and proofs, and suggest an additional axiom for these algebras. We also briefly compare axioms of JP-semilattices with those of nearlattices, another kind of meet semilattices with a partial join operation.
For a nonempty set S of vertices in a strong digraph D, the strong distance d(S) is the minimum size of a strong subdigraph of D containing the vertices of S. If S contains k vertices, then d(S) is referred to as the k-strong distance of S. For an integer k ≥ 2 and a vertex v of a strong digraph D, the k-strong eccentricity sek(v) of v is the maximum k-strong distance d(S) among all sets S of k vertices in D containing v. The minimum k-strong eccentricity among the vertices of D is its k-strong radius sradk D and the maximum k-strong eccentricity is its k-strong diameter sdiamk D. The k-strong center (k-strong periphery) of D is the subdigraph of D induced by those vertices of k-strong eccentricity sradk(D) (sdiamk(D)). It is shown that, for each integer k ≥ 2, every oriented graph is the k-strong center of some strong oriented graph. A strong oriented graph D is called strongly k-self-centered if D is its own k-strong center. For every integer r ≥ 6, there exist infinitely many strongly 3-self-centered oriented graphs of 3-strong radius r. The problem of determining those oriented graphs that are k-strong peripheries of strong oriented graphs is studied.
It is shown that a Korovkin type theorem for a sequence of linear positive operators acting in weighted space $L_{p,w}({\mathrm loc})$ does not hold in all this space and is satisfied only on some subspace.
In the paper we deal with the Kurzweil-Stieltjes integration of functions having values in a Banach space X. We extend results obtained by Štefan Schwabik and complete the theory so that it will be well applicable to prove results on the continuous dependence of solutions to generalized linear differential equations in a Banach space. By Schwabik, the integral ∫ b a d[F]g exists if F : [a, b] → L(X) has a bounded semi-variation on [a, b] and g : [a, b] → X is regulated on [a, b]. We prove that this integral has sense also if F is regulated on [a, b] and g has a bounded semi-variation on [a, b]. Furthermore, the integration by parts theorem is presented under the assumptions not covered by Schwabik (2001) and Naralenkov (2004), and the substitution formula is proved.
Let G be an undirected connected graph with n, n\geqslant 3, vertices and m edges with Laplacian eigenvalues µ^{1}\geqslant µ_{2}\geq ...\geq µ_{n-1> µ_{n}}=0. Denote by {\mu _I} = {\mu _{{r_1}}} + {\mu _{{r_2}}} + \ldots + {\mu _{{r_k}}}, 1\leq k\leq n-2, 1\leq r_{1}< r_{2}< ...< r_{k} \leq n-1, the sum of k arbitrary Laplacian eigenvalues, with {\mu _{{I_1}}} = {\mu _1} + {\mu _2} + \ldots + {\mu _k} and {\mu _{{I_n}}} = {\mu _{n - k}} + \ldots + {\mu _{n - 1}}. Lower bounds of graph invariants {\mu _{{I_1}}} - {\mu _{{I_n}}} and {\mu _{{I_1}}}/{\mu _{{I_n}}} are obtained. Some known inequalities follow as a special case., Igor Ž. Milovanović, Emina I. Milovanović, Edin Glogić., and Obsahuje seznam literatury
In this paper the equivalence $\tilde{\mathcal Q}^U$ on a semigroup $S$ in terms of a set $U$ of idempotents in $S$ is defined. A semigroup $S$ is called a $\mathcal U$-liberal semigroup with $U$ as the set of projections and denoted by $S(U)$ if every $\tilde{\mathcal Q}^U$-class in it contains an element in $U$. A class of $\mathcal U$-liberal semigroups is characterized and some special cases are considered.