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vol. 36 num. 2 lang. en<![CDATA[SciELO Logo]]>http://www.scielo.cl/img/en/fbpelogp.gif
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<![CDATA[A study on prime arithmetic integer additive set-indexers of graphs]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200195&lng=en&nrm=iso&tlng=en
Abstract: Let N0 be the set of all non-negative integers and P(N0) be its power set. An integer additive set-indexer (IASI) is defined as an injective function such that the induced function defined by is also injective, where N0 is the set of all non-negative integers. A graph G which admits an IASI is called an IASI graph. An IASI of a graph G is said to be an arithmetic IASI if the elements of the set-labels of all vertices and edges of G are in arithmetic progressions. In this paper, we discuss about a particular type of arithmetic IASI called prime arithmetic IASI.<![CDATA[The total detour monophonic number of a graph]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200209&lng=en&nrm=iso&tlng=en
Abstract: For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x − y monophonic path is called an x − y detour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x − y detour monophonic path for some x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G). A total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dm t (G). A total detour monophonic set of cardinality dm t (G) is called a dm t -set of G. We determine bounds for it and characterize graphs which realize the lower bound. It is shown that for positive integers r, d and k ≥ 6 with r < d there exists a connected graph G with monophonic radius r, monophonic diameter d and dm t (G) = k. For positive integers a, b such that 4 ≤ a ≤ b with b ≤ 2a, there exists a connected graph G such that dm(G) = a and dm t (G) = b. Also, if p, d and k are positive integers such that 2 ≤ d ≤ p − 2, 3 ≤ k ≤ p and p − d − k + 3 ≥ 0, there exists a connected graph G of order p, monophonic diameter d and dm t (G) = k.<![CDATA[A brief note on the existence of connections and covariant derivatives on modules]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200225&lng=en&nrm=iso&tlng=en
Abstract: In this note we make a review of the concepts of connection and covariant derivative on modules, in a purely algebraic context. Throughout the text, we consider algebras over an algebraically closed field of characteristic 0 and module will always mean left module. First, we concentrate our attention on a K-algebra A which is commutative, and use the Kähler differentials module, , to define connection (see Subsection 2.1). In this context, it is verified that the existence of connections implies the existence of covariant derivatives (cf. Prop. 2.3), and that every projective module admits a connection (cf. Prop. 2.5). Next (in Section 3), we focus our attention in the discussion of some counterexamples comparing these two notions. In fact, it is known that these two notions are equivalent when we consider regular K -algebras of finite type (see 18, Prop. 4.2). As well as, that there exists a connection on M if, and only if, the Atiyah-Kodaira-Spencer class of M, c(M), is zero (see 17 , Prop. 4.3). Finally, we take into account the case where A is (not necessarily commutative) and it is used the bimodule, , of noncommutative differentials introduces by Connes 9) ,10 in place of Kähler differentials to define a connection. In this case, it is proven that a module admits such connection if, and only if, it is a projective module (see 25, Theorem 5.2).<![CDATA[Fuzzy normed linear space valued sequence space]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200245&lng=en&nrm=iso&tlng=en
Abstract: In this article we define the notion of fuzzy normed linear space valued sequence space in a fuzzy normed linear space X and discuss some of its properties like completeness, monotone, solidness, convergence free and symmetricity. Also we prove some inclusion results.<![CDATA[Stability, Boundedness and periodic solutions to certain second order delay differential equations]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200257&lng=en&nrm=iso&tlng=en
Abstract: Stability, boundedness and existence of a unique periodic solution to certain second order nonlinear delay differential equations is discussed. By employing Lyapunov's direct (or second) method, a complete Lyapunov functional is constructed and used to establish sufficient conditions, on the nonlinear terms, that guarantee uniform asymptotic stability, uniform ultimate boundedness and existence of a unique periodic solution. Obtained results complement many outstanding recent results in the literature. Finally, examples are given to show the effectiveness of our method and correctness of our results.<![CDATA[Some new classes of vertex-mean graphs]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200283&lng=en&nrm=iso&tlng=en
Abstract: A vertex-mean labeling of a(p, q) graph G = ( V, E) is defined as an injective function such that the function defined by the rule satisfies the property that , where Ev denotes the set of edgesin G that are incident at v, N denotes the set of all natural numbers and Round is the it nearest integer function. A graph that has a vertex-mean labeling is called vertex-mean graph or V - mean graph. In this paper, we study V - mean behaviour of certain new classes of graphs and present a method to construct disconnected V - mean graphs.<![CDATA[On the toral rank conjecture and some consequences]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200299&lng=en&nrm=iso&tlng=en
Abstract The aim of this work is to improve the lower bound of the Puppe inequality. His theorem [15,Theorem 1.1] states that the sum of all Betti numbers of a well-behaved space X is at least equal to 2n, where n is rank of an n-torus T n acting almost freely on X.<![CDATA[An Algorithmic Approach to Equitable Total Chromatic Number of Graphs]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200307&lng=en&nrm=iso&tlng=en
Abstract: The equitable total coloring of a graph G is a combination of vertex and edge coloring whose color classes differs by atmost one. In this paper, we find the equitable total chromatic number for and Gn .<![CDATA[On some spaces of Lacunary I-convergent sequences of interval numbers defined by sequence of moduli]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200325&lng=en&nrm=iso&tlng=en
Abstract: In this article we introduce and study some spaces of I-convergent sequences of interval numbers with the help of a sequence of moduli, a bounded sequence of positive real numbers and a lacunary sequence of increasing integers. We study some topological and algebraic properties and some inclusion relations on these spaces.<![CDATA[Skolem difference mean labeling of disconnected graphs]]>
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200347&lng=en&nrm=iso&tlng=en
Abstract Let G = (V , E) be a graph with p vertices and q edges. G is said to have skolem difference mean labeling if it is possible to label the vertices with distinct elements ƒ (x) from 1,2,2,…, p + q in such a way that for each edge and the resulting labels of the edges are distinct and are from 1,2,2,…, q. A graph that admits a skolem difference mean labeling is called a skolem difference mean graph. In this paper, we prove that the graphs graphs.