OMS-Vol. 2 (2018), Issue 1, pp. 146–155 | Open Access Full-Text PDF

Abstract:In this paper, some inequalities related to Čebyšev’s functional are proved.

OMS-Vol. 2 (2018), Issue 1, pp. 134–145 | Open Access Full-Text PDF

Abstract:For any graph \(G=(V,E)\), lict graph \(\eta(G)\) of a graph \(G\) is the graph whose vertex set is the union of the set of edges and the set of cut-vertices of \(G\) in which two vertices are adjacent if and only if the corresponding edges are adjacent or the corresponding members of \(G\) are incident. A secure lict dominating set of a graph \(\eta(G)\) , is a dominating set \(F \subseteq V(\eta(G))\) with the property that for each \(v_{1} \in (V(\eta(G))-F)\), there exists \(v_{2} \in F\) adjacent to \(v_{1}\) such that \((F-\lbrace v_{2}\rbrace) \cup \lbrace v_{1} \rbrace\) is a dominating set of \(\eta(G)\). The secure lict dominating number \(\gamma_{se}(\eta(G))\) of \(G\) is a minimum cardinality of a secure lict dominating set of \(G\). In this paper many bounds on \(\gamma_{se}(\eta(G))\) are obtained and its exact values for some standard graphs are found in terms of parameters of \(G\). Also its relationship with other domination parameters is investigated.

OMS-Vol. 2 (2018), Issue 1, pp. 122–133 | Open Access Full-Text PDF

Abstract:In this article we discuss moduli and constants of quasi-Banach space and give some important properties of these moduli and constants. Moreover, we establish relationships of these moduli and constants with each other.

OMS-Vol. 2 (2018), Issue 1, pp. 115–121 | Open Access Full-Text PDF

Abstract:Let \(G=(V,E)\) be a~finite simple graph with \(|V(G)|\) vertices and \(|E(G)|\) edges. An edge-covering of \(G\) is a family of subgraphs \(H_1, H_2, \dots, H_t\) such that each edge of \(E(G)\) belongs to at least one of the subgraphs \(H_i\), \(i=1, 2, \dots, t\). If every subgraph \(H_i\) is isomorphic to a given graph \(H\), then the graph \(G\) admits an \(H\)-covering. A graph \(G\) admitting \(H\) covering is called an \((a,d)\)-\(H\)-antimagic if there is a bijection \(f:V\cup E \to \{1,2,\dots, |V(G)|+|E(G)| \}\) such that for each subgraph \(H’\) of \(G \) isomorphic to \(H\), the sum of labels of all the edges and vertices belonged to \(H’\) constitutes an arithmetic progression with the initial term \(a\) and the common difference \(d\). For \(f(V)= \{ 1,2,3,\dots,|V(G)|\}\), the graph \(G\) is said to be super \((a,d)\)-\(H\)-antimagic and for \(d=0\) it is called \(H\)-supermagic. In this paper, we investigate the existence of super \((a,d)\)-\(C_4\)-antimagic labeling of book graphs, for difference \(d=0,1\) and \(n\geq2\).

OMS-Vol. 2 (2018), Issue 1, pp. 93–114 | Open Access Full-Text PDF

Abstract:In this article, the authors obtain the boundedness of the fractional Marcinkiewicz integral with variable kernel on Morrey-Herz spaces with variable exponents \(\alpha\) and \(p\). The corresponding boundedness for commutators generalized by the Lipschitz function is also considered.

OMS-Vol. 2 (2018), Issue 1, pp. 84–92 | Open Access Full-Text PDF

Abstract:In this article we continue the investigations presented in our previous papers [1,2,3,4], presenting some, for the best of our knowledge, new transformations of the Gauss hypergeometric function (3) and (13). They have been obtained using only elementary methods and stem from a couple of integrals evaluated in terms of complete elliptic integral of first kind by Legendre in [5] Chapter XXVII, at sections II and III.