Tom 27 (2019): Zeszyt 1 (June 2019)

A set S ⊆ V is a neighborhood set of a graph G = (V, E), if G = ∪_v∈S 〈 N[v] 〉, where 〈 N[v] 〉 is the subgraph of a graph G induced by v and all vertices adjacent to v. A neighborhood set S is said to be a neighbor coloring set if it contains at least one vertex from each color class of a graph G, where color class of a colored graph is the set of vertices having one particular color. The neighbor chromatic number χ_n (G) is the minimum cardinality of a neighbor coloring set of a graph G. In this article, some results on neighbor chromatic number of Cartesian products of two paths (grid graph) and cycles (torus graphs) are explored.

Let X be a Banach bimodule over a Banach algebra 𝒜. We shall seek conditions under which (1) X and its dual factorize on the same side or (2) X factors on the left (or right) if and only if its second conjugate space has a left (or right) unit.

In this paper we consider two continuous functions f, g : [a, b] → ℝ and we study for these ones, under which circumstances the intermediate point function is four order di erentiable at the point x = a and we calculate its derivative.

Let A_p be the class of functions f(z) which are analytic in the open unit disk U. Applying some subordinations for functions, some interesting properties for f(z) ∈A_p are showed. Our results are generalizations for results given before.

Indulal and Balakrishnan (2016) have put forward the Indu-Bala product and based on this product four new operations are defined by the authors of this manuscript in the paper “Four new operations of graphs based on Indu-Bala product and the Zagreb indices”. In this paper we establish explicit formulas of the forgotten topological index and reduced second Zagreb index in connection with these new operations of graphs.

In this paper, we find accordance of some classical inequalities and fractional dynamic inequalities. We find inequalities such as Radon’s inequality, Bergström’s inequality, Rogers-Hölder’s inequality, Cauchy-Schwarz’s inequality, the weighted power mean inequality and Schlömilch’s inequality in generalized and extended form by using the Riemann-Liouville fractional integrals on time scales.

In this paper, we investigate the existence of solution of integro-differential equations (IDEs) with Hilfer-Hadamard fractional derivative. The main results are obtained by using Schaefer’s fixed point theorem. Some Ulam stability results are presented.

This paper mainly dedicated on overview of zero sets in Ideal topological spaces. We also introduce a new class of functions which generalizes the class of continuous functions and investigate its position in the hierarchy of continuous functions on Ideal topological spaces. Moreover, these new sets (zero*-ℐ-set) which is a pragmatic approach to characterize completely Hausdorff spaces.