Let \(0<k\in\mathbb{Z}\). A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graph \(M_k\) induced by the vertices of the \((2k+1)\)-cube representing the \(k\)- and \((k+1)\)-subsets of \(\{0,\ldots,2k\}\) is given via an associated dihedral quotient graph of \(M_k\) whose vertices represent the ordered (rooted) trees of order \(k+1\) and size \(k\).

Let \(G\) be a simple graph. A total dominator coloring of \(G\) is a proper coloring of the vertices of \(G\) in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number \(\chi_d^t(G)\) of \(G\) is the minimum number of colors among all total dominator coloring of \(G\). In this paper, we study the total dominator chromatic number of some graphs with specific construction. Also we compare \(\chi_d^t(G)\) with \(\chi_d^t(G-e)\), where \(e\in E(G)\).

As a generalization of KU-algebras, the notion of pseudo-KU algebras is introduced in 2020 by the author (D. A. Romano. Pseudo-UP algebras, An introduction. Bull. Int. Math. Virtual Inst., 10(2)(2020), 349-355). Some characterizations of pseudo-KU algebras are established in that article. In addition, it is shown that each pseudo-KU algebra is a pseudo-UP algebra. In this paper it is a concept developed of pseudo-KU algebras in more detail and it has identified some of the main features of this type of universal algebras such as the notions of pseudo-subalgebras, pseudo-ideals, pseudo-filters and pseudo homomorphisms. Also, it has been shown that every pseudo-KU algebra is a pseudo-BE algebra. In addition, a congruence was constructed on a pseudo-KU algebra generated by a pseudo-ideal and shown that the corresponding factor-structure is and pseudo-KU algebra as well.

In this article, modified difference-type estimator for the population mean in two-phase sampling scheme using two auxiliary variables has been proposed. The mean squared error of the proposed estimator has also been derived using large sample approximation. The efficiency comparison conditions for the proposed estimator in comparison with other existing estimators in which the proposed estimator performed better than the other relevant existing estimators have been given.

Let \(G = (V,E)\) be a simple connected undirected graph. In this paper, we define generalized the Liouville’s and Möbius functions of a graph \(G\) which are the sum of Liouville \(\lambda\) and Möbius \(\mu\) functions of the degree of the vertices of a graph denoted by \(\Lambda(G)=\sum\limits_{v\in V(G)}\lambda(deg(v))\) and \(M(G)=\sum\limits_{v\in V(G)}\mu(deg(v))\), respectively. We also determine the Liouville’s and Möbius functions of some standard graphs as well as determining the relationships between the two functions with their proofs. The sum of generalized the Liouville and Möbius functions extending over the divisor d of degree of vertices of graphs is also given.

In this paper, we introduce a symmetric function in order to derive a new generating functions of bivariate Pell Lucas polynomials. We define complete homogeneous symmetric functions and give generating functions for Gauss Fibonacci polynomials, Gauss Lucas polynomials, bivariate Fibonacci polynomials, bivariate Lucas polynomials, bivariate
Jacobsthal polynomials and bivariate Jacobsthal Lucas polynomials.

We establish the strong generalized solution of the second mixed problem for an Euler-Poisson-Darboux equation in which the free term has the form: \(\gamma(t) u(x_0,t_0)\) where \(u(x,t)\) is the unknown function sought at the point \((x_0,t_0).\)

In this paper, the exact value of covering radius of unit repetition codes and the bounds of covering radius of zero-divisor repetition codes have been determined by using Lee weight over the finite ring \(F_{2}+vF_{2}+v^2F_2\). Moreover the covering radius of different block repetition codes have been also studied.

In this work, by using \(t\)-conorm \(C\), we introduce anti fuzzy vector spaces and define sum, union, direct sum and normality of anti fuzzy vector spaces. We prove that sum, union, direct sum and normality of anti fuzzy vector spaces is also anti fuzzy vector space under \(t\)-conorm \(C.\) Moreover, we investigate linear transformations over anti fuzzy vector spaces (normal anti fuzzy vector spaces) under \(t\)-conorms and prove that image and pre image of them is also anti fuzzy vector space (normal anti fuzzy vector space) under \(t\)-conorms.

Radial Basis Function (RBF) is a real valued function whose value rests only on the distance from some other points called a center, so that a linear combination of radial basis functions are typically used to approximate given functions or differential equations. Radial Basis Function (RBF) approximation has the ability to give an accurate approximation for large data sites which gives smooth solution for a given number of knots points; particularly, when the RBFs are scaled to the nearly flat and the shape parameter is chosen wisely. In this research work, an algorithm for solving partial differential equations is written and implemented on some selected problems, inverse multiquadric (IMQ) function was considered among other RBFs. Preference is given to the choice of shape parameter, which need to be wisely chosen. The strategy is written as an algorithm to perform number of interpolation experiments by changing the interval of the shape parameters and consequently select the best shape parameter that give small root means square error (RMSE). All the computational work has been done using Matlab. The interpolant for the selected problems and its corresponding root means square errors (RMSEs) are tabulated and plotted.