Open Journal of Discrete Applied Mathematics (ODAM)

In this paper, we study the existence of nontrivial solution for the fractional differential equation of order \(\alpha\) with three point boundary conditions having the following form
$$
D^{\alpha}u(t)=f(t,v(t),D^{\nu}v(t)),\quad t\in(0,T)$$
$$u(0)=0,\quad u(T)=au(\xi),$$
where \(1<\alpha<2\), \(\nu, a>0\), \(\xi\in (0,T)\), \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\). \(D\) is the standard Riemann-Liouville fractional derivative operator and \(f\in C([0,1]\times\mathbf{R}^{2},\mathbf{R})\). Applying the Leray-Schauder nonlinear alternative we prove the existence of at least one solution. As an application, we also given some examples to illustrate the results obtained.

Fibonacci polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions and by varying the recurrence relation and maintaining the initial conditions. In this paper, both the recurrence relation and initial conditions of generalized Fibonacci polynomials are varied and defined by recurrence relation as \(R_n(x)=axR_{n-1}(x)+bR_{n-2}(x)\) for all \(n\geq2,\) with initial conditions \(R_0(x)=2p\) and \(R_1(x)=px+q\) where \(a\) and \(b\) are positive integers and \(p\) and \(q\) are non-negative integers. Further some fundamental properties of these generalized polynomials such as explicit sum formula, sum of first \(n\) terms, sum of first \(n\) terms with (odd or even) indices and generalized identity are derived by Binet’s formula and generating function only.

The degree tolerant number of the power graph of the finite Albenian group, \(\mathbb{Z}_n\) under addition modulo \(n\), \(n\in \mathbb{N}\) is investigated. A surprising simple result, \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = k\) for the product of primes, \(n=p_1p_2p_3\cdots p_k\) is presented.

This paper serves as the first extension of the topic of dominator colorings of graphs to the setting of digraphs. We establish the dominator chromatic number over all possible orientations of paths and cycles. In this endeavor we discover that there are infinitely many counterexamples of a graph and subgraph pair for which the subgraph has a larger dominator chromatic number than the larger graph into which it embeds. Most importantly, we use these results to characterize all digraph families for which the dominator chromatic number is two. Finally, a new graph invariant measuring the difference between the dominator chromatic number of a graph and the chromatic number of that graph is established and studied. The paper concludes with some of the possible avenues for extending this line of research.

This work presents study on regularized and non-regularized versions of perceptron learning and least squares algorithms for classification problems. The Fréchet derivatives for least squares and perceptron algorithms are derived. Different Tikhonov’s regularization techniques for choosing the regularization parameter are discussed. Numerical experiments demonstrate performance of perceptron and least squares algorithms to classify simulated and experimental data sets.

A new recursion in only one variable allows very simple verifications of Bressoud’s polynomial identities, which lead to the Rogers-Ramanujan identities. This approach might be compared with an earlier approach due to Chapman. Applying the \(q\)-Chu-Vandermonde convolution, as suggested by Cigler, makes the computations particularly simple and elementary. The same treatment is also applied to the Santos polynomials and perhaps more polynomials from a list of Rogers-Ramanujan like polynomials [1].

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 \(e\in E(G)\).

This paper proposes the concept of direct product of fuzzy multigroups under \(t\)-norms and some of their basic properties are obtained. Next, we investigate and obtain some new results of strong upper alpha-cut, weak upper alpha-cut, strong lower alpha-cut and weak lower alpha-cut of them. Later, we prove conjugation and commutation between them. Finally, the notion of homomorphism in the context of fuzzy multigroups was defined and some homomorphic properties of fuzzy multigroups under \(t\)-norms in terms of homomorphic images and homomorphic preimages, respectively, were presented.

The concept of fuzzy set theory is of paramount relevance to tackling the issues of uncertainties in real-life problems. In a quest to having a reasonable means of curbing imprecision, the idea of fuzzy sets had been generalized to intuitionistic fuzzy sets, fuzzy multisets, Pythagorean fuzzy sets among others. The notion of intuitionistic fuzzy multisets (IFMS) came into the limelight naturally because there are instances when repetitions of both membership and non-membership degrees cannot be ignored like in the treatment of patients, where each consultations are key in diagnosis and therapy. In IFMS theory, the sum of the degrees of membership and non-membership is less than or equals one at each levels. Supposing the sum of the degrees of membership and non-membership is greater than or equal to one at any level, then the concept of Pythagorean fuzzy multisets (PFMS) is appropriate to handling such scenario. In this paper, the idea of PFMS is proposed as an extensional Pythagorean fuzzy sets proposed by R. R. Yager. In fact, PFMS is a Pythagorean fuzzy set in the framework of multiset. The main objectives of this paper are to expatiate the operations under PFMSs and discuss some of their algebraic properties with some related results. The concepts of level sets, cuts, accuracy and score functions, and modal operators are established in the setting of PFMSs with a number of results. Finally, to demonstrate the applicability of the proposed soft computing technique, a course placements scenario is discussed via PFMS framework using composite relation defined on PFMSs. This soft computing technique could find expression in other multi-criteria decision-making (MCDM) problems.