k-generalized Fibonacci numbers which are concatenations of two repdigits

We show that the k-generalized Fibonacci numbers that are concatenations of two repdigits have at most four digits

Rough action on topological rough groups

[EN] In this paper we explore the interrelations between rough set theory and group theory. To this end, we first define a topological rough group homomorphism and its kernel. Moreover, we introduce rough action and topological rough group homeomorphisms, providing several examples. Next, we combine these two notions in order to define topological rough homogeneous spaces, discussing results concerning open subsets in topological rough groups.The authors wish to thank the Deanship for Scientific Research (DSR) at King Abdulaziz University for financially funding this project under grant no. KEP-PhD-2-130-39. Also, we would like to thank the editor and referees for their valuable suggestions which have improved the presentation of the paper.Altassan, A.; Alharbi, N.; Aydi, H.; Özel, C. (2020). Rough action on topological rough groups. Applied General Topology. 21(2):295-304. https://doi.org/10.4995/agt.2020.13156OJS295304212S. Akduman, E. Zeliha, A. Zemci and S. 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On ABC energy and its application to anticancer drugs

For a simple connected graph $\Gamma$ with node set $V(\Gamma) = \{w_{1}, w_{2}, \dots, w_{n}\}$ and degree sequence $d_{i}$, the atom-bond connectivity ($ABC$) matrix of $\Gamma$ has an $(ij)$-th entry $\sqrt{\frac{d_{i}+d_{j}-2}{d_{i}d_{j}}}$ if $w_{i}$ is adjacent to $w_{j}$ and $0$, otherwise. The multiset of all eigenvalues of $ABC$ matrix is known as the $ABC$ spectrum and their absolute sum is known as the $ABC$ energy of $\Gamma.$ Two graphs of same order are known as $ABC$ equienergetic if they have the same $ABC$ energy but share different $ABC$ spectrum. We describe the $ABC$ spectrum of some special graph operations and as an application, we construct the $ABC$ equienergetic graphs. Further, we give linear regression analysis of $ABC$ index/energy with the physical properties of anticancer drugs. We observe that they are better correlated with $ABC$-energy

Generalized Quasi Trees with Respect to Degree Based Topological Indices and Their Applications to COVID-19 Drugs

The l-generalized quasi tree is a graph G for which we can find W⊂V(G) with |W|=l such that G−W is a tree but for an arbitrary Y⊂V(G) with |Y|l, G−Y is not a tree. In this paper, inequalities with respect to zeroth-order Randić and hyper-Zagreb indices are studied in the class of l-generalized quasi trees. The corresponding extremal graphs corresponding to these indices in the class of l-generalized quasi trees are also obtained. In addition, we carry QSPR analysis of COVID-19 drugs with zeroth-order Randić and hyper-Zagreb indices (energy)

Topological Properties of Polymeric Networks Modelled by Generalized Sierpiński Graphs

In this article, we compute the irregularity measures of generalized Sierpiński graphs and obtain some bounds on these irregularities. Moreover, we discuss some bounds on connectivity indices for generalized Sierpiński graphs of any arbitrary graph H along with classification of the extremal graphs used to attain them

On linear equations in free Lie algebras

We investigate equations of the form $[x_1,u_1]+\ldots+[x_k,u_k]=0$ over a free Lie algebra $L$. In the case where the coefficients $u_1,\ldots,u_k$ are free generators of $L$, we generalize a number of earlier results on equations with two variables to equations with an arbitrary number of indeterminates. Our main results refer to the case where the coefficients coincide with the free generators of $L$. We give a detailed description of the solution space and we obtain an explicit basis for its multilinear fine homogeneous component

Almost Repdigit <i>k</i>-Fibonacci Numbers with an Application of <i>k</i>-Generalized Fibonacci Sequences

In this paper, we define the notion of almost repdigit as a positive integer whose digits are all equal except for at most one digit, and we search all terms of the k-generalized Fibonacci sequence which are almost repdigits. In particular, we find all k-generalized Fibonacci numbers which are powers of 10 as a special case of almost repdigits. In the second part of the paper, by using the roots of the characteristic polynomial of the k-generalized Fibonacci sequence, we introduce k-generalized tiny golden angles and show the feasibility of this new type of angles in application to magnetic resonance imaging

On Fundamental Theorems of Fuzzy Isomorphism of Fuzzy Subrings over a Certain Algebraic Product

In this study, we define the concept of an ω-fuzzy set ω-fuzzy subring and show that the intersection of two ω-fuzzy subrings is also an ω-fuzzy subring of a given ring. Moreover, we give the notion of an ω-fuzzy ideal and investigate different fundamental results of this phenomenon. We extend this ideology to propose the notion of an ω-fuzzy coset and develop a quotient ring with respect to this particular fuzzy ideal analog into a classical quotient ring. Additionally, we found an ω-fuzzy quotient subring. We also define the idea of a support set of an ω-fuzzy set and prove various important characteristics of this phenomenon. Further, we describe ω-fuzzy homomorphism and ω-fuzzy isomorphism. We establish an ω-fuzzy homomorphism between an ω-fuzzy subring of the quotient ring and an ω-fuzzy subring of this ring. We constitute a significant relationship between two ω-fuzzy subrings of quotient rings under the given ω-fuzzy surjective homomorphism and prove some more fundamental theorems of ω-fuzzy homomorphism for these specific fuzzy subrings. Finally, we present three fundamental theorems of ω-fuzzy isomorphism

The Eccentric-Distance Sum Polynomials of Graphs by Using Graph Products

The correlations between the physico-chemical properties of a chemical structure and its molecular structure-properties are used in quantitative structure-activity and property relationship studies (QSAR/QSPR) by using graph-theoretical analysis and techniques. It is well known that some structure-activity and quantitative structure-property studies, using eccentric distance sum, are better than the corresponding values obtained by using the Wiener index. In this article, we give precise expressions for the eccentric distance sum polynomial of some graph products such as join, Cartesian, lexicographic, corona and generalized hierarchical products. We implement our outcomes to calculate this polynomial for some significant families of molecular graphs in the form of the above graph products