16 results
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2. Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number
- Author
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Chang Liu and Jianping Li
- Subjects
Combinatorics ,Index (economics) ,Domination analysis ,extremal trees ,domination number ,General Mathematics ,the zeroth-order general randić index ,QA1-939 ,Mathematics ,Zeroth order - Abstract
The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by $ ^0R_{\alpha}(G) $, is the sum of items $ (d_{v})^{\alpha} $ over all vertices $ v\in V_G $, where $ \alpha $ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $ ^0R_{\alpha} $ of trees with a given domination number $ \gamma $, for $ \alpha\in(-\infty, 0)\cup(1, \infty) $ and $ \alpha\in(0, 1) $, respectively. The corresponding extremal graphs of these bounds are also characterized.
- Published
- 2022
3. Varieties of a class of elementary subalgebras
- Author
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Yang Pan and Yanyong Hong
- Subjects
Physics ,General Mathematics ,Dimension (graph theory) ,Subalgebra ,elementary subalgebras ,commuting roots ,Type (model theory) ,Combinatorics ,Restricted Lie algebra ,Algebraic group ,Lie algebra ,QA1-939 ,Variety (universal algebra) ,Algebraically closed field ,Mathematics::Representation Theory ,irreducible components ,Mathematics - Abstract
Let $ G $ be a connected standard simple algebraic group of type $ C $ or $ D $ over an algebraically closed field $ \Bbbk $ of positive characteristic $ p > 0 $, and $ \mathfrak{g}: = \mathrm{Lie}(G) $ be the Lie algebra of $ G $. Motivated by the variety of $ \mathbb{E}(r, \mathfrak{g}) $ of $ r $-dimensional elementary subalgebras of a restricted Lie algebra $ \mathfrak{g} $, in this paper we characterize the irreducible components of $ \mathbb{E}(\mathrm{rk}_{p}(\mathfrak{g})-1, \mathfrak{g}) $ where the $ p $-rank $ \mathrm{rk}_{p}(\mathfrak{g}) $ is defined to be the maximal dimension of an elementary subalgebra of $ \mathfrak{g} $.
- Published
- 2022
4. Note on r-central Lah numbers and r-central Lah-Bell numbers
- Author
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Hye Kyung Kim
- Subjects
Combinatorics ,lah-bell numbers ,General Mathematics ,QA1-939 ,r-lah-bell polynomials ,central factorial numbers of the second kind ,Lah number ,lah numbers ,Mathematics ,Bell number ,r-lah numbers - Abstract
The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.
- Published
- 2022
5. Semilattice strongly regular relations on ordered n-ary semihypergroups
- Author
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Sorasak Leeratanavalee and Jukkrit Daengsaen
- Subjects
Combinatorics ,hyperideal ,hyperfilter ,General Mathematics ,QA1-939 ,Congruence (manifolds) ,Semilattice ,n-ary semihypergroup ,ordered semihypergroup ,Prime (order theory) ,Mathematics ,Counterexample - Abstract
In this paper, we introduce the concept of $ j $-hyperfilters, for all positive integers $ 1\leq j \leq n $ and $ n \geq 2 $, on (ordered) $ n $-ary semihypergroups and establish the relationships between $ j $-hyperfilters and completely prime $ j $-hyperideals of (ordered) $ n $-ary semihypergroups. Moreover, we investigate the properties of the relation $ \mathcal{N} $, which is generated by the same principal hyperfilters, on (ordered) $ n $-ary semihypergroups. As we have known from [21] that the relation $ \mathcal{N} $ is the least semilattice congruence on semihypergroups, we illustrate by counterexample that the similar result is not necessarily true on $ n $-ary semihypergroups where $ n\geq 3 $. However, we provide a sufficient condition that makes the previous conclusion true on $ n $-ary semihypergroups and ordered $ n $-ary semihypergroups where $ n\geq 3 $. Finally, we study the decomposition of prime hyperideals and completely prime hyperideals by means of their $ \mathcal{N} $-classes. As an application of the results, a related problem posed by Tang and Davvaz in [31] is solved.
- Published
- 2022
6. On the eccentric connectivity coindex in graphs
- Author
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Ber-Lin Yu, Xianhao Shi, and Hongzhuan Wang
- Subjects
Physics ,Vertex (graph theory) ,General Mathematics ,eccentric connectivity coindex ,Unicyclic graphs ,Order (ring theory) ,trees ,Graph ,Combinatorics ,unicyclic graphs ,Topological index ,QA1-939 ,Astrophysics::Earth and Planetary Astrophysics ,cactus ,Eccentricity (mathematics) ,diameter ,Connectivity ,Mathematics - Abstract
The well-studied eccentric connectivity index directly consider the contribution of all edges in a graph. By considering the total eccentricity sum of all non-adjacent vertex, Hua et al. proposed a new topological index, namely, eccentric connectivity coindex of a connected graph. The eccentric connectivity coindex of a connected graph $ G $ is defined as \begin{document}$ \overline{\xi}^{c}(G) = \sum\limits_{uv\notin E(G)} (\varepsilon_{G}(u)+\varepsilon_{G}(v)). $\end{document} Where $ \varepsilon_{G}(u) $ (resp. $ \varepsilon_{G}(v) $) is the eccentricity of the vertex $ u $ (resp. $ v $). In this paper, some extremal problems on the $ \overline{\xi}^{c} $ of graphs with given parameters are considered. We present the sharp lower bounds on $ \overline{\xi}^{c} $ for general connecteds graphs. We determine the smallest eccentric connectivity coindex of cacti of given order and cycles. Also, we characterize the graph with minimum and maximum eccentric connectivity coindex among all the trees with given order and diameter. Additionally, we determine the smallest eccentric connectivity coindex of unicyclic graphs with given order and diameter and the corresponding extremal graph is characterized as well.
- Published
- 2022
7. A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $
- Author
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Jinchao Yu, Dan Yang, Xiaoying Zhu, and Jingjing Zhang
- Subjects
Physics ,Class (set theory) ,Mean curvature ,Conjecture ,principal curvatures ,General Mathematics ,linear weingarten hypersurfaces ,Diagonalizable matrix ,Space (mathematics) ,Lambda ,Combinatorics ,Hypersurface ,Principal curvature ,QA1-939 ,Mathematics::Differential Geometry ,proper mean curvature vector ,Nuclear Experiment ,Mathematics - Abstract
A nondegenerate hypersurface in a pseudo-Euclidean space $ \mathbb{E}^{n+1}_{s} $ is called to have proper mean curvature vector if its mean curvature $ \vec{H} $ satisfies $ \Delta \vec{H} = \lambda \vec{H} $ for a constant $ \lambda $. In 2013, Arvanitoyeorgos and Kaimakamis conjectured [1]: any hypersurface satisfying $ \Delta \vec{H} = \lambda \vec{H} $ in pseudo-Euclidean space $ \mathbb E_{s}^{n+1} $ has constant mean curvature. This paper will give further support evidences to this conjecture by proving that a linear Weingarten hypersurface $ M^{n}_{r} $ in $ \mathbb E^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda \vec{H} $ has constant mean curvature if $ M^{n}_{r} $ has diagonalizable shape operator with less than seven distinct principal curvatures.
- Published
- 2022
8. Exceptional set in Waring–Goldbach problem for sums of one square and five cubes
- Author
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Ran Song, Jinjiang Li, Yiyang Pan, and Min Zhang
- Subjects
waring–goldbach problem ,Combinatorics ,Set (abstract data type) ,General Mathematics ,Waring–Goldbach problem ,QA1-939 ,Square (unit) ,exceptional set ,Mathematics ,circle method - Abstract
Let $ N $ be a sufficiently large integer. In this paper, it is proved that, with at most $ O\big(N^{4/9+\varepsilon}\big) $ exceptions, all even positive integers up to $ N $ can be represented in the form $ p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3 $, where $ p_1, p_2, p_3, p_4, p_5, p_6 $ are prime numbers.
- Published
- 2022
9. The linear $ k $-arboricity of symmetric directed trees
- Author
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Xiaoling Zhou, Weihua He, and Chao Yang
- Subjects
Connected component ,General Mathematics ,Arboricity ,Digraph ,directed trees ,linear arboricity ,Combinatorics ,Computer Science::Discrete Mathematics ,symmetric directed trees ,Path (graph theory) ,QA1-939 ,linear k-arboricity ,Computer Science::Data Structures and Algorithms ,Computer Science::Distributed, Parallel, and Cluster Computing ,Mathematics ,digraphs - Abstract
A linear $ k $-diforest is a directed forest in which every connected component is a directed path of length at most $ k $. The linear $ k $-arboricity of a digraph $ D $ is the minimum number of arc-disjoint linear $ k $-diforests whose union covers all the arcs of $ D $. In this paper, we study the linear $ k $-arboricity for symmetric directed trees and fully determine the linear $ 2 $-arboricity for all symmetric directed trees.
- Published
- 2022
10. On the stability of two functional equations for (S,N)-implications
- Author
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Dapeng Lang, Xinyu Han, Sizhao Li, and Songsong Dai
- Subjects
Stability study ,General Mathematics ,lcsh:Mathematics ,functional equations ,stability ,lcsh:QA1-939 ,Fuzzy logic ,Stability (probability) ,humanities ,Combinatorics ,(s,n)-implication ,law of importation ,iterative boolean-like law ,Product (mathematics) ,fuzzy implications ,Functional equation ,Beta (velocity) ,Law of importation ,Fuzzy negation ,Mathematics - Abstract
The iterative functional equation $ \alpha\rightarrow(\alpha\rightarrow \beta) = \alpha\rightarrow \beta $ and the law of importation $ (\alpha\wedge \beta)\rightarrow \gamma = \alpha\rightarrow (\beta\rightarrow \gamma) $ are two tautologies in classical logic. In fuzzy logics, they are two important properties, and are respectively formulated as $ I(\alpha, \beta) = I(\alpha, I(\alpha, \beta)) $ and $ I(T(\alpha, \beta), \gamma) = I(\alpha, I(\beta, \gamma)) $ where $ I $ is a fuzzy implication and $ T $ is a $ t $-norm. Over the past several years, solutions to these two functional equations involving different classes of fuzzy implications have been studied. However, there are no results about stability study of fuzzy functional equations involving fuzzy implication. This paper discusses fuzzy implications that do not strictly satisfying these equations, but approximately satisfy these equations. Then we establish the Hyers-Ulam stability of the iterative functional equation involving the $ (S, N) $-implication, where the $ (S, N) $-implication is a common class of fuzzy implications generated by a continuous $ t $-conorm $ S $ and a continuous fuzzy negation $ N $. Furthermore, given a fixed $ t $-norm (the minimum $ t $-norm or the product $ t $-norm) the Hyers-Ulam stability of the law of importation involving the $ (S, N) $-implication is studied.
- Published
- 2021
11. On the first general Zagreb eccentricity index
- Author
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Aisha Javed, Muhammad Imran, Muhammad Jamil, and Roslan Hasni
- Subjects
Combinatorics ,eccentricity of vertices ,first general zagreb eccentricity index ,General Mathematics ,extremal graphs ,lcsh:Mathematics ,Shortest path problem ,Bipartite graph ,lcsh:QA1-939 ,Upper and lower bounds ,Graph ,Vertex (geometry) ,Mathematics - Abstract
In a graph G, the distance between two vertices is the length of the shortest path between them. The maximum distance between a vertex to any other vertex is considered as the eccentricity of the vertex. In this paper, we introduce the first general Zagreb eccentricity index and found upper and lower bounds on this index in terms of order, size and diameter. Moreover, we characterize the extremal graphs in the class of trees, trees with pendant vertices and bipartite graphs. Results on some famous topological indices can be presented as the corollaries of our main results.
- Published
- 2021
12. Effect of edge and vertex addition on Albertson and Bell indices
- Author
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Ismail Naci Cangul and Sadik Delen
- Subjects
Vertex (graph theory) ,Vertex deletion ,General Mathematics ,lcsh:Mathematics ,omega invariant ,Topological graph ,vertex addition ,lcsh:QA1-939 ,albertson index ,Graph ,Combinatorics ,bell index ,Computer Science::Discrete Mathematics ,edge addition ,Mathematics ,irregularity index - Abstract
Topological graph indices have been of great interest in the research of several properties of chemical substances as it is possible to obtain these properties only by using mathematical calculations. The irregularity indices are the ones to determine the degree of irregularity of a graph. Albertson and Bell indices are two of them. Edge and vertex deletion and addition are important and useful methods in calculating several properties of a given graph. In this paper, the effects of adding a new edge or a new vertex to a graph on the Albertson and Bell indices are determined.
- Published
- 2021
13. A census of critical sets based on non-trivial autotopisms of Latin squares of order up to five
- Author
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Raúl M. Falcón, Laura Johnson, Stephanie Perkins, Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII), and Junta de Andalucía
- Subjects
Class (set theory) ,critical set ,Enumeration ,Mathematics::General Mathematics ,General Mathematics ,Order up to ,Structure (category theory) ,enumeration ,Combinatorics ,Set (abstract data type) ,cycle structure ,Latin square ,Mathematics ,Autotopism ,Mathematics::Combinatorics ,Group (mathematics) ,Cycle structure ,lcsh:Mathematics ,Mathematics::History and Overview ,Census ,lcsh:QA1-939 ,autotopism ,latin square ,Critical set - Abstract
This paper delves into the study of critical sets of Latin squares having a given isotopism in their autotopism group. Particularly, we prove that the sizes of these critical sets only depend on both the main class of the Latin square and the cycle structure of the isotopism under consideration. Keeping then in mind that the autotopism group of a Latin square acts faithfully on the set of entries of the latter, we enumerate all the critical sets based on autotopisms of Latin squares of order up to five. Junta de Andalucía FQM-016
- Published
- 2021
14. Necessary and sufficient conditions on the Schur convexity of a bivariate mean
- Author
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Bai-Ni Guo, Hong-Ping Yin, Xi-Min Liu, and Jing-Yu Wang
- Subjects
Mathematics::Combinatorics ,inequality ,General Mathematics ,lcsh:Mathematics ,Regular polygon ,Bivariate analysis ,lcsh:QA1-939 ,Convexity ,Combinatorics ,schur harmonically convex function ,schur convex function ,majorization ,Majorization ,Mathematics::Representation Theory ,bivariate mean ,Mathematics ,Schur-convex function ,necessary and sufficient condition - Abstract
In the paper, the authors find and apply necessary and sufficient conditions for a bivariate mean of two positive numbers with three parameters to be Schur convex or Schur harmonically convex respectively.
- Published
- 2021
15. Multiple solutions for a class of boundary value problems of fractional differential equations with generalized Caputo derivatives
- Author
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Yansheng Liu and Yating Li
- Subjects
Physics ,generalized caputo derivatives ,Class (set theory) ,multiple solutions ,boundary value problems ,General Mathematics ,fixed point theory ,Fixed-point theorem ,Combinatorics ,Cone (topology) ,fractional differential equation ,QA1-939 ,Boundary value problem ,Fractional differential ,Mathematics - Abstract
This paper is mainly concerned with the existence of multiple solutions for the following boundary value problems of fractional differential equations with generalized Caputo derivatives: \begin{document}$ \hskip 3mm \left\{ \begin{array}{lll} ^{C}_{0}D^{\alpha}_{g}x(t)+f(t, x) = 0, \ 0 where $ 2 < \alpha < 3 $, $ 1 < \nu < 2 $, $ \alpha-\nu-1 > 0 $, $ f\in C([0, 1]\times \mathbb{R}^{+}, \mathbb{R}^{+}) $, $ g' > 0 $, $ h\in C([0, 1], \mathbb{R}^{+}) $, $ \mathbb{R}^{+} = [0, +\infty) $. Applying the fixed point theorem on cone, the existence of multiple solutions for considered system is obtained. The results generalize and improve existing conclusions. Meanwhile, the Ulam stability for considered system is also considered. Finally, three examples are worked out to illustrate the main results.
- Published
- 2021
16. Further results on permutation polynomials and complete permutation polynomials over finite fields
- Author
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Jian Zou, Qian Liu, Jianrui Xie, and Ximeng Liu
- Subjects
Combinatorics ,Permutation ,Finite field ,General Mathematics ,agw criterion ,QA1-939 ,complete permutation polynomial ,finite field ,permutation polynomial ,Mathematics - Abstract
In this paper, by employing the AGW criterion and determining the number of solutions to some equations over finite fields, we further investigate nine classes of permutation polynomials over $ \mathbb{F}_{p^n} $ with the form $ (x^{p^m}-x+\delta)^{s_1}+(x^{p^m}-x+\delta)^{s_2}+x $ and propose five classes of complete permutation polynomials over $ \mathbb{F}_{p^{2m}} $ with the form $ ax^{p^m}+bx+h(x^{p^m}-x) $.
- Published
- 2021
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