For the given bipartite graphs G 1 , . . . , G n , the bipartite Ramsey number BR ( G 1 , . . . , G n ) is the least positive integer b such that any complete bipartite graph K b,b having edges coloured with 1 , 2 , . . . , n , contains a copy of some G i ( 1 ≤ i ≤ n ), where all the edges of G i have colour i . For the given bipartite graphs G 1 , . . . , G n and a positive integer m , the m -bipartite Ramsey number BR m ( G 1 , . . . , G n ) is defined as the least positive integer b ( b ≥ m ) such that any complete bipartite graph K m,b having edges coloured with 1 , 2 , . . . , n , contains a copy of some G i ( 1 ≤ i ≤ n ), where all the edges of G i have colour i . The values of BR m ( G 1 , G 2 ) (for each m ), BR m ( K 3 , 3 , K 3 , 3 ) and BR m ( K 2 , 2 , K 5 , 5 ) (for particular values of m ) have already been determined in several articles, where G 1 = K 2 , 2 and G 2 ∈ { K 3 , 3 , K 4 , 4 } . In this article, the value of BR m ( K 2 , 2 , K 6 , 6 ) is computed for each m ∈ { 2 , 3 , . . . , 8 } .
对于给定的二部图g1，…， G n，二部拉姆齐数BR (g1)，…， G n)是最小的正整数b，使得任何完全二部图K b,b的边有1,2，…， n，包含一个G i(1≤i≤n)的副本，其中G i的所有边的颜色都是i。对于给定的二部图g1，…， G n和正整数m, m -二部拉姆齐数BR m (g1，…)， G n)被定义为最小正整数b (b≥m)，使得任何完全二部图K m,b的边有1,2，…， n，包含一个G i(1≤i≤n)的副本，其中G i的所有边的颜色都是i。BR m (g1, g2)(对于每个m)， BR m (k3,3, k3,3)和BR m (k2,2, k5,5)(对于m的特定值)的值已经在几篇文章中确定，其中g1 = k2,2并且g2∈{k3,3, k4,4}。在本文中，对于每个m∈{2,3，…，计算BR m (k2,2, k6,6)的值。， 8}。
{"title":"The m-Bipartite Ramsey Number of the K{2,2} Versus K{6,6}","authors":"Yaser Rowshan","doi":"10.47443/cm.2022.011","DOIUrl":"https://doi.org/10.47443/cm.2022.011","url":null,"abstract":"For the given bipartite graphs G 1 , . . . , G n , the bipartite Ramsey number BR ( G 1 , . . . , G n ) is the least positive integer b such that any complete bipartite graph K b,b having edges coloured with 1 , 2 , . . . , n , contains a copy of some G i ( 1 ≤ i ≤ n ), where all the edges of G i have colour i . For the given bipartite graphs G 1 , . . . , G n and a positive integer m , the m -bipartite Ramsey number BR m ( G 1 , . . . , G n ) is defined as the least positive integer b ( b ≥ m ) such that any complete bipartite graph K m,b having edges coloured with 1 , 2 , . . . , n , contains a copy of some G i ( 1 ≤ i ≤ n ), where all the edges of G i have colour i . The values of BR m ( G 1 , G 2 ) (for each m ), BR m ( K 3 , 3 , K 3 , 3 ) and BR m ( K 2 , 2 , K 5 , 5 ) (for particular values of m ) have already been determined in several articles, where G 1 = K 2 , 2 and G 2 ∈ { K 3 , 3 , K 4 , 4 } . In this article, the value of BR m ( K 2 , 2 , K 6 , 6 ) is computed for each m ∈ { 2 , 3 , . . . , 8 } .","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76539049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Relating Energy and Sombor Energy","authors":"","doi":"10.47443/cm.2021.0054","DOIUrl":"https://doi.org/10.47443/cm.2021.0054","url":null,"abstract":"","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73128553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, the notion of weakly irreducible filters in strong quasi-ordered residuated systems is introduced and analyzed. It is shown that any weakly irreducible filter is a prime (and therefore, irreducible) filter. It is also proved that if the lattice F(A) of all filters in a strong quasi-ordered residuated system A is distributive, then any irreducible filter in A is weakly irreducible in A.
{"title":"Weakly Irreducible Filter in Strong Quasi-Ordered Residuated Systems","authors":"D. Romano","doi":"10.47443/cm.2021.0032","DOIUrl":"https://doi.org/10.47443/cm.2021.0032","url":null,"abstract":"In this article, the notion of weakly irreducible filters in strong quasi-ordered residuated systems is introduced and analyzed. It is shown that any weakly irreducible filter is a prime (and therefore, irreducible) filter. It is also proved that if the lattice F(A) of all filters in a strong quasi-ordered residuated system A is distributive, then any irreducible filter in A is weakly irreducible in A.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76178784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
where d(u) and d(u, v) are the degree of the vertex u and the distance between the vertices u and v in G, respectively. This new index is useful in predicting physico-chemical properties with high accuracy compared to some classic topological indices. Mathematical relations between the Harary-Albertson index and other classic topological indices are established. The extremal values of the Harary-Albertson index for trees of given order are also determined.
{"title":"Harary-Albertson index of graphs","authors":"Zhen Lin","doi":"10.47443/cm.2021.0051","DOIUrl":"https://doi.org/10.47443/cm.2021.0051","url":null,"abstract":"where d(u) and d(u, v) are the degree of the vertex u and the distance between the vertices u and v in G, respectively. This new index is useful in predicting physico-chemical properties with high accuracy compared to some classic topological indices. Mathematical relations between the Harary-Albertson index and other classic topological indices are established. The extremal values of the Harary-Albertson index for trees of given order are also determined.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80276925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A graph is said to be Hamiltonian (respectively, traceable) if it has a Hamiltonian cycle (respectively, Hamiltonian path), where a Hamiltonian cycle (respectively, Hamiltonian path) is a cycle (respectively, path) containing all the vertices of the graph. In this short note, sufficient conditions involving the clique number for the Hamiltonian and traceable graphs are presented.
{"title":"The Clique Number and Some Hamiltonian Properties of Graphs","authors":"Rao Li","doi":"10.47443/cm.2021.0038","DOIUrl":"https://doi.org/10.47443/cm.2021.0038","url":null,"abstract":"Abstract A graph is said to be Hamiltonian (respectively, traceable) if it has a Hamiltonian cycle (respectively, Hamiltonian path), where a Hamiltonian cycle (respectively, Hamiltonian path) is a cycle (respectively, path) containing all the vertices of the graph. In this short note, sufficient conditions involving the clique number for the Hamiltonian and traceable graphs are presented.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77976419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anestis G. Hatzimichailidis, G. Souliotis, B. Papadopoulos
In this paper we introduce fuzzy implications stemming from a class of strong negations, which are generated via conical sections. The strong negations form a structural element in the production of fuzzy implications
{"title":"Fuzzy implications based on strong negations","authors":"Anestis G. Hatzimichailidis, G. Souliotis, B. Papadopoulos","doi":"10.47443/cm.2021.0040","DOIUrl":"https://doi.org/10.47443/cm.2021.0040","url":null,"abstract":"In this paper we introduce fuzzy implications stemming from a class of strong negations, which are generated via conical sections. The strong negations form a structural element in the production of fuzzy implications","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80346753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this short note, an alternative proof of a harmonic mean inequality involving Nielsen’s beta function is provided. This inequality was first posed as a conjecture by Nantomah [Bull. Int. Math. Virtual Inst. 9 (2019) 263–269] and subsequently proved by Matejı́čka [Probl. Anal. Issues Anal. 8(26) (2019) 105–111]. The present proof is more compact and relatively simple.
在这个简短的笔记中，提供了涉及Nielsen的beta函数的调和平均不等式的另一种证明。这个不等式最初是由Nantomah [Bull]提出的一个猜想。Int。数学。虚拟研究所，9(2019)263-269]，随后由matejyi æ ka [Probl.]证明。分析的议题通报。8(26)(2019)105-111]。现在的证明更紧凑，也相对简单。
{"title":"An alternative proof of a harmonic mean inequality for Nielsen’s beta function","authors":"K. Nantomah","doi":"10.47443/cm.2021.0028","DOIUrl":"https://doi.org/10.47443/cm.2021.0028","url":null,"abstract":"Abstract In this short note, an alternative proof of a harmonic mean inequality involving Nielsen’s beta function is provided. This inequality was first posed as a conjecture by Nantomah [Bull. Int. Math. Virtual Inst. 9 (2019) 263–269] and subsequently proved by Matejı́čka [Probl. Anal. Issues Anal. 8(26) (2019) 105–111]. The present proof is more compact and relatively simple.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77351521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There are misprints in the paper [1]. On the fourth line in the section of Introduction on Page 54, the “Define G1(n, k) := Kk ∨K k+1 where k ≥ 2 and G2(n, k) := Kk ∨K k+2 where k ≥ 1” should be “Define G1(n, k) := Kn−k−1 ∨K k+1 where k ≥ 2 and G2(n, k) := Kn−k−2 ∨K k+2 where k ≥ 1”. Because of the above changes, the following changes should be made, accordingly. • At the end of Proof of Theorem 1.1 on Page 55, “G is G1(n, k)” should be changed into “G is Kk ∨K k+1”. • At the end of Proof of Theorem 1.2 on Page 56, “G is G2(n, k)” should be changed into “G is Kk ∨K k+2”.
{"title":"Corrigendum to “Combinations of some spectral invariants and Hamiltonian properties of\u0000graphs, Contrib. Math. 1 (2020) 54–56”","authors":"Rao Li","doi":"10.47443/cm.2021.n2","DOIUrl":"https://doi.org/10.47443/cm.2021.n2","url":null,"abstract":"There are misprints in the paper [1]. On the fourth line in the section of Introduction on Page 54, the “Define G1(n, k) := Kk ∨K k+1 where k ≥ 2 and G2(n, k) := Kk ∨K k+2 where k ≥ 1” should be “Define G1(n, k) := Kn−k−1 ∨K k+1 where k ≥ 2 and G2(n, k) := Kn−k−2 ∨K k+2 where k ≥ 1”. Because of the above changes, the following changes should be made, accordingly. • At the end of Proof of Theorem 1.1 on Page 55, “G is G1(n, k)” should be changed into “G is Kk ∨K k+1”. • At the end of Proof of Theorem 1.2 on Page 56, “G is G2(n, k)” should be changed into “G is Kk ∨K k+2”.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80768009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper provides the unification of several continuity-like properties of functions and relations in the framework of relator spaces. Motivated by Galois connections, we consider an ordered pair of relations instead of a single relation. A family R of relations on a set X to another set Y is called a relator on X to Y . All reasonable generalizations of the usual topological structures (such as proximities, closures, topologies, filters and convergences, for instance) can be derived from relators. Therefore, they should not be studied separately. From the various topological and algebraic structures (such as lower bounds, minimum and infimum, for instance) derived from relators, by using Pataki connections, we can obtain several closure and projection operations for relators. Each of them will lead to four continuity-like properties of an ordered pair of relators.
{"title":"Basic tools and continuity-like properties in relator spaces","authors":"M. Rassias, Á. Száz","doi":"10.47443/cm.2021.0016","DOIUrl":"https://doi.org/10.47443/cm.2021.0016","url":null,"abstract":"This paper provides the unification of several continuity-like properties of functions and relations in the framework of relator spaces. Motivated by Galois connections, we consider an ordered pair of relations instead of a single relation. A family R of relations on a set X to another set Y is called a relator on X to Y . All reasonable generalizations of the usual topological structures (such as proximities, closures, topologies, filters and convergences, for instance) can be derived from relators. Therefore, they should not be studied separately. From the various topological and algebraic structures (such as lower bounds, minimum and infimum, for instance) derived from relators, by using Pataki connections, we can obtain several closure and projection operations for relators. Each of them will lead to four continuity-like properties of an ordered pair of relators.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88166538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Corrigendum to “Theory of hyper-singular integrals and its application to the Navier-Stokes problem, Contrib. Math. 2 (2020) 47–54”","authors":"A. Ramm","doi":"10.47443/cm.2021.n1","DOIUrl":"https://doi.org/10.47443/cm.2021.n1","url":null,"abstract":"","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76814220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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