The series (cid:80) ∞ k =0 G N ( k ) (2 k +1) r and (cid:80) ∞ k =1 H N ( k ) k r are considered, where G N ( k ) and H N ( k ) are the Borwein-Chamberland sums appeared in the expansions of integer powers of the arcsine reported in the paper [D. Borwein, M. Chamberland, Int. J. Math. Math. Sci. 2007 (2007) #1981]. For 3 ≤ r ∈ N , representations for these series in terms of zeta values are derived, extending a theorem proved in the paper [J. Ewell, Canad. Math. Bull. 34 (1991) 60–66]. Several corollaries (especially for the case r = 3 ) are obtained, extending some known representations, including Euler’s famous rapidly converging series for ζ (3) . The technique can be applied to the case r = 2 and it yields generalizations of the formulas (cid:80) ∞ k =0 1 (2 k +1
考虑级数（cid:80）∞k=0 G N（k）（2k+1）r和（cid:80%）∞k=1 H N（k。对于3≤r∈N，导出了这些级数在ζ值方面的表示，扩展了[J.Ewell，Canad.Math.Bull.34（1991）60–66]中证明的定理。得到了几个推论（特别是对于r＝3的情况），扩展了一些已知的表示，包括欧拉著名的ζ（3）的快速收敛级数。该技术可应用于r＝2的情况，并得到公式（cid:80）∞k＝0 1（2k+1）的推广
{"title":"Two series which generalize Dirichlet’s lambda and Riemann’s zeta functions at positive integer arguments","authors":"Lubomir Markov","doi":"10.47443/dml.2023.051","DOIUrl":"https://doi.org/10.47443/dml.2023.051","url":null,"abstract":"The series (cid:80) ∞ k =0 G N ( k ) (2 k +1) r and (cid:80) ∞ k =1 H N ( k ) k r are considered, where G N ( k ) and H N ( k ) are the Borwein-Chamberland sums appeared in the expansions of integer powers of the arcsine reported in the paper [D. Borwein, M. Chamberland, Int. J. Math. Math. Sci. 2007 (2007) #1981]. For 3 ≤ r ∈ N , representations for these series in terms of zeta values are derived, extending a theorem proved in the paper [J. Ewell, Canad. Math. Bull. 34 (1991) 60–66]. Several corollaries (especially for the case r = 3 ) are obtained, extending some known representations, including Euler’s famous rapidly converging series for ζ (3) . The technique can be applied to the case r = 2 and it yields generalizations of the formulas (cid:80) ∞ k =0 1 (2 k +1","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46912466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Archibald, A. Blecher, A. Knopfmacher, T. Mansour
Motivated by cut points in graph theory, we consider a similar notion in compositions and bargraphs. This is equivalent to counting r -chimneys (a single column extending beyond its immediate neighbours by at least r cells in a bargraph). We establish generating functions for compositions that avoid or count 2 -chimneys. Thereafter, in the case of bargraphs we provide two methods for obtaining these generating functions as well as asymptotic estimates for the more general r -chimneys where r ≥ 1
{"title":"Chimneys in compositions and bargraphs","authors":"M. Archibald, A. Blecher, A. Knopfmacher, T. Mansour","doi":"10.47443/dml.2023.103","DOIUrl":"https://doi.org/10.47443/dml.2023.103","url":null,"abstract":"Motivated by cut points in graph theory, we consider a similar notion in compositions and bargraphs. This is equivalent to counting r -chimneys (a single column extending beyond its immediate neighbours by at least r cells in a bargraph). We establish generating functions for compositions that avoid or count 2 -chimneys. Thereafter, in the case of bargraphs we provide two methods for obtaining these generating functions as well as asymptotic estimates for the more general r -chimneys where r ≥ 1","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42489012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For an arbitrary invariant ρ ( G ) of a graph G the ρ -edge stability number es ρ ( G ) of G is the minimum number of edges of G whose removal results in a graph H ⊆ G with ρ ( H ) (cid:54) = ρ ( G ) . If such an edge set does not exist, then es ρ ( G ) = ∞ . Gallai’s Theorem states that α (cid:48) ( G ) + β (cid:48) ( G ) = n ( G ) for a graph G without isolated vertices, where α (cid:48) ( G ) is the matching number, β (cid:48) ( G ) the edge covering number, and n ( G ) the order of G . We prove a corresponding result for invariants that are based on the edge stability number es ρ ( G )
{"title":"A Gallai’s Theorem type result for the edge stability of graphs","authors":"A. Kemnitz, M. Marangio","doi":"10.47443/dml.2023.088","DOIUrl":"https://doi.org/10.47443/dml.2023.088","url":null,"abstract":"For an arbitrary invariant ρ ( G ) of a graph G the ρ -edge stability number es ρ ( G ) of G is the minimum number of edges of G whose removal results in a graph H ⊆ G with ρ ( H ) (cid:54) = ρ ( G ) . If such an edge set does not exist, then es ρ ( G ) = ∞ . Gallai’s Theorem states that α (cid:48) ( G ) + β (cid:48) ( G ) = n ( G ) for a graph G without isolated vertices, where α (cid:48) ( G ) is the matching number, β (cid:48) ( G ) the edge covering number, and n ( G ) the order of G . We prove a corresponding result for invariants that are based on the edge stability number es ρ ( G )","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49404332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Erratum to: A common approach to three open problems in number theory","authors":"A. Tyszka","doi":"10.47443/dml.2023.049e","DOIUrl":"https://doi.org/10.47443/dml.2023.049e","url":null,"abstract":"","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46982086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tipaluck Krityakierne, Poohrich Siriputcharoen, T. Thanatipanonda, Chaloemkiat Yapolha
We consider card guessing with no feedback, a variant of the game previously studied by Ciucu in 1998. In this study, we derive an exact, closed-form formula for the asymptotic (in the number of cards, n ) expected number of correct guesses, as well as higher moments, for a one-time riffle shuffle game under the optimal strategy. The problem is tackled using two different approaches: one approach utilizes a fast generating function based on a recurrence relation to obtain numerical moments, while the other is the symbolic approach employing the method of indicators for finding expected counts. The results obtained contribute to the existing literature on card guessing with no feedback.
{"title":"Moments of the one-shuffle no-feedback card guessing game","authors":"Tipaluck Krityakierne, Poohrich Siriputcharoen, T. Thanatipanonda, Chaloemkiat Yapolha","doi":"10.47443/dml.2023.119","DOIUrl":"https://doi.org/10.47443/dml.2023.119","url":null,"abstract":"We consider card guessing with no feedback, a variant of the game previously studied by Ciucu in 1998. In this study, we derive an exact, closed-form formula for the asymptotic (in the number of cards, n ) expected number of correct guesses, as well as higher moments, for a one-time riffle shuffle game under the optimal strategy. The problem is tackled using two different approaches: one approach utilizes a fast generating function based on a recurrence relation to obtain numerical moments, while the other is the symbolic approach employing the method of indicators for finding expected counts. The results obtained contribute to the existing literature on card guessing with no feedback.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47915555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A weak-friendship graph is a connected induced subgraph of a friendship graph. The unique graphs attaining the first two smallest eccentricity spread in the class of weak-friendship graphs of given order are determined in this paper
弱友谊图是友谊图的连通诱导子图。本文确定了一类给定阶的弱友图中偏心距扩展前两个最小的唯一图
{"title":"The eccentricity spread of weak-friendship graphs","authors":"Xuanshi Jia, Xiaohong Li, M. Brunetti","doi":"10.47443/dml.2023.099","DOIUrl":"https://doi.org/10.47443/dml.2023.099","url":null,"abstract":"A weak-friendship graph is a connected induced subgraph of a friendship graph. The unique graphs attaining the first two smallest eccentricity spread in the class of weak-friendship graphs of given order are determined in this paper","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42456363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A classical theorem independently due to Gallai and Roy states that a graph G has a proper k -coloring if and only if G has an orientation without coherent paths of length k . An analogue of this result for signed graphs is proved in this article.
{"title":"On colorings and orientations of signed graphs","authors":"Daniel C. Slilaty","doi":"10.47443/dml.2023.080","DOIUrl":"https://doi.org/10.47443/dml.2023.080","url":null,"abstract":"A classical theorem independently due to Gallai and Roy states that a graph G has a proper k -coloring if and only if G has an orientation without coherent paths of length k . An analogue of this result for signed graphs is proved in this article.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46493381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a distribution of pebbles on the vertices of a graph, a rubbling move places one pebble at a vertex and removes a pebble each at two not necessarily distinct adjacent vertices. One pebble is the cost of transportation. A vertex is t -reachable if at least t pebbles can be moved to the vertex using rubbling moves. The optimal t -rubbling number of a graph is the minimum number of pebbles in a pebble distribution that makes every vertex t -reachable. The optimal t -rubbling numbers of complete graphs and paths are determined.
{"title":"Optimal t-rubbling on complete graphs and paths","authors":"Nándor Sieben","doi":"10.47443/dml.2023.089","DOIUrl":"https://doi.org/10.47443/dml.2023.089","url":null,"abstract":"Given a distribution of pebbles on the vertices of a graph, a rubbling move places one pebble at a vertex and removes a pebble each at two not necessarily distinct adjacent vertices. One pebble is the cost of transportation. A vertex is t -reachable if at least t pebbles can be moved to the vertex using rubbling moves. The optimal t -rubbling number of a graph is the minimum number of pebbles in a pebble distribution that makes every vertex t -reachable. The optimal t -rubbling numbers of complete graphs and paths are determined.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47905824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let D be a digraph with n vertices and let σ 1 ( D ) , σ 2 ( D ) , . . . , σ n ( D ) be the singular values of the adjacency matrix of D , where σ 1 ( D ) ≥ σ 2 ( D ) ≥ · · · ≥ σ n ( D ) . The spectral norm of D is σ 1 ( D ) . In this paper, we determine the orientations of graphs with the first three largest values of the spectral norm over the family of all orientations of bicyclic graphs with at least 12 vertices.
设D是一个有向图，有n个顶点，设σ 1 (D)， σ 2 (D)，…， σ n (D)为D的邻接矩阵的奇异值，其中σ 1 (D)≥σ 2 (D)≥···≥σ n (D)。D的谱范数为σ 1 (D)。在至少有12个顶点的双环图的所有方向族上，我们确定了谱范数前三个最大值的图的方向。
{"title":"The first three largest values of the spectral norm of oriented bicyclic graphs","authors":"Kun Wei, Jianping Li","doi":"10.47443/dml.2023.082","DOIUrl":"https://doi.org/10.47443/dml.2023.082","url":null,"abstract":"Let D be a digraph with n vertices and let σ 1 ( D ) , σ 2 ( D ) , . . . , σ n ( D ) be the singular values of the adjacency matrix of D , where σ 1 ( D ) ≥ σ 2 ( D ) ≥ · · · ≥ σ n ( D ) . The spectral norm of D is σ 1 ( D ) . In this paper, we determine the orientations of graphs with the first three largest values of the spectral norm over the family of all orientations of bicyclic graphs with at least 12 vertices.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45376352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate several measures of peripherality for vertices and edges in networks. We improve asymptotic bounds on the maximum value achieved by edge peripherality, edge sum peripherality, and the Trinajsti'c index over $n$ vertex graphs. We also prove similar results on the maxima over $n$-vertex bipartite graphs, trees, and graphs with a fixed diameter. Finally, we refute two conjectures of Furtula, the first on necessary conditions for minimizing the Trinajsti'c index and the second about maximizing the Trinajsti'c index.
我们研究了网络中顶点和边的几种外围度量。我们改进了在 $n$ 顶点图中，边外围性、边外围性总和以及 Trinajsti'c 指数所达到的最大值的渐进约束。我们还证明了关于 $n$ 顶点双方形图、树和具有固定直径的图的最大值的类似结果。最后，我们反驳了 Furtula 的两个猜想，第一个是关于 Trinajsti'c index 最小化的必要条件，第二个是关于 Trinajsti'c index 最大化的必要条件。
{"title":"Extremal bounds on peripherality measures","authors":"L. Tang","doi":"10.47443/dml.2023.148","DOIUrl":"https://doi.org/10.47443/dml.2023.148","url":null,"abstract":"We investigate several measures of peripherality for vertices and edges in networks. We improve asymptotic bounds on the maximum value achieved by edge peripherality, edge sum peripherality, and the Trinajsti'c index over $n$ vertex graphs. We also prove similar results on the maxima over $n$-vertex bipartite graphs, trees, and graphs with a fixed diameter. Finally, we refute two conjectures of Furtula, the first on necessary conditions for minimizing the Trinajsti'c index and the second about maximizing the Trinajsti'c index.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"77 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139368242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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