Discrete Mathematics最新文献

{"title":"Weak star-Drazin and Drazin-star matrices","authors":"Dijana Mosić ,&nbsp;Daochang Zhang","doi":"10.1016/j.disc.2024.114386","DOIUrl":"10.1016/j.disc.2024.114386","url":null,"abstract":"<div><div>Some systems of matrix equations weaker than existing are considered using a minimal rank weak Drazin inverse and a minimal rank right weak Drazin inverse. In order to solve new systems, we define weak star-Drazin and Drazin-star matrices which are new kinds of square matrices and generalizations of star-Drazin and Drazin-star matrices. Many characterizations and expressions of weak star-Drazin and Drazin-star matrices are proposed. As consequences, we recover known results and present new results about star-Drazin and Drazin-star matrices. Interesting special cases of weak star-Drazin and Drazin-star matrices are studied for the first time in the literature. We apply weak star-Drazin and Drazin-star matrices to solve some linear equations and present their general solution forms.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114386"},"PeriodicalIF":0.7,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Graphs that admit a Hamilton path are cup-stackable","authors":"Petr Gregor ,&nbsp;Arturo Merino ,&nbsp;Torsten Mütze ,&nbsp;Francesco Verciani","doi":"10.1016/j.disc.2024.114375","DOIUrl":"10.1016/j.disc.2024.114375","url":null,"abstract":"<div><div>Fay, Hurlbert and Tennant recently introduced a one-player game on a finite connected graph <em>G</em>, which they called cup stacking. Stacks of cups are placed at the vertices of <em>G</em>, and are transferred between vertices via stacking moves, subject to certain constraints, with the goal of stacking all cups at a single target vertex. If this is possible for every target vertex of <em>G</em>, then <em>G</em> is called <em>stackable</em>. In this paper, we prove that if <em>G</em> admits a Hamilton path, then <em>G</em> is stackable, which confirms several of the conjectures raised by Fay, Hurlbert and Tennant. Furthermore, we prove stackability for certain powers of bipartite graphs, and we construct graphs of arbitrarily large minimum degree and connectivity that do not allow stacking onto any of their vertices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114375"},"PeriodicalIF":0.7,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the diameter of a super-order-commuting graph","authors":"Janko Bračič ,&nbsp;Bojan Kuzma","doi":"10.1016/j.disc.2024.114385","DOIUrl":"10.1016/j.disc.2024.114385","url":null,"abstract":"<div><div>We answer a question about the diameter of an order-super-commuting graph on a symmetric group by studying the number-theoretical concept of <em>d</em>-complete sequences of primes in arithmetic progression.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114385"},"PeriodicalIF":0.7,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Subchromatic numbers of powers of graphs with excluded minors","authors":"Pedro P. Cortés ,&nbsp;Pankaj Kumar ,&nbsp;Benjamin Moore ,&nbsp;Patrice Ossona de Mendez ,&nbsp;Daniel A. Quiroz","doi":"10.1016/j.disc.2024.114377","DOIUrl":"10.1016/j.disc.2024.114377","url":null,"abstract":"<div><div>A <em>k-subcolouring</em> of a graph <em>G</em> is a function <span><math><mi>f</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span> such that the set of vertices coloured <em>i</em> induce a disjoint union of cliques. The <em>subchromatic number</em>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>sub</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the minimum <em>k</em> such that <em>G</em> admits a <em>k</em>-subcolouring. Nešetřil, Ossona de Mendez, Pilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>sub</mtext></mrow></msub><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> when <em>G</em> is planar. We show that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>sub</mtext></mrow></msub><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>≤</mo><mn>43</mn></math></span> when <em>G</em> is planar, improving their bound of 135. We give even better bounds when the planar graph <em>G</em> has larger girth. Moreover, we show that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>sub</mtext></mrow></msub><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo><mo>≤</mo><mn>95</mn></math></span>, improving the previous bound of 364. For these we adapt some recent techniques of Almulhim and Kierstead (2022), while also extending the decompositions of triangulated planar graphs of Van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich and Siebertz (2017), to planar graphs of arbitrary girth. Note that these decompositions are the precursors of the graph product structure theorem of planar graphs.</div><div>We give improved bounds for <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>sub</mtext></mrow></msub><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>)</mo></math></span> for all <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span>, whenever <em>G</em> has bounded treewidth, bounded simple treewidth, bounded genus, or excludes a clique or biclique as a minor. For this we introduce a family of parameters which form a gradation between the strong and the weak colouring numbers. We give upper bounds for these parameters for graphs coming from such classes.</div><div>Finally, we give a 2-approximation algorithm for the subchromatic number of graphs having a layering in which each layer has bounded cliquewidth and this layering is computable in polynomial time (like the class of all <span><math><msup><mrow><mi>d</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math></span> powers of planar graphs, for fixed <em>d</em>). This algorithm works even if the power <em>p</em> and the graph <em>G</em> is unknown.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114377"},"PeriodicalIF":0.7,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Characterizing forbidden pairs for the existence of even factors","authors":"Lei Zhang ,&nbsp;Liming Xiong","doi":"10.1016/j.disc.2024.114384","DOIUrl":"10.1016/j.disc.2024.114384","url":null,"abstract":"<div><div>Let <span><math><mi>H</mi></math></span> be a class of given graphs. We say that a graph is <span><math><mi>H</mi></math></span>-<span><math><mi>f</mi><mi>r</mi><mi>e</mi><mi>e</mi></math></span> if it contains no induced subgraph isomorphic to <em>H</em> for every <span><math><mi>H</mi><mo>∈</mo><mi>H</mi></math></span>. In 2017, the second author characterized all connected graphs <em>H</em> with order at least three such that every <em>H</em>-free graph <em>G</em> has an even factor if and only if <span><math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span> and every odd branch-bond of <em>G</em> has an edge branch. In this paper, we consider the case <em>H</em> is disconnected and characterize all the pairs <span><math><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></math></span> of connected graphs such that every <span><math><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></math></span>-free graph <em>G</em> has an even factor if and only if <span><math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span> and every odd branch-bond of <em>G</em> has an edge branch.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114384"},"PeriodicalIF":0.7,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some bounds on the Laplacian eigenvalues of token graphs","authors":"C. Dalfó ,&nbsp;M.A. Fiol ,&nbsp;A. Messegué","doi":"10.1016/j.disc.2024.114382","DOIUrl":"10.1016/j.disc.2024.114382","url":null,"abstract":"<div><div>The <em>k</em>-token graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> on <em>n</em> vertices is the graph whose vertices are the <span><math><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span> <em>k</em>-subsets of vertices from <em>G</em>, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in <em>G</em>.</div><div>It is known that the algebraic connectivity (or second Laplacian eigenvalue) of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> equals the algebraic connectivity <span><math><mi>α</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em>.</div><div>In this paper, we give some bounds on the (Laplacian) eigenvalues of the <em>k</em>-token graph (including the algebraic connectivity) in terms of the <em>h</em>-token graph, with <span><math><mi>h</mi><mo>≤</mo><mi>k</mi></math></span>. For instance, we prove that if <em>λ</em> is an eigenvalue of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, but not of <em>G</em>, then<span><span><span><math><mi>λ</mi><mo>≥</mo><mi>k</mi><mi>α</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>.</mo></math></span></span></span> As a consequence, we conclude that if <span><math><mi>α</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>k</mi></math></span>, then <span><math><mi>α</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>α</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for every <span><math><mi>h</mi><mo>≤</mo><mi>k</mi></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114382"},"PeriodicalIF":0.7,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Linked partition ideals and overpartitions","authors":"Nancy S.S. Gu,&nbsp;Kuo Yu","doi":"10.1016/j.disc.2024.114380","DOIUrl":"10.1016/j.disc.2024.114380","url":null,"abstract":"<div><div>Linked partition ideals which were first introduced by Andrews have recently appeared in a series of works to study generating functions for partitions. Recently, Andrews found some relations between a certain kind of overpartitions and 4-regular partitions into distinct parts. Then with the aid of linked partition ideals for overpartitions, Andrews and Chern established a general relation between these two sets of partitions. Motivated by their work, we consider the overpatitions denoted by <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> satisfying the following conditions: (1) Only odd parts may be overlined; (2) The difference between any two parts is <span><math><mo>⩾</mo><mn>2</mn><mi>k</mi></math></span> where the inequality is strict if the larger one is overlined. Let <em>S</em> be a set of given parts. Then <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math></span> denotes the subset of overpartitions in <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> where parts from <em>S</em> are forbidden. Combining linked partition ideals and a recurrence relation for a family of multiple series given by Chern, we study the generating functions for <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math></span> for some given <em>S</em>. Furthermore, by establishing a <em>q</em>-series identity, we find a relation between <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mo>{</mo><mover><mrow><mn>1</mn></mrow><mo>‾</mo></mover><mo>}</mo></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> and distinct partitions. Meanwhile, some statistics on partitions are discussed.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114380"},"PeriodicalIF":0.7,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On 2-distance 16-coloring of planar graphs with maximum degree at most five","authors":"Zakir Deniz","doi":"10.1016/j.disc.2024.114379","DOIUrl":"10.1016/j.disc.2024.114379","url":null,"abstract":"<div><div>A vertex coloring of a graph <em>G</em> is called a 2-distance coloring if any two vertices at a distance at most 2 from each other receive different colors. Suppose that <em>G</em> is a planar graph with a maximum degree at most 5. We prove that <em>G</em> admits a 2-distance 16-coloring, which improves the result given by Zou et al. (2024) <span><span>[13]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114379"},"PeriodicalIF":0.7,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143169279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Determinantal decomposition of graphs with a unique perfect matching","authors":"Daniel A. Jaume ,&nbsp;Diego G. Martinez ,&nbsp;Cristian Panelo","doi":"10.1016/j.disc.2024.114370","DOIUrl":"10.1016/j.disc.2024.114370","url":null,"abstract":"<div><div>In this work, a structural decomposition of graphs with a unique perfect matching is introduced. The decomposition is given by the barbell subgraphs: even subdivisions of two <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> graphs joined by an edge such that the unique perfect matching of <em>G</em> induces a perfect matching in the subgraphs. The decomposition breaks a graph <em>G</em>, with a unique perfect matching, into two subgraphs, one of which is a Kőnig-Egerváry graph. Furthermore, the decomposition is shown to be multiplicative with respect to determinantal-type (Schur) functions of the adjacency matrix of graphs with a unique perfect matching. Additionally, in this work, Godsil's formula for the determinant of trees with perfect matching is extended to all graphs with a unique perfect matching.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114370"},"PeriodicalIF":0.7,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143170181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Further results and questions on S-packing coloring of subcubic graphs","authors":"Maidoun Mortada ,&nbsp;Olivier Togni","doi":"10.1016/j.disc.2024.114376","DOIUrl":"10.1016/j.disc.2024.114376","url":null,"abstract":"<div><div>For a non-decreasing sequence of integers <span><math><mi>S</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, an <em>S</em>-packing coloring of <em>G</em> is a partition of <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> into <em>k</em> subsets <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that the distance between any two distinct vertices <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is at least <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mn>1</mn></math></span>, <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span>. We consider the <em>S</em>-packing coloring problem on subclasses of subcubic graphs: For <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>3</mn></math></span>, a subcubic graph <em>G</em> is said to be <em>i</em>-saturated if every vertex of degree 3 is adjacent to at most <em>i</em> vertices of degree 3. Furthermore, a vertex of degree 3 in a subcubic graph is called heavy if all its three neighbors are of degree 3, and <em>G</em> is said to be <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mi>i</mi><mo>)</mo></math></span>-saturated if every heavy vertex is adjacent to at most <em>i</em> heavy vertices. We prove that every 1-saturated subcubic graph is <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>-packing colorable and <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-packing colorable. We also prove that every <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-saturated subcubic graph is <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-packing colorable.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114376"},"PeriodicalIF":0.7,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143169797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}