Discrete Applied Mathematics最新文献

{"title":"A note on the generalized Turán number of star forests","authors":"Yu-Yue Zhang, Jian-Hua Yin","doi":"10.1016/j.dam.2025.12.032","DOIUrl":"10.1016/j.dam.2025.12.032","url":null,"abstract":"<div><div>The generalized Turán number <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is defined to be the maximum number of copies of a complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> in any <span><math><mi>H</mi></math></span>-free graph on <span><math><mi>n</mi></math></span> vertices. Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denote the star on <span><math><mrow><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices, and let <span><math><mrow><mi>k</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></mrow></math></span> denote the disjoint union of <span><math><mi>k</mi></math></span> copies of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span>. Gan et al. and Chase determined <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for all integers <span><math><mrow><mi>s</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. Recently, Liu and Yin further investigated the problem of determining the generalized Turán number of star forests. They determined <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>2</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>4</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>3</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>5</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. Moreover, they also determined <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>4</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>6</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1<","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 94-112"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792204","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":"Forbidden configurations and dominating bicliques in undirected 2-quasi best match graphs","authors":"Annachiara Korchmaros ,&nbsp;Peter F. Stadler","doi":"10.1016/j.dam.2026.01.001","DOIUrl":"10.1016/j.dam.2026.01.001","url":null,"abstract":"<div><div>2-quasi best match graphs (2qBMGs) are directed graphs that capture a notion of close relatedness in phylogenetics. Here, we investigate the underlying undirected graph of a 2qBMG (un2qBMG) and show that it contains neither a path <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> nor a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> of length <span><math><mrow><mi>l</mi><mo>≥</mo><mn>6</mn></mrow></math></span> as an induced subgraph. This property guarantees the existence of specific vertex decompositions with dominating bicliques that provide further insights into their structure.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 308-314"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927070","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":"Non-adaptive prophet inequalities for minor-closed classes of matroids","authors":"Kanstantsin Pashkovich,&nbsp;Alice Sayutina","doi":"10.1016/j.dam.2025.12.001","DOIUrl":"10.1016/j.dam.2025.12.001","url":null,"abstract":"<div><div>We consider the matroid prophet inequality problem. This problem has been extensively studied in the case of adaptive mechanisms. In particular, there is a tight 2-competitive mechanism for all matroids (Kleinberg and Weinberg, 2012).</div><div>However, it is not known what classes of matroids admit non-adaptive mechanisms with constant guarantee. Recently, in Chawla et al. (2024) it was shown that there are constant-competitive non-adaptive mechanisms for graphic matroids. In this work, we show that various known classes of matroids admit constant-competitive non-adaptive mechanisms.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 26-43"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145748180","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":"Even cycle decompositions of the line graphs of cubic graphs","authors":"Chuixiang Zhou,&nbsp;Yuxi Zou","doi":"10.1016/j.dam.2025.12.052","DOIUrl":"10.1016/j.dam.2025.12.052","url":null,"abstract":"<div><div>An even cycle decomposition of a graph is defined as a partition of its edges into cycles of even length. Let <span><math><mi>G</mi></math></span> be a 2-connected cubic graph. Markström conjectured that the line graph <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span> admits an even cycle decomposition. Suppose that <span><math><mrow><mi>F</mi><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> is a 2-factor of <span><math><mi>G</mi></math></span>, where each <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>\u0000 (<span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></math></span>) is a cycle. Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> be the graph obtained from <span><math><mi>G</mi></math></span> by contracting each <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>\u0000 (<span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></math></span>) to a vertex and deleting the resulting loops and parallel edges. In this paper, we prove that Markström’s conjecture is true if <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> is isomorphic to a tree for some 2-factor <span><math><mi>F</mi></math></span> of <span><math><mi>G</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 300-307"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927069","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":"Non-inclusive g-extra diagnosability of interconnection networks under MM* model","authors":"Weixing Zheng ,&nbsp;Shuming Zhou ,&nbsp;Eddie Cheng","doi":"10.1016/j.dam.2025.12.069","DOIUrl":"10.1016/j.dam.2025.12.069","url":null,"abstract":"<div><div>With the popularization and deepening of applications on high-performance computing platforms, which are often built upon large-scale multiprocessor systems, system-level diagnosis has become essential to ensure system reliability and fault tolerability. Classical diagnosability and various conditional diagnosabilities are vital metrics to evaluate a system’s capability to accurately identify faulty processors. However, the upper bounds of these metrics commonly rely on the conventional assumptions of inclusiveness between faulty sets—one situation that rarely holds in practical scenarios. To overcome this deficiency, the non-inclusive diagnosability has been introduced and explored under various diagnostic models. Despite significant progress, the study of non-inclusive <span><math><mi>g</mi></math></span>-extra diagnosability under the MM* model remains up in the air. This work addresses this gap by determining the non-inclusive 2-extra diagnosability of general interconnection networks under the MM* model. As applications, we derive the non-inclusive 2-extra diagnosabilities of several well-known networks, including hypercube, <span><math><mi>k</mi></math></span>-ary <span><math><mi>n</mi></math></span>-cube, and bubble sort graph. In addition, we propose a novel diagnosis algorithm, NFDAM, for non-inclusive <span><math><mi>g</mi></math></span>-extra diagnosis, which runs in polynomial time with a complexity of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo><mi>⋅</mi><msup><mrow><mrow><mo>(</mo><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denotes the maximum degree of graph <span><math><mi>G</mi></math></span>. Simulation results demonstrate the effectiveness of the proposed algorithm in fault identification.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 315-326"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977710","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 tree–wheel Ramsey numbers","authors":"Yanbo Zhang ,&nbsp;Yaojun Chen ,&nbsp;Yunqing Zhang","doi":"10.1016/j.dam.2025.12.007","DOIUrl":"10.1016/j.dam.2025.12.007","url":null,"abstract":"<div><div>Given two graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, the Ramsey number <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the smallest positive integer <span><math><mi>r</mi></math></span> such that every graph on <span><math><mi>r</mi></math></span> vertices contains <span><math><mi>G</mi></math></span> as a subgraph or its complement contains <span><math><mi>H</mi></math></span> as a subgraph. Let <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote a tree on <span><math><mi>n</mi></math></span> vertices, and let <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> denote a wheel on <span><math><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices. Chen et al. (2004) asked for a characterization of those trees <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfying <span><math><mrow><mi>R</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> for <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span>. Subsequently, Hafidh and Baskoro (2021) conjectured that if <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a tree with maximum degree at most <span><math><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>2</mn></mrow></math></span>, then <span><math><mrow><mi>R</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> for <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></math></span>. More recently, Britz et al. (2025) verified this conjecture partially for <span><math><mrow><mi>m</mi><mo>=</mo><mn>4</mn></mrow></math></span> and large <span><math><mi>n</mi></math></span>.</div><div>In this note, we resolve the conjecture of Hafidh and Baskoro for all large <span><math><mi>n</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 80-84"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792203","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":"ILP and CP models for normalized integer weighted voting game design problem","authors":"Roger Kameugne ,&nbsp;Pierre Wambo ,&nbsp;Yves Pascal Ndjopnang Wantiep","doi":"10.1016/j.dam.2025.11.051","DOIUrl":"10.1016/j.dam.2025.11.051","url":null,"abstract":"<div><div>Weighted voting games are a family of cooperative games, where each voter or player carries a weight. If the sum of the weights of the players in favor of a proposal is larger than or equal to a given quota, then this proposal is accepted. In the weighted game field, the problem of normalized integer weighted voting game representation (NIWVG), and the integer weighted voting game design problem (IWVGD) or inverse power index problem have been subjects of several studies. A solution to the last problem is an integer-weighted voting game with a specific power index, as close as possible (following a norm) to a given influence vector. Finding a normalized integer-weighted voting game solution for the inverse power index problem is a challenging task. The nature of the power index and the norm awarded in the problem can impact the approach used to solve this problem. In this paper, exact approaches based on integer linear programming (ILP) and constraint programming (CP) are proposed for this problem. The models proposed in this paper combine the Shapley–Shubik index or Banzhaf–Coleman index with Manhattan and Chebyshev norms respectively. Computational results show that ILP is an adequate approach when the power index and the norm can be linearized while CP appears to be suitable for the remaining cases of voting games with a small number of players.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 197-209"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685318","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":"IP.LSH.DBSCAN: Integrated parallel density-based clustering by locality-sensitive hashing","authors":"Amir Keramatian,&nbsp;Vincenzo Gulisano,&nbsp;Marina Papatriantafilou,&nbsp;Philippas Tsigas","doi":"10.1016/j.dam.2025.11.047","DOIUrl":"10.1016/j.dam.2025.11.047","url":null,"abstract":"<div><div>Locality-sensitive hashing (LSH) is an established method for fast data indexing and approximate similarity search, with useful parallelism properties. Although indexes and similarity measures are key for data clustering, little has been investigated on the multifaceted benefits of LSH in the problem. We show how approximate DBSCAN clustering can be <em>fused</em> into the process of creating an LSH index, and, through parallelization and fine-grained synchronization, also utilize efficiently available computing capacity. The resulting algorithm, <span><math><mstyle><mi>I</mi><mi>P</mi><mo>.</mo><mi>L</mi><mi>S</mi><mi>H</mi><mo>.</mo><mi>D</mi><mi>B</mi><mi>S</mi><mi>C</mi><mi>A</mi><mi>N</mi></mstyle></math></span>, described in this article, can support a wide range of applications with diverse distance functions, as well as data distributions and dimensionality. We analyse the algorithm’s asymptotic completion time and provide an open-source prototype implementation. We also conduct a detailed evaluation measuring latency and accuracy metrics of <span><math><mstyle><mi>I</mi><mi>P</mi><mo>.</mo><mi>L</mi><mi>S</mi><mi>H</mi><mo>.</mo><mi>D</mi><mi>B</mi><mi>S</mi><mi>C</mi><mi>A</mi><mi>N</mi></mstyle></math></span>, on a 36-core machine with 2-way hyper threading on massive data-sets with various numbers of dimensions. The analysis and the empirical study of <span><math><mstyle><mi>I</mi><mi>P</mi><mo>.</mo><mi>L</mi><mi>S</mi><mi>H</mi><mo>.</mo><mi>D</mi><mi>B</mi><mi>S</mi><mi>C</mi><mi>A</mi><mi>N</mi></mstyle></math></span> show how it complements the landscape of established state-of-the-art methods, by offering up to several orders of magnitude speed-up on higher dimensional datasets, with tunable high clustering accuracy.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 183-196"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685323","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":"Lines on digraphs of low diameter","authors":"Gabriela Araujo-Pardo ,&nbsp;Martín Matamala ,&nbsp;Juan Pablo Peña ,&nbsp;José Zamora","doi":"10.1016/j.dam.2025.12.015","DOIUrl":"10.1016/j.dam.2025.12.015","url":null,"abstract":"<div><div>A set of <span><math><mi>n</mi></math></span> non-collinear points in the Euclidean plane defines at least <span><math><mi>n</mi></math></span> different lines. Chen and Chvátal in 2008 conjectured that the same result is true in metric spaces for an adequate definition of line. More recently, it was conjectured in 2018 by Aboulker et al. that any large enough bridgeless graph on <span><math><mi>n</mi></math></span> vertices defines a metric space that has at least <span><math><mi>n</mi></math></span> lines.</div><div>We study the natural extension of Aboulker et al.’s conjecture into the context of quasi-metric spaces defined by digraphs of low diameter. We prove that it is valid for quasi-metric spaces defined by bipartite digraphs of diameter at most three, oriented graphs of diameter two and, digraphs of diameter three and directed girth four.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 293-300"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737341","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":"Note on two Kirchhoff index Turán problems","authors":"Haiyan Chen ,&nbsp;Wei Qiu","doi":"10.1016/j.dam.2025.11.007","DOIUrl":"10.1016/j.dam.2025.11.007","url":null,"abstract":"<div><div>The Kirchhoff index of a connected graph is the sum of the resistance distance between all pairs of vertices. Bu et al. (<em>Bounds on the resistance distance and Kirchhoff index of graphs, Discrete Appl. Math. 2025, 234-241</em>) proved that (i) if <span><math><mi>n</mi></math></span> is divisible by <span><math><mi>t</mi></math></span>, then the Turán graph <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is the unique extremal graph with the minimum Kirchhoff index among all connected <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graphs with <span><math><mi>n</mi></math></span> vertices; (ii) if <span><math><mi>n</mi></math></span> is even, then the Turán graph <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is the unique extremal graph with the minimum Kirchhoff index among all connected <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graphs with <span><math><mi>n</mi></math></span> vertices <span><math><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>4</mn><mi>t</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></math></span>. They conjectured that the above two results hold for all <span><math><mi>n</mi></math></span>. In this note, we confirm their conjectures.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 28-32"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145555234","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}