Discrete Applied Mathematics最新文献

{"title":"On partitioning a bipartite graph into cycles and degenerated cycles","authors":"Shuya Chiba ,&nbsp;Koshin Yoshida","doi":"10.1016/j.dam.2025.09.003","DOIUrl":"10.1016/j.dam.2025.09.003","url":null,"abstract":"<div><div>For a bipartite graph <span><math><mi>G</mi></math></span>, let <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the minimum degree sum of two non-adjacent vertices in different partite sets of <span><math><mi>G</mi></math></span>. We prove the following result: If <span><math><mi>G</mi></math></span> is a balanced bipartite graph of order <span><math><mrow><mn>2</mn><mi>n</mi><mo>≥</mo><mi>k</mi></mrow></math></span>, and if <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>n</mi><mo>−</mo><mfenced><mrow><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></math></span>, then one of the following (i)–(iv) holds: (i) <span><math><mi>G</mi></math></span> contains <span><math><mi>k</mi></math></span> vertex-disjoint subgraphs <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> such that <span><math><mrow><msub><mrow><mo>⋃</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></msub><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and for each <span><math><mi>i</mi></math></span>, <span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></math></span>, <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a cycle or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>; (ii) <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span>; (iii) <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>8</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math></span>; (iv) <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>10</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>=</mo><mn>4</mn></mrow></math></span>. This result is a bipartite graph version of the result of Enomoto and Li (2004). We actually prove a stronger result which gives us control on the number of cycles in the <span><math><mi>k</mi></math></span> vertex-disjoint subgraphs of (i).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 635-646"},"PeriodicalIF":1.0,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145105020","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":"Cover numbers by certain graph families","authors":"Márton Marits","doi":"10.1016/j.dam.2025.09.009","DOIUrl":"10.1016/j.dam.2025.09.009","url":null,"abstract":"<div><div>We define the <em>cover number</em> of a graph <span><math><mi>G</mi></math></span> by a graph class <span><math><mi>P</mi></math></span> as the minimum number of graphs of class <span><math><mi>P</mi></math></span> required to cover the edge set of <span><math><mi>G</mi></math></span>. Taking inspiration from a paper by Harary et al. (1977), we find an exact formula for the cover number by the graph classes <span><math><mrow><mo>{</mo><mi>G</mi><mo>∣</mo><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>f</mi><mrow><mo>(</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>}</mo></mrow></math></span> for all non-decreasing functions <span><math><mi>f</mi></math></span>.</div><div>After this, we establish a chain of inequalities between five cover numbers, the one by the class <span><math><mrow><mo>{</mo><mi>G</mi><mo>∣</mo><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>}</mo></mrow></math></span>, by the class of perfect graphs, generalized split graphs, co-unipolar graphs and finally the cover number by bipartite graphs. We prove that at each inequality, the difference between the two sides can grow arbitrarily large. We also prove that the cover number by unipolar graphs cannot be expressed in terms of the chromatic or the clique number.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 400-404"},"PeriodicalIF":1.0,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145105085","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 the tractability of defensive alliance problem","authors":"Sangam Balchandar Reddy,&nbsp;Anjeneya Swami Kare","doi":"10.1016/j.dam.2025.09.008","DOIUrl":"10.1016/j.dam.2025.09.008","url":null,"abstract":"<div><div>Given a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span>, a non-empty set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> is a defensive alliance if, for every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>S</mi></mrow></math></span>, the majority of vertices in its closed neighbourhood belong to <span><math><mi>S</mi></math></span>; that is, <span><math><mrow><mrow><mo>|</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><mo>∩</mo><mi>S</mi><mo>|</mo></mrow><mo>≥</mo><mrow><mo>|</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><mo>∖</mo><mi>S</mi><mo>|</mo></mrow></mrow></math></span>. The Defensive Alliance problem (<span>Defensive Alliance</span>) asks for a defensive alliance of minimum cardinality. The decision version of the problem is known to be NP-complete even when restricted to split graphs and bipartite graphs. From a parameterized complexity perspective, the <span>Defensive Alliance</span> is known to be fixed-parameter tractable (FPT) when parameterized by the solution size, the vertex cover number, or the neighbourhood diversity of the input graph. In contrast, the problem is W[1]-hard when parameterized by the treewidth or the feedback vertex set number.</div><div>In this paper, we investigate the complexity of the <span>Defensive Alliance</span> on bounded degree graphs. We prove that the problem is <em>polynomial-time solvable</em> on graphs with maximum degree at most 5 but becomes NP-complete when the maximum degree is 6. This result rules out fixed-parameter tractability with respect to the maximum degree. Additionally, we analyse the problem from the perspective of parameterized complexity and present an FPT algorithm parameterized by twin cover number, thereby resolving an open question posed in Gaikwad and Maity (2022).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 116-127"},"PeriodicalIF":1.0,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097619","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":"The toll walk transit function of a graph: Axiomatic characterizations and first-order non-definability","authors":"Manoj Changat ,&nbsp;Jeny Jacob ,&nbsp;Lekshmi Kamal K. Sheela ,&nbsp;Iztok Peterin","doi":"10.1016/j.dam.2025.09.006","DOIUrl":"10.1016/j.dam.2025.09.006","url":null,"abstract":"<div><div>A walk <span><math><mrow><mi>W</mi><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span>, <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, is called a toll walk if <span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> are the only neighbors of <span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> on <span><math><mi>W</mi></math></span> in a graph <span><math><mi>G</mi></math></span>. A toll walk interval <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, contains all the vertices that belong to a toll walk between <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span>. The toll walk intervals yield a toll walk transit function <span><math><mrow><mi>T</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>×</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msup></mrow></math></span>. We represent several axioms that characterize the toll walk transit function among chordal graphs, trees, asteroidal triple-free graphs, Ptolemaic graphs, and distance-hereditary graphs. We also show that the toll walk transit function cannot be described in the language of first-order logic for an arbitrary graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 128-145"},"PeriodicalIF":1.0,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097642","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":"Berge coalitional stabilities in the graph model for conflict resolution","authors":"Giannini Italino Alves Vieira ,&nbsp;Leandro Chaves Rêgo","doi":"10.1016/j.dam.2025.09.012","DOIUrl":"10.1016/j.dam.2025.09.012","url":null,"abstract":"<div><div>Altruism is a behavior that is commonly observed in human interactions. The concept of Berge stability has been introduced in game theory and, more recently, in the graph model for conflict resolution, to represent decision makers (DMs) that act altruistically expecting others to reciprocate. However, this stability concept has only been introduced from the point of view of individual DMs. This paper incorporates the concept of Berge stability into the framework of coalition analysis within the graph model for conflict resolution. In particular, it introduces nine novel concepts for coalition analysis and thoroughly examines the relationships among these new concepts, as well as their connections to other coalition concepts found in the existing literature on this model. To illustrate the use in practice of the proposed stability concepts, a coalitional Berge stability analysis is conducted in the Elmira groundwater contamination conflict.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 405-418"},"PeriodicalIF":1.0,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145105087","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":"Odd coloring of 2-boundary planar graphs and beyond","authors":"Weichan Liu ,&nbsp;Mengke Qi ,&nbsp;Xin Zhang","doi":"10.1016/j.dam.2025.08.064","DOIUrl":"10.1016/j.dam.2025.08.064","url":null,"abstract":"<div><div>In this paper, we introduce the notion of 2-boundary planar graphs. A graph is 2-boundary planar if it has an embedding in the plane so that all vertices lie on the boundary of at most two faces and no edges are crossed. A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. Petruševski and Škrekovski conjectured in 2022 that every planar graph admits an odd 5-coloring. We confirm this conjecture for 2-boundary planar graphs. Moreover, we present several questions regarding 2-boundary planar graphs that are of independent interest.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 68-79"},"PeriodicalIF":1.0,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097643","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":"The maximum number of cliques in graphs with given fractional matching number and minimum degree","authors":"Chengli Li,&nbsp;Yurui Tang","doi":"10.1016/j.dam.2025.09.010","DOIUrl":"10.1016/j.dam.2025.09.010","url":null,"abstract":"<div><div>Recently, Ma, Qian and Shi determined the maximum size of an <span><math><mi>n</mi></math></span>-vertex graph with given fractional matching number <span><math><mi>s</mi></math></span> and maximum degree at most <span><math><mi>d</mi></math></span>. Motivated by this result, we determine the maximum number of <span><math><mi>ℓ</mi></math></span>-cliques in a graph with given fractional matching number and minimum degree, which generalizes Shi and Ma’s result about the maximum size of a graph with given fractional matching number and minimum degree at least one. We also determine the maximum number of complete bipartite graphs in a graph with prescribed fractional matching number and minimum degree.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 390-399"},"PeriodicalIF":1.0,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145105088","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":"Single machine scheduling with a restricted rate-modifying activity to minimize the weighted makespan","authors":"Lili Zuo,&nbsp;Jing Zhang,&nbsp;Zhan Shi,&nbsp;Bingbing Fan","doi":"10.1016/j.dam.2025.09.013","DOIUrl":"10.1016/j.dam.2025.09.013","url":null,"abstract":"<div><div>We investigate the single machine scheduling problem with a restricted rate-modifying activity (RMA) aimed at minimizing the weighted makespan. The RMA is an activity that modifies the machine’s production rate while occupying it for a specified duration. Importantly, its starting time is constrained to fall within a predetermined interval <span><math><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></math></span>. Our results demonstrate that the special case where parameter <span><math><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow></math></span> admits a polynomial-time solution, while the cases with <span><math><mrow><mi>b</mi><mo>=</mo><mi>∞</mi></mrow></math></span> or <span><math><mrow><mi>b</mi><mo>=</mo><mi>a</mi></mrow></math></span> are proven to be binary NP-hard. For these NP-hard problems, we develop pseudo-polynomial time dynamic programming solutions. Notably, in the scenario where splitting is permitted, we establish a polynomial-time algorithm for the <span><math><mrow><mi>b</mi><mo>=</mo><mi>a</mi></mrow></math></span> case, which in turn enables the derivation of a 2-approximation algorithm for the non-split version. Additionally, by combining our dynamic programming approach with the vector trimming technique, we achieve fully polynomial-time approximation schemes (FPTAS) for all NP-hard variants under consideration.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 89-100"},"PeriodicalIF":1.0,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097617","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 sign-invertible graphs","authors":"Isaiah Osborne ,&nbsp;Dong Ye","doi":"10.1016/j.dam.2025.09.005","DOIUrl":"10.1016/j.dam.2025.09.005","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph and <span><math><mi>A</mi></math></span> be its adjacency matrix. A graph <span><math><mi>G</mi></math></span> is invertible if its adjacency matrix <span><math><mi>A</mi></math></span> is invertible and the inverse of <span><math><mi>G</mi></math></span> is a weighted graph with adjacency matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. A signed graph <span><math><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></mrow></math></span> is a weighted graph with a special weight function <span><math><mrow><mi>σ</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>. A graph is sign-invertible if its inverse is a signed graph. A sign-invertible graph is always unimodular. The inverses of graphs have interesting combinatorial interests. In this paper, we study inverses of graphs and provide a combinatorial description for sign-invertible graphs, which provides a tool to characterize sign-invertible graphs. As applications, we completely characterize sign-invertible bipartite graphs with a unique perfect matching, and sign-invertible graphs with cycle rank at most two. As corollaries of these characterizations, some early results on trees (Buckley, Doty and Harary in 1982) and unicyclic graphs with a unique perfect matching (Kalita and Sarma in 2022) follow directly.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 101-115"},"PeriodicalIF":1.0,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097644","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":"Interlacing properties of Laplacian eigenvalues of chain graphs","authors":"Milica Anđelić ,&nbsp;Zoran Stanić ,&nbsp;Fernando C. Tura","doi":"10.1016/j.dam.2025.09.007","DOIUrl":"10.1016/j.dam.2025.09.007","url":null,"abstract":"<div><div>Chain graphs are <span><math><mrow><mo>{</mo><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>}</mo></mrow></math></span>-free graphs. The Laplacian spectrum of a chain graph of order <span><math><mi>n</mi></math></span> consists of <span><math><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>h</mi></mrow></math></span> integer eigenvalues and <span><math><mrow><mn>2</mn><mi>h</mi></mrow></math></span> possibly non-integer eigenvalues that correspond to the associated quotient matrix of order <span><math><mrow><mn>2</mn><mi>h</mi></mrow></math></span>. We show that <span><math><mrow><mn>2</mn><mi>h</mi></mrow></math></span> complementary eigenvalues interlace vertex degrees. As an application, we confirm that the Brouwer’s conjecture holds for chain graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 80-88"},"PeriodicalIF":1.0,"publicationDate":"2025-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145061296","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}