{"title":"Perfect matchings of (4,6)-fullerenes with largest forcing number","authors":"Yaxian Zhang, Heping Zhang","doi":"10.1016/j.dam.2025.04.004","DOIUrl":"10.1016/j.dam.2025.04.004","url":null,"abstract":"<div><div>Clar number (or resonant number) is a thoroughly investigated parameter of plane graphs emerging from mathematical chemistry to measure stability of some organic molecules. It was shown that the Clar number of a <span><math><mrow><mo>(</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></math></span>-fullerene graph <span><math><mi>G</mi></math></span>, a plane cubic graph with only hexagonal and quadrilateral faces, is equal to its maximum forcing number <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mi>M</mi></math></span> be any perfect matching of <span><math><mi>G</mi></math></span> attaining the maximum forcing number. We use <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> to denote two largest sets of disjoint <span><math><mi>M</mi></math></span>-alternating cycles and <span><math><mi>M</mi></math></span>-alternating facial cycles respectively. Then <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≥</mo><mrow><mo>|</mo><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>. In this paper, we consider when <span><math><mrow><mrow><mo>|</mo><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> holds. First we show that every cycle in <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> has length 4, 6, 8 or 12. Then we construct two types of tubular <span><math><mrow><mo>(</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></math></span>-fullerene graphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> so that there is a <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> containing a 12-cycle if and only if <span><math><mrow><mi>G</mi><mo>∈</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>, and there is a <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> containing a 8-cycle if and only if <span><math><mrow><mi>G</mi><mo>∈</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>. As a consequence, we obtain the following three equivalent statements: (i) Each <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 10-25"},"PeriodicalIF":1.0,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143851791","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":"Matching polynomials of path-trees of a complete bipartite graph","authors":"Haiyan Chen , Yinxia Yuan","doi":"10.1016/j.dam.2025.04.014","DOIUrl":"10.1016/j.dam.2025.04.014","url":null,"abstract":"<div><div>Suppose that the vertex set of the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> is <span><math><mrow><mi>X</mi><mo>∪</mo><mi>Y</mi></mrow></math></span>, where <span><math><mrow><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mi>s</mi><mo>≤</mo><mi>t</mi><mo>=</mo><mo>|</mo><mi>Y</mi><mo>|</mo></mrow></math></span>. Let <span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>w</mi></mrow></msubsup></math></span> denote the path-tree of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> corresponding to vertex <span><math><mrow><mi>w</mi><mo>∈</mo><mi>X</mi><mo>∪</mo><mi>Y</mi></mrow></math></span>. Then we show that the matching polynomial <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><msubsup><mrow><mi>T</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>w</mi></mrow></msubsup><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is equal to <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mi>⋅</mi></mrow></math></span>\u0000 <span><math><mfenced><mrow><mtable><mtr><mtd><msup><mrow><mfenced><mrow><munderover><mrow><mo>∏</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mrow><mfenced><mrow><mi>μ</mi><msup><mrow><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>α</mi><mo>+</mo><mi>k</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>μ</mi><msup><mrow><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>α</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mfrac><mrow><mi>α</mi><mo>+</mo><mi>k</mi></mrow><mrow><mi>α</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></msup></mrow></mfenced></mrow><mrow><msubsup><mrow><mi>β</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></msup></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mo>,</mo></mtd><mtd><mspace></mspace><mtext> if </mtext><mi>w</mi><mo>∈</mo><mi>X</mi><mo>;</mo></mtd></mtr><mtr><mtd><mi>μ</mi><msup><mrow><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mfenced><mrow><munderover><mrow><mo>∏</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mrow><mfenced><mrow><mi>μ</mi><msup><mrow><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>α</mi><mo>+</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 173-179"},"PeriodicalIF":1.0,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143851864","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":"Unit dual quaternion directed graphs, formation control and general weighted directed graphs","authors":"Liqun Qi , Chunfeng Cui , Chen Ouyang","doi":"10.1016/j.dam.2025.04.027","DOIUrl":"10.1016/j.dam.2025.04.027","url":null,"abstract":"<div><div>We study the multi-agent formation control problem in a directed graph. The relative configurations are expressed by unit dual quaternions (UDQs). We call such a weighted directed graph a unit dual quaternion directed graph (UDQDG). We show that a desired relative configuration scheme is reasonable or balanced in a UDQDG if and only if there is a diagonal matrix with UDQ diagonal elements such that the dual quaternion Laplacian is similar to the unweighted Laplacian of the underlying directed graph. A direct method and a unit gain graph method are proposed to solve the balance problem of general unit weighted directed graphs. We then study the balance problem of general non-unit weighted directed graphs. Numerical experiments for UDQDG are reported.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 197-209"},"PeriodicalIF":1.0,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143856013","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":"An (F3,F4)-partition of planar graphs without 4- and 6-cycles","authors":"Kaiyang Hu, Mingfang Huang","doi":"10.1016/j.dam.2025.04.018","DOIUrl":"10.1016/j.dam.2025.04.018","url":null,"abstract":"<div><div>An <span><math><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>)</mo></mrow></math></span>-partition of a graph <span><math><mi>G</mi></math></span> is a partition of its vertices set into <span><math><mi>k</mi></math></span> subsets <span><math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> where each <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> induces a forest with maximum degree at most <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span>. Cho et al.(2021) proved that every planar graph without 4- and 5-cycles admits an <span><math><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></mrow></math></span>-partition. In this paper, we show that every planar graph without 4- and 6-cycles admits an <span><math><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></mrow></math></span>-partition, which strengthens a previous result due to Nakprasit et al. (2024) in a stronger form.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 1-9"},"PeriodicalIF":1.0,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143851792","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":"Two-disjoint-cycle-cover pancyclicity of augmented cubes","authors":"Dongqin Cheng","doi":"10.1016/j.dam.2025.04.033","DOIUrl":"10.1016/j.dam.2025.04.033","url":null,"abstract":"<div><div>A graph <span><math><mi>G</mi></math></span> is two-disjoint-cycle-cover <span><math><mrow><mo>[</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo></mrow></math></span>-pancyclic if there is a collection of two vertex disjoint cycles <span><math><mi>C</mi></math></span> of length <span><math><mrow><mi>l</mi><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of length <span><math><mrow><mi>l</mi><mrow><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> such that <span><math><mrow><mi>l</mi><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow><mo>+</mo><mi>l</mi><mrow><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mi>l</mi><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>. The augmented cube is one of the variations of hypercube and possesses many good properties that the hypercube does not have. In this paper, we prove that the <span><math><mi>n</mi></math></span>-dimensional augmented cube <span><math><mrow><mi>A</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> is two-disjoint-cycle-cover <span><math><mrow><mo>[</mo><mn>3</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></mrow></math></span>-pancyclic, where <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 240-246"},"PeriodicalIF":1.0,"publicationDate":"2025-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143850568","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":"Proof of a conjecture on isolation of graphs dominated by a vertex","authors":"Peter Borg","doi":"10.1016/j.dam.2025.04.009","DOIUrl":"10.1016/j.dam.2025.04.009","url":null,"abstract":"<div><div>A copy of a graph <span><math><mi>F</mi></math></span> is called an <span><math><mi>F</mi></math></span>-copy. For any graph <span><math><mi>G</mi></math></span>, the <span><math><mi>F</mi></math></span>-isolation number of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>ι</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, is the size of a smallest subset <span><math><mi>D</mi></math></span> of the vertex set of <span><math><mi>G</mi></math></span> such that the closed neighbourhood <span><math><mrow><mi>N</mi><mrow><mo>[</mo><mi>D</mi><mo>]</mo></mrow></mrow></math></span> of <span><math><mi>D</mi></math></span> in <span><math><mi>G</mi></math></span> intersects the vertex sets of the <span><math><mi>F</mi></math></span>-copies contained by <span><math><mi>G</mi></math></span> (equivalently, <span><math><mrow><mi>G</mi><mo>−</mo><mi>N</mi><mrow><mo>[</mo><mi>D</mi><mo>]</mo></mrow></mrow></math></span> contains no <span><math><mi>F</mi></math></span>-copy). Thus, <span><math><mrow><mi>ι</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> is the domination number <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span>, and <span><math><mrow><mi>ι</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> is the vertex-edge domination number of <span><math><mi>G</mi></math></span>. We prove that if <span><math><mi>F</mi></math></span> is a <span><math><mi>k</mi></math></span>-edge graph, <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> (that is, <span><math><mi>F</mi></math></span> has a vertex that is adjacent to all the other vertices of <span><math><mi>F</mi></math></span>), and <span><math><mi>G</mi></math></span> is a connected <span><math><mi>m</mi></math></span>-edge graph, then <span><math><mrow><mi>ι</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌊</mo><mrow><mfrac><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow><mo>⌋</mo></mrow></mrow></math></span> unless <span><math><mi>G</mi></math></span> is an <span><math><mi>F</mi></math></span>-copy or <span><math><mi>F</mi></math></span> is a 3-path and <span><math><mi>G</mi></math></span> is a 6-cycle. This was recently posed as a conjecture by Zhang and Wu, who settled the extreme case where <span><math><mi>F</mi></math></span> is a star. The result for the other extreme case where <span><math><mi>F</mi></math></span> is a clique had been obtained by Fenech, Kaemawichanurat and the present author. The bound is attainable for any <span><math><mrow><mi>m</mi><mo>≥</mo><mn>0</mn></mrow></math></span> unless <span><math><mrow><mn>1</mn>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 247-253"},"PeriodicalIF":1.0,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143847942","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}
Guoyan Ao , Ruifang Liu , Jinjiang Yuan , C.T. Ng , T.C.E. Cheng
{"title":"Sufficient conditions for k-factors and spanning trees of graphs","authors":"Guoyan Ao , Ruifang Liu , Jinjiang Yuan , C.T. Ng , T.C.E. Cheng","doi":"10.1016/j.dam.2025.04.015","DOIUrl":"10.1016/j.dam.2025.04.015","url":null,"abstract":"<div><div>For any integer <span><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></math></span> a graph <span><math><mi>G</mi></math></span> has a <span><math><mi>k</mi></math></span>-factor if it contains a <span><math><mi>k</mi></math></span>-regular spanning subgraph. In this paper, we present a sufficient condition in terms of the number of <span><math><mi>r</mi></math></span>-cliques to guarantee the existence of a <span><math><mi>k</mi></math></span>-factor in a graph with minimum degree at least <span><math><mi>δ</mi></math></span>, which improves the sufficient condition of O (2021) based on the number of edges. For any integer <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn><mo>,</mo></mrow></math></span> a spanning <span><math><mi>k</mi></math></span>-tree of a connected graph <span><math><mi>G</mi></math></span> is a spanning tree in which every vertex has degree at most <span><math><mi>k</mi></math></span>. Motivated by the technique of Li and Ning (2016), we present a tight spectral condition for an <span><math><mi>m</mi></math></span>-connected graph to have a spanning <span><math><mi>k</mi></math></span>-tree, which extends the result of Fan et al. (2022) from <span><math><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></math></span> to general <span><math><mi>m</mi></math></span>. Let <span><math><mi>T</mi></math></span> be a spanning tree of a connected graph. The leaf degree of <span><math><mi>T</mi></math></span> is the maximum number of leaves adjacent to <span><math><mi>v</mi></math></span> in <span><math><mi>T</mi></math></span> for any <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>. We provide a tight spectral condition for the existence of a spanning tree with leaf degree at most <span><math><mi>k</mi></math></span> in a connected graph with minimum degree <span><math><mi>δ</mi></math></span>, where <span><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span> is an integer.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 124-135"},"PeriodicalIF":1.0,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143848585","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 largest eigenvalue of C−k-free signed graphs","authors":"Yongang Wang, Huiqiu Lin","doi":"10.1016/j.dam.2025.04.037","DOIUrl":"10.1016/j.dam.2025.04.037","url":null,"abstract":"<div><div>Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>−</mo></mrow></msubsup></math></span> be the set of all negative <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. For odd cycle, Wang, Hou and Li (2024) gave a spectral condition for the existence of negative <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> in unbalanced signed graphs. For even cycle, we determine the maximum index among all <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>−</mo></mrow></msubsup></math></span>-free unbalanced signed graphs and completely characterize the extremal signed graph in this paper. This could be regarded as a signed graph version of the results by Nikiforov (2007) and Zhai and Wang (2012).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 164-172"},"PeriodicalIF":1.0,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143848586","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":"Nordhaus–Gaddum decompositions for group coloring and DP coloring","authors":"Allan Bickle , Lucian Mazza","doi":"10.1016/j.dam.2025.04.034","DOIUrl":"10.1016/j.dam.2025.04.034","url":null,"abstract":"<div><div>A Nordhaus–Gaddum theorem states bounds on <span><math><mrow><mi>p</mi><mfenced><mrow><mi>G</mi></mrow></mfenced><mo>+</mo><mi>p</mi><mfenced><mrow><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover></mrow></mfenced></mrow></math></span> and <span><math><mrow><mi>p</mi><mfenced><mrow><mi>G</mi></mrow></mfenced><mi>⋅</mi><mi>p</mi><mfenced><mrow><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover></mrow></mfenced></mrow></math></span> for some graph parameter <span><math><mrow><mi>p</mi><mfenced><mrow><mi>G</mi></mrow></mfenced></mrow></math></span>. We consider the sum upper bound for DP coloring and group coloring, and prove that for a graph <span><math><mi>G</mi></math></span> with order <span><math><mi>n</mi></math></span>, <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>D</mi><mi>P</mi></mrow></msub><mfenced><mrow><mi>G</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>D</mi><mi>P</mi></mrow></msub><mfenced><mrow><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover></mrow></mfenced><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mfenced><mrow><mi>G</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mfenced><mrow><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover></mrow></mfenced><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span>. Viewing <span><math><mfenced><mrow><mi>G</mi><mo>,</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover></mrow></mfenced></math></span> as a decomposition of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, we determine the extremal decompositions for these bounds.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 264-269"},"PeriodicalIF":1.0,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143847944","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 expected values for the Kirchhoff indices in the random hexagonal-quadrilateral chain and its spiro chain","authors":"Wei Qin, Xiaoling Ma","doi":"10.1016/j.dam.2025.04.028","DOIUrl":"10.1016/j.dam.2025.04.028","url":null,"abstract":"<div><div>A hexagonal-quadrilateral chain is composed of <span><math><mi>n</mi></math></span> hexagons and <span><math><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> quadrilaterals connected by cut edges, which is a class of polycyclic aromatic hydrocarbons. The Kirchhoff index <span><math><mrow><mi>K</mi><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span> is the sum of resistance distances between all pairs of vertices in <span><math><mi>G</mi></math></span>. In this paper, we obtain exact analytical expressions of the expected values for the Kirchhoff indices of the random hexagonal-quadrilateral chain and its corresponding spiro chain with <span><math><mi>n</mi></math></span> hexagons and <span><math><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> quadrilaterals, respectively. Moreover, we also discuss the average values for the Kirchhoff indices of the random hexagonal-quadrilateral chain and its corresponding random spiro chain, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 150-163"},"PeriodicalIF":1.0,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143848584","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}