Hongwei Zhu , Shitao Li , Minjia Shi , Shu-Tao Xia , Patrick Solé
{"title":"Some bounds on the cardinality of the b-symbol weight spectrum of codes","authors":"Hongwei Zhu , Shitao Li , Minjia Shi , Shu-Tao Xia , Patrick Solé","doi":"10.1016/j.disc.2025.114678","DOIUrl":"10.1016/j.disc.2025.114678","url":null,"abstract":"<div><div>The size of the Hamming distance spectrum of a code has received great attention in recent research. The main objective of this paper is to extend these significant theories to the <em>b</em>-symbol distance spectrum. We examine this question for various types of codes, including unrestricted codes, additive codes, linear codes, and cyclic codes, successively. For the first three cases, we determine the maximum size of the <em>b</em>-symbol distance spectra of these codes smoothly. For the case of cyclic codes, we introduce three approaches to characterize the upper bound for the cardinality of the <em>b</em>-symbol weight spectrum of cyclic codes, namely the period distribution approach, the primitive idempotent approach, and the <em>b</em>-symbol weight formula approach. As two by-products of this paper, the maximum number of symplectic weights of linear codes is determined, and a basic inequality among the parameters <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> of cyclic codes is provided.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114678"},"PeriodicalIF":0.7,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633180","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":"Distributions of mesh patterns of short lengths on king permutations","authors":"Dan Li, Philip B. Zhang","doi":"10.1016/j.disc.2025.114681","DOIUrl":"10.1016/j.disc.2025.114681","url":null,"abstract":"<div><div>Brändén and Claesson introduced the concept of mesh patterns in 2011, and since then, these patterns have attracted significant attention in the literature. Subsequently, in 2015, Hilmarsson et al. initiated the first systematic study of avoidance of mesh patterns, while Kitaev and Zhang conducted the first systematic study of the distribution of mesh patterns in 2019. A permutation <span><math><mi>σ</mi><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is called a king permutation if <span><math><mrow><mo>|</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow><mo>></mo><mn>1</mn></math></span> for each <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. Riordan derived a recurrence relation for the number of such permutations in 1965. The generating function for king permutations was obtained by Flajolet and Sedgewick in 2009. In this paper, we initiate a systematic study of the distribution of mesh patterns on king permutations by finding distributions for 22 mesh patterns of short lengths.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114681"},"PeriodicalIF":0.7,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633179","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":"Symmetric covers and pseudocovers of complete graphs","authors":"Yan Zhou Zhu","doi":"10.1016/j.disc.2025.114677","DOIUrl":"10.1016/j.disc.2025.114677","url":null,"abstract":"<div><div>We first characterize all faithful arc-transitive covers of complete graphs and we give a general construction of such covers.</div><div>A graph Γ is a pseudocover of its quotient Σ if they have the same valency and Γ is not a cover of Σ. As the second result of this paper, we prove that the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> has a connected arc-transitive pseudocover if and only if <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> is not a prime.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114677"},"PeriodicalIF":0.7,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632499","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 optimal placement delivery arrays","authors":"Lijun Ji , Ruizhong Wei , Liying Yu","doi":"10.1016/j.disc.2025.114680","DOIUrl":"10.1016/j.disc.2025.114680","url":null,"abstract":"<div><div>A placement delivery array (PDA) is a combinatorial configuration derived from coded caching schemes which are used to reduce computer network traffics during peak usage periods. In this paper, we introduce the concept of optimal PDAs (OPDAs) as fundamental combinatorial objects and explore their key combinatorial properties. Furthermore, we present several new infinite families of OPDAs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114680"},"PeriodicalIF":0.7,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632498","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 odd and strong odd colorings of graphs","authors":"Jing-Ru Pang , Lian-Ying Miao , Yi-Zheng Fan","doi":"10.1016/j.disc.2025.114683","DOIUrl":"10.1016/j.disc.2025.114683","url":null,"abstract":"<div><div>An odd <em>k</em>-coloring of a graph <em>G</em> is a proper <em>k</em>-coloring such that for every non-isolated vertex <em>v</em> there is a color that occurs an odd number of times in the neighborhood of <em>v</em>. A strong odd <em>k</em>-coloring of <em>G</em> is a proper <em>k</em>-coloring such that for every vertex <em>v</em> every color occurs an odd number of times or 0 times in the neighborhood of <em>v</em>, which is a strengthened version of odd coloring and also a relaxation of square coloring. The odd chromatic number (or the strong odd chromatic number) of a graph <em>G</em>, denoted by <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (or <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>s</mi><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>), is the minimum number of colors in any odd coloring (or strong odd coloring) of the graph <em>G</em>. In this paper, we prove that for any <span><math><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>></mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, there exists a <span><math><mi>Δ</mi><mo>(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> such that if <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>Δ</mi><mo>(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> and <span><math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>Δ</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math></span>, then <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mo>⌈</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⌉</mo></math></span>, where <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the chromatic number of <em>G</em>, and <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are the maximum degree and minimum degree of <em>G</em> respectively. In addition, we construct a planar graph with strong odd chromatic number 13, which answers a question asked by Caro, Petruševski, Škrekovski and Tuza in negative.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114683"},"PeriodicalIF":0.7,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632533","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":"Minimum bull-saturated graphs","authors":"Xinying Hua, Yuejian Peng","doi":"10.1016/j.disc.2025.114674","DOIUrl":"10.1016/j.disc.2025.114674","url":null,"abstract":"<div><div>For a given graph family <span><math><mi>F</mi></math></span>, a graph <em>G</em> is said to be <span><math><mi>F</mi></math></span><em>-saturated</em> if <em>G</em> contains no member of <span><math><mi>F</mi></math></span> as a subgraph but the addition of any edge to <em>G</em> produces a member of <span><math><mi>F</mi></math></span>. If <span><math><mi>F</mi><mo>=</mo><mo>{</mo><mi>F</mi><mo>}</mo></math></span>, then <em>G</em> is said to be an <em>F-saturated graph</em>. The <em>saturation number</em> of <em>F</em>, denoted by <span><math><mrow><mi>sat</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mspace></mspace><mi>F</mi><mo>)</mo></math></span>, is the minimum number of edges in an <em>F</em>-saturated graph with <em>n</em> vertices. A <em>bull</em> is the graph obtained by attaching two leaves to two vertices of a triangle. In this paper, we determine the saturation number of a bull and characterize corresponding minimum bull-saturated graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114674"},"PeriodicalIF":0.7,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144623302","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":"Spectral extremal problems for non-bipartite graphs without odd cycles","authors":"Lantao Zou, Lihua Feng, Yongtao Li","doi":"10.1016/j.disc.2025.114670","DOIUrl":"10.1016/j.disc.2025.114670","url":null,"abstract":"<div><div>A well-known result of Mantel asserts that every <em>n</em>-vertex triangle-free graph <em>G</em> has at most <span><math><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>4</mn><mo>⌋</mo></math></span> edges. Moreover, Erdős proved that if <em>G</em> is further non-bipartite, then <span><math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>⌊</mo><msup><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>4</mn><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span>. Recently, Lin, Ning and Wu (2021) <span><span>[35]</span></span> established a spectral version by showing that if <em>G</em> is a triangle-free non-bipartite graph on <em>n</em> vertices, then <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>λ</mi><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo></math></span>, with equality if and only if <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo></math></span> is obtained from <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></span> by subdividing an edge. In this paper, we investigate the maximum spectral radius of a non-bipartite graph without some short odd cycles. Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>ℓ</mi><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo></math></span> be the graph obtained by identifying a vertex of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> and a vertex of the smaller partite set of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>ℓ</mi><mo>,</mo><mn>2</mn></mrow></msub></math></span>. We prove that for <span><math><mn>1</mn><mo>≤</mo><mi>ℓ</mi><mo><</mo><mi>k</mi></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>187</mn><mi>k</mi></math></span>, if <em>G</em> is an <em>n</em>-vertex <span><math><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo></math></span>-free non-bipartite graph, then <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114670"},"PeriodicalIF":0.7,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144597516","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}
Endre Boros , Vladimir Gurvich , Martin Milanič , Dmitry Tikhanovsky , Yushi Uno
{"title":"Conformality of minimal transversals of maximal cliques","authors":"Endre Boros , Vladimir Gurvich , Martin Milanič , Dmitry Tikhanovsky , Yushi Uno","doi":"10.1016/j.disc.2025.114657","DOIUrl":"10.1016/j.disc.2025.114657","url":null,"abstract":"<div><div>Given a hypergraph <span><math><mi>H</mi></math></span>, the dual hypergraph of <span><math><mi>H</mi></math></span> is the hypergraph of all minimal transversals of <span><math><mi>H</mi></math></span>. A hypergraph is conformal if it is the family of maximal cliques of a graph. In a recent work, Boros, Gurvich, Milanič, and Uno (Journal of Graph Theory, 2025) studied conformality of dual hypergraphs and proved several results related to this property, leading in particular to a polynomial-time algorithm for recognizing graphs in which, for any fixed <em>k</em>, all minimal transversals of maximal cliques have size at most <em>k</em>. In this follow-up work, we provide a novel aspect to the study of graph clique transversals, by considering the dual conformality property from the perspective of graphs. More precisely, we study graphs for which the family of minimal transversals of maximal cliques is conformal. Such graphs are called clique dually conformal (CDC for short). It turns out that the class of CDC graphs is a rich generalization of the class of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graphs. As our main results, we completely characterize CDC graphs within the families of triangle-free graphs and split graphs. Both characterizations lead to polynomial-time recognition algorithms. Generalizing the fact that every <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graph is CDC, we also show that the class of CDC graphs is closed under substitution, in the strong sense that substituting a graph <em>H</em> for a vertex of a graph <em>G</em> results in a CDC graph if and only if both <em>G</em> and <em>H</em> are CDC.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114657"},"PeriodicalIF":0.7,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579926","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":"Ternary near-extremal self-dual codes of lengths 36, 48 and 60","authors":"Masaaki Harada","doi":"10.1016/j.disc.2025.114675","DOIUrl":"10.1016/j.disc.2025.114675","url":null,"abstract":"<div><div>For lengths 36, 48 and 60, we construct new ternary near-extremal self-dual codes with weight enumerators for which no ternary near-extremal self-dual codes were previously known to exist.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114675"},"PeriodicalIF":0.7,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579928","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":"Spectral extremal problems on outerplanar and planar graphs","authors":"Xilong Yin, Dan Li","doi":"10.1016/j.disc.2025.114673","DOIUrl":"10.1016/j.disc.2025.114673","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mtext>spex</mtext></mrow><mrow><mi>OP</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mtext>spex</mtext></mrow><mrow><mi>P</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be the maximum spectral radius over all <em>n</em>-vertex <em>F</em>-free outerplanar graphs and planar graphs, respectively. Define <span><math><mi>t</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> as <em>t</em> vertex-disjoint <em>l</em>-cycles, <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>l</mi></mrow></msub></math></span> as the graph obtained by sharing a common vertex among <em>t</em> edge-disjoint <em>l</em>-cycles and <span><math><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> as the disjoint union of <span><math><mi>t</mi><mo>+</mo><mn>1</mn></math></span> copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. In the 1990s, Cvetković and Rowlinson conjectured <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> maximizes spectral radius in outerplanar graphs on <em>n</em> vertices, while Boots and Royle (independently, Cao and Vince) conjectured <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∨</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></math></span> does so in planar graphs. Tait and Tobin (2017) <span><span>[22]</span></span> determined the fundamental structure as the key to confirming these two conjectures for sufficiently large <em>n</em>. Recently, Fang et al. (2024) <span><span>[12]</span></span> characterized the extremal graph with <span><math><msub><mrow><mtext>spex</mtext></mrow><mrow><mi>P</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>t</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>)</mo></math></span> in planar graphs by using this key. In this paper, we first focus on outerplanar graphs and adopt a similar approach to describe the key structure of the connected extremal graph with <span><math><msub><mrow><mtext>spex</mtext></mrow><mrow><mi>OP</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, where <em>F</em> is contained in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> but not in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><mo>(</mo><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∪</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><msub><mrow><","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114673"},"PeriodicalIF":0.7,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579927","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}
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