Pub Date : 2026-01-07DOI: 10.1016/j.dam.2025.12.045
Pingshan Li, Ke Yang, Wei Jin
A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order and characterized the corresponding extremal graphs. They also proposed a question regarding the second-largest number of dissociation sets among all connected graphs of order and the corresponding extremal graphs. In this paper, we give a positive answer to this question.
{"title":"A note on the second-largest number of dissociation sets in connected graphs","authors":"Pingshan Li, Ke Yang, Wei Jin","doi":"10.1016/j.dam.2025.12.045","DOIUrl":"10.1016/j.dam.2025.12.045","url":null,"abstract":"<div><div>A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order <span><math><mi>n</mi></math></span> and characterized the corresponding extremal graphs. They also proposed a question regarding the second-largest number of dissociation sets among all connected graphs of order <span><math><mi>n</mi></math></span> and the corresponding extremal graphs. In this paper, we give a positive answer to this question.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 34-40"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928977","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}
Pub Date : 2026-01-07DOI: 10.1016/j.dam.2025.12.046
Zi-Ming Zhou , Zhi-Bin Du , Chang-Xiang He
For a graph , let be the sum of the first two largest signless Laplacian eigenvalues of , and . Very recently, Zhou et al. (2024) proved that (the star graph with an additional edge) is the unique graph with minimum value of among the graphs on vertices. In this paper, we prove that the vertex-disjoint union of and possibly some isolated vertices is the unique graph with minimum value of among the graphs with edges.
{"title":"Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues","authors":"Zi-Ming Zhou , Zhi-Bin Du , Chang-Xiang He","doi":"10.1016/j.dam.2025.12.046","DOIUrl":"10.1016/j.dam.2025.12.046","url":null,"abstract":"<div><div>For a graph <span><math><mi>G</mi></math></span>, let <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the sum of the first two largest signless Laplacian eigenvalues of <span><math><mi>G</mi></math></span>, and <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>−</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Very recently, Zhou et al. (2024) proved that <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> (the star graph with an additional edge) is the unique graph with minimum value of <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> among the graphs on <span><math><mi>n</mi></math></span> vertices. In this paper, we prove that the vertex-disjoint union of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> and possibly some isolated vertices is the unique graph with minimum value of <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> among the graphs with <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> edges.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 139-144"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928987","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}
Pub Date : 2026-01-07DOI: 10.1016/j.dam.2025.12.051
Junqing Cai , Meirun Chen , Cheng-Kuan Lin
The hypercube and folded hypercube serve as fundamental interconnection network topologies in parallel computing, valued for their efficient communication and inherent fault tolerance. This paper investigates their resilience to dimensional-edge faults with respect to three critical Hamiltonian properties: Hamiltonicity, Hamiltonian laceability, and hyper Hamiltonian laceability. We establish precise bounds for fault tolerance in these structures, proving that: (1) For , both the dimensional-edge fault-tolerant Hamiltonicity and Hamiltonian laceability equal , while hyper Hamiltonian laceability tolerates up to ; (2) For , the dimensional-edge fault-tolerant Hamiltonicity is ; (3) For odd-dimensional , the dimensional-edge fault-tolerant Hamiltonian laceability and hyper Hamiltonian laceability are and , respectively. These results significantly advance our understanding of fault tolerance in cube-based network topologies and provide rigorous theoretical guarantees for their reliable operation in practical systems.
{"title":"Dimensional edge fault-tolerant Hamiltonicity of (folded) hypercubes","authors":"Junqing Cai , Meirun Chen , Cheng-Kuan Lin","doi":"10.1016/j.dam.2025.12.051","DOIUrl":"10.1016/j.dam.2025.12.051","url":null,"abstract":"<div><div>The hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and folded hypercube <span><math><mrow><mi>F</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> serve as fundamental interconnection network topologies in parallel computing, valued for their efficient communication and inherent fault tolerance. This paper investigates their resilience to dimensional-edge faults with respect to three critical Hamiltonian properties: Hamiltonicity, Hamiltonian laceability, and hyper Hamiltonian laceability. We establish precise bounds for fault tolerance in these structures, proving that: (1) For <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, both the dimensional-edge fault-tolerant Hamiltonicity and Hamiltonian laceability equal <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mi>n</mi></mrow></math></span>, while hyper Hamiltonian laceability tolerates up to <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></math></span>; (2) For <span><math><mrow><mi>F</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, the dimensional-edge fault-tolerant Hamiltonicity is <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>n</mi></mrow></math></span>; (3) For odd-dimensional <span><math><mrow><mi>F</mi><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></math></span>, the dimensional-edge fault-tolerant Hamiltonian laceability and hyper Hamiltonian laceability are <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>4</mn><mi>n</mi></mrow></math></span>, respectively. These results significantly advance our understanding of fault tolerance in cube-based network topologies and provide rigorous theoretical guarantees for their reliable operation in practical systems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 154-164"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928893","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}
Pub Date : 2026-01-07DOI: 10.1016/j.dam.2025.12.048
Yahui Zhang, Yan Li, Changxiang He
For graphs , and , let signify that any red-blue edge-coloring of contains either a red or a blue . The Ramsey number is defined as . In this note, we show that and under some bounds on , where and is a given graph with minimum color class of size 1. Our construction of edge-deleted balanced complete bipartite graphs removes more edges than earlier results concerning the critical Ramsey number of paths.
对于图F， G和H，设F→（G，H）表示F的任何红蓝边着色包含一个红色G或一个蓝色H，拉姆齐数R（G，H）定义为min{R |Kr→（G，H）}。在这篇文章中，我们证明了Kr∈Pr→（G,Pn）和Kr∈K≤≤n2²−1，≤n2²−1→（G,Pn）在n上的某些界下，其中r= r (G,Pn)，并且G是一个最小色类大小为1的给定图。我们构造的边删除平衡完全二部图比先前关于路径临界拉姆齐数的结果删除了更多的边。
{"title":"On some critical Ramsey numbers involving paths","authors":"Yahui Zhang, Yan Li, Changxiang He","doi":"10.1016/j.dam.2025.12.048","DOIUrl":"10.1016/j.dam.2025.12.048","url":null,"abstract":"<div><div>For graphs <span><math><mi>F</mi></math></span>, <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, let <span><math><mrow><mi>F</mi><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> signify that any red-blue edge-coloring of <span><math><mi>F</mi></math></span> contains either a red <span><math><mi>G</mi></math></span> or a blue <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 defined as <span><math><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo>{</mo><mi>r</mi><mspace></mspace><mo>|</mo><mspace></mspace><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>. In this note, we show that <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>K</mi></mrow><mrow><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn><mo>,</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> under some bounds on <span><math><mi>n</mi></math></span>, where <span><math><mrow><mi>r</mi><mo>=</mo><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>G</mi></math></span> is a given graph with minimum color class of size 1. Our construction of edge-deleted balanced complete bipartite graphs removes more edges than earlier results concerning the critical Ramsey number of paths.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 145-153"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928982","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}
Pub Date : 2026-01-07DOI: 10.1016/j.dam.2025.12.053
Ming Jiang , Xiaogang Liu , Jing Wang
Quantum state transfer, first introduced by Bose in 2003, is an important physical phenomenon in quantum networks, which plays a vital role in quantum communication and quantum computing. In 2004, Christandl et al. proposed the concept of perfect state transfer on graphs by modeling the quantum network using graphs, and unveiled the feasibility of applying graph theory to quantum state transfer. In 2018, Chen and Godsil proposed the definition of Laplacian perfect pair state transfer on graphs, which is a brilliant generalization of perfect state transfer. In this paper, we investigate the existence of Laplacian perfect pair state transfer in tensor product and double cover of two regular graphs, respectively, and reveal fundamental connections between perfect state transfer and Laplacian perfect pair state transfer. We give necessary and sufficient conditions for the tensor product of two regular graphs to admit Laplacian perfect pair state transfer, where one of the two regular graphs admits perfect state transfer or Laplacian perfect pair state transfer. Additionally, we characterize the existence of Laplacian perfect pair state transfer in the double cover of two regular graphs. By our results, a variety of families of graphs admitting Laplacian perfect pair state transfer can be constructed.
{"title":"Pair state transfer in tensor product and double cover","authors":"Ming Jiang , Xiaogang Liu , Jing Wang","doi":"10.1016/j.dam.2025.12.053","DOIUrl":"10.1016/j.dam.2025.12.053","url":null,"abstract":"<div><div>Quantum state transfer, first introduced by Bose in 2003, is an important physical phenomenon in quantum networks, which plays a vital role in quantum communication and quantum computing. In 2004, Christandl et al. proposed the concept of perfect state transfer on graphs by modeling the quantum network using graphs, and unveiled the feasibility of applying graph theory to quantum state transfer. In 2018, Chen and Godsil proposed the definition of Laplacian perfect pair state transfer on graphs, which is a brilliant generalization of perfect state transfer. In this paper, we investigate the existence of Laplacian perfect pair state transfer in tensor product and double cover of two regular graphs, respectively, and reveal fundamental connections between perfect state transfer and Laplacian perfect pair state transfer. We give necessary and sufficient conditions for the tensor product of two regular graphs to admit Laplacian perfect pair state transfer, where one of the two regular graphs admits perfect state transfer or Laplacian perfect pair state transfer. Additionally, we characterize the existence of Laplacian perfect pair state transfer in the double cover of two regular graphs. By our results, a variety of families of graphs admitting Laplacian perfect pair state transfer can be constructed.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 165-176"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928983","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}
Pub Date : 2026-01-05DOI: 10.1016/j.dam.2025.12.049
Yifan Li , Shuming Zhou , Qifan Zhang
For a network , the subversion at the vertex set (resp., edge set) of is defined as the removal of the closed neighborhood of the vertex set (resp., all end vertices of the edge set) from , where the vertex set (resp., edge set) is referred as subverted vertices (resp., edges). Neighbor connectivity and edge neighbor connectivity serve as key indicators for assessing the subversion of spy networks and network disruptions throughout the deletion of closed neighborhood. The neighbor connectivity (resp., edge neighbor connectivity ) of a network is defined as the minimum number of subverted vertices (resp., edges) required to disconnect it, make it empty or complete (resp., trivial). Gu et al. (IEEE Trans. Netw. Sci. Eng. 11 (5) (2024) 1-13) conjectured that whether holds for all compound graphs constructed by the underlying block . In this paper, we solve this conjecture and determine the (edge) neighbor connectivity of a class of hypercube-based compound network, including half hypercube, hierarchical hypercube, hierarchical cubic network and dual-cube-like network. In addition, we present network vulnerability analysis algorithms based on neighborhood fault pattern. To evaluate their effectiveness, taking the half hypercube, hierarchical cubic network and real-world network dwt-918 as examples, we perform experimental simulations to analyze both the cardinality distribution of subverted vertices and topological configurations of survival graph.
对于网络G，在顶点集(p。，边集)定义为顶点集（resp.）的闭邻域的移除。，边集的所有端点)来自G，其中顶点集(resp。，边集)被称为颠覆顶点(如。,边缘)。邻居连通性和边缘邻居连通性是评估间谍网络颠覆和网络中断的关键指标。邻居连通性κNB(G)。，网络G的边缘邻居连通性λNB(G)定义为颠覆顶点的最小个数(p。需要断开它，使其为空或完整(参见。琐碎的)。Gu et al. （IEEE译）Netw。科学。Eng. 11（5)(2024) 1-13）推测κNB(G)=δ(G)−12+1是否对所有由底层块Qn构造的复合图G成立。本文解决了这一猜想，并确定了一类基于超立方体的复合网络（包括半超立方体、分层超立方体、分层立方网络和双类立方体网络）的（边）邻居连通性。此外，提出了基于邻域故障模式的网络漏洞分析算法。为了评估它们的有效性，我们以半超立方体、分层立方网络和现实世界网络dwt-918为例，进行了实验模拟，分析了颠覆顶点的基数分布和生存图的拓扑构型。
{"title":"Neighbor connectivity of hypercube-based compound network","authors":"Yifan Li , Shuming Zhou , Qifan Zhang","doi":"10.1016/j.dam.2025.12.049","DOIUrl":"10.1016/j.dam.2025.12.049","url":null,"abstract":"<div><div>For a network <span><math><mi>G</mi></math></span>, the subversion at the vertex set (resp., edge set) of <span><math><mi>G</mi></math></span> is defined as the removal of the closed neighborhood of the vertex set (resp., all end vertices of the edge set) from <span><math><mi>G</mi></math></span>, where the vertex set (resp., edge set) is referred as subverted vertices (resp., edges). Neighbor connectivity and edge neighbor connectivity serve as key indicators for assessing the subversion of spy networks and network disruptions throughout the deletion of closed neighborhood. The neighbor connectivity <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>N</mi><mi>B</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> (resp., edge neighbor connectivity <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>N</mi><mi>B</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>) of a network <span><math><mi>G</mi></math></span> is defined as the minimum number of subverted vertices (resp., edges) required to disconnect it, make it empty or complete (resp., trivial). Gu et al. (IEEE Trans. Netw. Sci. Eng. 11 (5) (2024) 1-13) conjectured that whether <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>NB</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></math></span> holds for all compound graphs <span><math><mi>G</mi></math></span> constructed by the underlying block <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In this paper, we solve this conjecture and determine the (edge) neighbor connectivity of a class of hypercube-based compound network, including half hypercube, hierarchical hypercube, hierarchical cubic network and dual-cube-like network. In addition, we present network vulnerability analysis algorithms based on neighborhood fault pattern. To evaluate their effectiveness, taking the half hypercube, hierarchical cubic network and real-world network dwt-918 as examples, we perform experimental simulations to analyze both the cardinality distribution of subverted vertices and topological configurations of survival graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 1-15"},"PeriodicalIF":1.0,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898206","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}
Pub Date : 2026-01-05DOI: 10.1016/j.dam.2025.12.044
Bilal Ahmad Rather , Hilal Ahmad Ganie , Jainfeng Wang
In a connected graph , the distance signless Laplacian is defined as , where is the diagonal matrix of vertex transmissions and is the distance matrix indexed by the vertices of , such that , where represents the distance between the vertices and . Motivated by the Laplacian and signless Laplacian matrices of , Aouchiche and Hensen (2013) developed the idea of distance (signless) Laplacian matrix, which has attracted the interest among numerous spectral graph theory researchers in the field of algebraic graph theory. The spectral investigation of resulted in numerous articles. In this paper, we present a review of research on the distance signless Laplacian of connected graphs.
{"title":"Distance signless Laplacian spectra of graphs: A survey","authors":"Bilal Ahmad Rather , Hilal Ahmad Ganie , Jainfeng Wang","doi":"10.1016/j.dam.2025.12.044","DOIUrl":"10.1016/j.dam.2025.12.044","url":null,"abstract":"<div><div>In a connected graph <span><math><mi>G</mi></math></span>, the distance signless Laplacian is defined as <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>Q</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>D</mi><mi>i</mi><mi>a</mi><mi>g</mi><mrow><mo>(</mo><mi>T</mi><mi>r</mi><mo>)</mo></mrow><mo>+</mo><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>D</mi><mi>i</mi><mi>a</mi><mi>g</mi><mrow><mo>(</mo><mi>T</mi><mi>r</mi><mo>)</mo></mrow></mrow></math></span> is the diagonal matrix of vertex transmissions and <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></mrow></math></span> is the distance matrix indexed by the vertices of <span><math><mi>G</mi></math></span>, such that <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mi>d</mi><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>d</mi><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> represents the distance between the vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>. Motivated by the Laplacian and signless Laplacian matrices of <span><math><mi>G</mi></math></span>, Aouchiche and Hensen (2013) developed the idea of distance (signless) Laplacian matrix, which has attracted the interest among numerous spectral graph theory researchers in the field of algebraic graph theory. The spectral investigation of <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>Q</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> resulted in numerous articles. In this paper, we present a review of research on the distance signless Laplacian of connected graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 41-138"},"PeriodicalIF":1.0,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898207","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}
Pub Date : 2026-01-05DOI: 10.1016/j.dam.2025.12.050
Gideon Amir , Yaakov Malinovsky
In this note, we extend a recent result on the uniqueness of the maximum score in a classical round-robin tournament to general round-robin tournament models with equally strong players, where the scores take values in .
{"title":"Uniqueness of maximum scores in countable-outcome round-robin tournaments","authors":"Gideon Amir , Yaakov Malinovsky","doi":"10.1016/j.dam.2025.12.050","DOIUrl":"10.1016/j.dam.2025.12.050","url":null,"abstract":"<div><div>In this note, we extend a recent result on the uniqueness of the maximum score in a classical round-robin tournament to general round-robin tournament models with equally strong players, where the scores take values in <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mspace></mspace><mn>1</mn><mo>]</mo></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 16-22"},"PeriodicalIF":1.0,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898208","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}
Pub Date : 2026-01-05DOI: 10.1016/j.dam.2025.12.040
Chenke Zhang , Qing Cui , Jinze Hu , Erfei Yue , Shengjin Ji
Let be a graph and be a family of graphs. We say a graph is -saturated if does not contain any member in and for any , creates a copy of some member in . The saturation number of is the minimum number of edges of an -saturated graphs of vertices, denoted by . If , then we write it as for short. In this paper, we determine the exact value of , and as its application, we obtain two bounds on for and sufficiently large . Furthermore, is determined, where is a linear forest without isolated vertices.
{"title":"Some results on minimum saturated graphs","authors":"Chenke Zhang , Qing Cui , Jinze Hu , Erfei Yue , Shengjin Ji","doi":"10.1016/j.dam.2025.12.040","DOIUrl":"10.1016/j.dam.2025.12.040","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph and <span><math><mi>F</mi></math></span> be a family of graphs. We say a graph <span><math><mi>G</mi></math></span> is <span><math><mi>F</mi></math></span>-saturated if <span><math><mi>G</mi></math></span> does not contain any member in <span><math><mi>F</mi></math></span> and for any <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>G</mi><mo>+</mo><mi>e</mi></mrow></math></span> creates a copy of some member in <span><math><mi>F</mi></math></span>. The saturation number of <span><math><mi>F</mi></math></span> is the minimum number of edges of an <span><math><mi>F</mi></math></span>-saturated graphs of <span><math><mi>n</mi></math></span> vertices, denoted by <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>. If <span><math><mrow><mi>F</mi><mo>=</mo><mrow><mo>{</mo><mi>F</mi><mo>}</mo></mrow></mrow></math></span>, then we write it as <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> for short. In this paper, we determine the exact value of <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>, and as its application, we obtain two bounds on <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>10</mn></mrow></math></span> and sufficiently large <span><math><mi>n</mi></math></span>. Furthermore, <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> is determined, where <span><math><mi>F</mi></math></span> is a linear forest without isolated vertices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 23-33"},"PeriodicalIF":1.0,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898209","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}
Pub Date : 2025-12-29DOI: 10.1016/j.dam.2025.12.043
Yong-De Feng, Yawen Chen, Baoyindureng Wu
An odd graph is a graph for which every vertex satisfies . An odd spanning tree of is a spanning tree such that for all . It is known that for any complete graph of even order has an odd spanning tree. In this paper, we establish the exact number of labeled odd spanning trees in . By employing the classical Prüfer sequence and constructing the corresponding generating function, we prove that the number of labeled odd spanning trees in is given by (where is even).
{"title":"The number of odd spanning trees in the complete graphs","authors":"Yong-De Feng, Yawen Chen, Baoyindureng Wu","doi":"10.1016/j.dam.2025.12.043","DOIUrl":"10.1016/j.dam.2025.12.043","url":null,"abstract":"<div><div>An odd graph is a graph <span><math><mi>G</mi></math></span> for which every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> satisfies <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>≡</mo><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. An odd spanning tree <span><math><mi>T</mi></math></span> of <span><math><mi>G</mi></math></span> is a spanning tree such that <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>T</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>≡</mo><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>2</mn><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>. It is known that for any complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of even order has an odd spanning tree. In this paper, we establish the exact number of labeled odd spanning trees in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. By employing the classical Prüfer sequence and constructing the corresponding generating function, we prove that the number of labeled odd spanning trees in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is given by <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></mfenced><msup><mrow><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>−</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span>\u0000 <!--> <!-->(where <span><math><mi>n</mi></math></span> is even).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 295-299"},"PeriodicalIF":1.0,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885019","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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