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邻接树图是哈密尔顿图猜想的一个等价命题 总被引:1,自引:0,他引:1
本文给出了简单图的邻接树图是哈密尔顿图”猜想的等价命题,阐明只需证明该猜想对2-连通图成立即可,另外,我们给出了该猜想一种特殊情形的构造性证明。 相似文献
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设$G$是简单无向图. 对于实数$\alpha \in [0,1]$, Nikiforov于2017年定义图的$A_\alpha$-矩阵为$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, 其中$A(G)$和$D(G)$分别为图$G$的邻接矩阵和度对角矩阵. 图的$A_\alpha$-矩阵可以看着是图的邻接矩阵和无符号拉普拉斯矩阵的共同推广, 其最大特征值称为图的$A_\alpha$- 谱半径. 对于$\alpha\in[0,1)$, 本文确定了不含三角形图的$A_\alpha$-谱半径的一个下界;对于$\alpha \in[1/2, 1)$, 本文确定了不含三角形$k$圈图的$A_\alpha$-谱半径的一个上界. 相似文献
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For a graph G, a path cover is a set of vertex disjoint paths covering all the vertices of G, and a path cover number of G, denoted by p(G), is the minimum number of paths in a path cover among all the path covers of G. In this paper, we prove that if G is a K_(1,4)-free graph of order n and σ_(k+1)(G) ≥ n-k, then p(G) ≤ k, where σ_(k+1)(G) = min{∑v∈S d(v) : S is an independent set of G with |S| = k + 1}. 相似文献
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本文给出了$2$为完美匹配单圈图的无符号拉普拉斯特征值的充分必要条件. 相似文献
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定向图Gσ是一个不含有环(loop)和重边的有向图,其中G称作它的基图.S(Gσ)是Gσ的斜邻接矩阵.S(Gσ)的秩称为Gσ的斜秩,记为sr(Gσ).定向图的斜邻接矩阵是斜对称的,因而,它的斜秩是偶数.本文主要考虑简单定向图的斜秩,首先给出斜秩的一些简单基本知识,紧接着分别刻画斜秩是2的定向图和斜秩是4的带有悬挂点的定向图;其次利用匹配数给出具有n个顶点、围长是k的单圈图的斜秩表达式;作为推论,列出斜秩是4的所有单圈图和带有悬挂点的双圈图;另外研究具有n个顶点、围长是k的单圈图的图类中斜秩的最小值,并刻画了极图;最后研究斜邻接矩阵是非奇异的定向单圈图. 相似文献
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Let Γ be a signed graph and A(Γ) be the adjacency matrix of Γ. The nullity ofΓ is the multiplicity of eigenvalue zero in the spectrum of A(Γ). In this paper, the connected bicyclic signed graphs(including simple bicyclic graphs) of order n with nullity n-7 are completely characterized. 相似文献
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令$\eta(\Gamma)$和$c(\Gamma)$是符号图$\Gamma$的零度和基本圈数. 一个符号圈拼接图是指每个块都是圈的连通符号图. 本文证明了对任意符号拼接图$\eta(\Gamma)\le c(\Gamma)+1$成立, 并且刻画了等号成立的极图, 推广了王登银等人(2022)在简单圈拼接图上的结果. 此外, 我们证明了任意的符号拼接图$\eta(\Gamma)\neq c(\Gamma)$, 给出了满足$\eta(\Gamma)=c(\Gamma)-1$的符号拼接图的一些性质并刻画处$\eta(\Gamma)=c(\Gamma)-1$的二部符号拼接图. 相似文献
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A mixed graph means a graph containing both oriented edges and undirected edges. The nullity of the Hermitian-adjacency matrix of a mixed graph G, denoted by ηH(G),is referred to as the multiplicity of the eigenvalue zero. In this paper, for a mixed unicyclic graph G with given order and matching number, we give a formula on ηH(G), which combines the cases of undirected and oriented unicyclic graphs and also corrects an error in Theorem 4.2 of [Xueliang LI, Guihai YU. The skew-rank of oriented graphs. Sci. Sin. Math., 2015, 45:93-104(in Chinese)]. In addition, we characterize all the n-vertex mixed graphs with nullity n-3, which are determined by the spectrum of their Hermitian-adjacency matrices. 相似文献
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研究了含有多个圈的图的邻接矩阵的秩.将k(k≥2)条点不交的路,首和尾分别粘合得到的图称为Θ-图.用Γ(k-1)表示含有Θ-图作为导出子图的(k-1)-圈图的集合,而用C(η,k)表示含有n个顶点和k个边不交的圈的图的集合.确定了Γ(k-1)中秩等于5和6的图以及C(n,k)中秩等于4,5和6的图. 相似文献
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The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time O(n 4) whether three given vertices of a graph belong to an induced tree. Here, we study four-in- a-tree for triangle-free graphs. We give a structural answer to the following question: what does a triangle-free graph look like if no induced tree covers four given vertices? Our main result says that any such graph must have the “same structure”, in a sense to be defined precisely, as a square or a cube. We provide an O(nm)-time algorithm that given a triangle-free graph G together with four vertices outputs either an induced tree that contains them or a partition of V(G) certifying that no such tree exists. We prove that the problem of deciding whether there exists a tree T covering the four vertices such that at most one vertex of T has degree at least 3 is NP-complete. 相似文献
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In this paper we present an efficient algorithm for generating maximal triangle-free graphs. A program based on this algorithm has been used to check a conjecture of Erdo´´s about the local density of triangle-free graphs and turned out to be very powerful for the computation of triangle Ramsey numbers. 相似文献
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图$G$的$(\mathcal{O}_{k_1}, \mathcal{O}_{k_2})$-划分是将$V(G)$划分成两个非空子集$V_{1}$和$V_{2}$, 使得$G[V_{1}]$和$G[V_{2}]$分别是分支的阶数至多$k_1$和$k_2$的图.在本文中,我们考虑了有围长限制的平面图的点集划分问题,使得每个部分导出一个具有有界大小分支的图.我们证明了每一个围长至少为6并且$i$-圈不与$j$-圈相交的平面图允许$(\mathcal{O}_{2}$, $\mathcal{O}_{3})$-划分,其中$i\in\{6,7,8\}$和$j\in\{6,7,8,9\}$. 相似文献
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The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle‐free graphs with a given rank and characterize all such graphs achieving the maximum order. 相似文献