混合图的Hermitian邻接矩阵的零维数

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.     

On strong metric dimension of graphs and their complements

A vertex x in a graph G strongly resolves a pair of vertices v, w if there exists a shortest x-w path containing v or a shortest x-v path containing w in G. A set of vertices S■V(G) is a strong resolving set of G if every pair of distinct vertices of G is strongly resolved by some vertex in S. The strong metric dimension of G, denoted by sdim(G), is the minimum cardinality over all strong resolving sets of G. For a connected graph G of order n≥2, we characterize G such that sdim(G) equals 1, n-1, or n-2, respectively. We give a Nordhaus-Gaddum-type result for the strong metric dimension of a graph and its complement: for a graph G and its complement G, each of order n≥4 and connected, we show that 2≤sdim(G)+sdim(G)≤2( n-2). It is readily seen that sdim(G)+sdim(G)=2 if and only if n=4; we show that, when G is a tree or a unicyclic graph, sdim(G)+sdim(G)=2(n 2) if and only if n=5 and G ～=G ～=C5, the cycle on five vertices. For connected graphs G and G of order n≥5, we show that 3≤sdim(G)+sdim(G)≤2(n-3) if G is a tree; we also show that 4≤sdim(G)+sdim(G)≤2(n-3) if G is a unicyclic graph of order n≥6. Furthermore, we characterize graphs G satisfying sdim(G)+sdim(G)=2(n-3) when G is a tree or a unicyclic graph.     

单圈图的零度的一个注记

The number of zero eigenvalues in the spectrum of the graph G is called its nullity and is denoted by η(G).In this paper,we determine the all extremal unicyclic graphs achieving the fifth upper bound n-6 and the sixth upperbound n-7.     

A characterization of PM-compact Hamiltonian bipartite graphs

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. A graph is called perfect matching compact(shortly, PM-compact), if its perfect matching polytope has diameter one. This paper gives a complete characterization of simple PM-compact Hamiltonian bipartite graphs. We first define two families of graphs, called the H2C-bipartite graphs and the H23-bipartite graphs, respectively. Then we show that, for a simple Hamiltonian bipartite graph G with |V(G)| ≥ 6, G is PM-compact if and only if G is K3,3, or G is a spanning Hamiltonian subgraph of either an H2C-bipartite graph or an H23-bipartite graph.     

The limit case of a domination property

The domination number γ(G) of a connected graph G of order n is bounded below by(n+2-e(G))/ 3 , where (G) denotes the maximum number of leaves in any spanning tree of G. We show that (n+2-e(G))/ 3 = γ(G) if and only if there exists a tree T ∈ T ( G) ∩ R such that n1(T ) = e(G), where n1(T ) denotes the number of leaves of T1, R denotes the family of all trees in which the distance between any two distinct leaves is congruent to 2 modulo 3, and T (G) denotes the set composed by the spanning trees of G. As a consequence of the study, we show that if (n+2-e(G))/ 3 = γ(G), then there exists a minimum dominating set in G whose induced subgraph is an independent set. Finally, we characterize all unicyclic graphs G for which equality (n+2-e(G))/ 3= γ(G) holds and we show that the length of the unique cycle of any unicyclic graph G with (n+2-e(G))/ 3= γ(G) belongs to {4} ∪ {3 , 6, 9, . . . }.     

Extremal Problems on Components and Loops in Graphs

The authors recently defined a new graph invariant denoted by Ω(G) only in terms of a given degree sequence which is also related to the Euler characteristic. It has many important combinatorial applications in graph theory and gives direct information compared to the better known Euler characteristic on the realizability, connectedness, cyclicness, components, chords, loops etc. Many similar classification problems can be solved by means of Ω. All graphs G so that Ω(G) ≤-4 are shown to be disconnected, and if Ω(G) ≥-2, then the graph is potentially connected. It is also shown that if the realization is a connected graph and Ω(G) =-2, then certainly the graph should be a tree.Similarly, it is shown that if the realization is a connected graph G and Ω(G) ≥ 0, then certainly the graph should be cyclic. Also, when Ω(G) ≤-4, the components of the disconnected graph could not all be cyclic and if all the components of G are cyclic, then Ω(G) ≥ 0. In this paper, we study an extremal problem regarding graphs. We find the maximum number of loops for three possible classes of graphs.We also state a result giving the maximum number of components amongst all possible realizations of a given degree sequence.     

INDEPENDENT-SET-DELETABLE FACTOR-CRITICAL POWER GRAPHS

It is said that a graph G is independent-set-deletable factor-critical (in short, ID-factor-critical), if, for every independent set 7 which has the same parity as |V(G)|, G-I has a perfect matching. A graph G is strongly IM-extendable, if for every spanning supergraph H of G, every induced matching of H is included in a perfect matching of H. The k-th power of G, denoted by Gk, is the graph with vertex set V(G) in which two vertices are adjacent if and only if they have distance at most k in G. ID-factor-criticality and IM-extendability of power graphs are discussed in this article. The author shows that, if G is a connected graph, then G3 and T(G) (the total graph of G) are ID-factor-critical, and G4 (when |V(G)| is even) is strongly IM-extendable; if G is 2-connected, then D2 is ID-factor-critical.     

Induced Matching Number of the Plane Grid Graph

An induced matching M in a graph G is a matching such that V(M) induces a 1-regular subgraph of G. The induced matching number of a graph G, denoted by I M(G), is the maximum number r such that G has an induced matching of r edges. Induced matching number of Pm×Pn is investigated in this paper. The main results are as follows:(1) If at least one of m and n is even, then IM(Pm×Pn=[(mn)/4].(2) If m is odd, then     

零度为$n-7$的双圈符号图

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.     

关于双圈图的谱半径

如果G是连通的并且G的边数是n 1,那么n阶图G叫做双圈图,设B(n)是所有的阶为n的双圈图构成的集合,本文给出了B(n)(n(?)9)中前三大的邻接谱半径以及它们对应的图.     

图的无符号拉普拉斯和距离无符号拉普拉斯特征值

Let G be a simple graph. We first show that ■, where δiand di denote the i-th signless Laplacian eigenvalue and the i-th degree of vertex in G, respectively.Suppose G is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from G. In addition, for the distance signless Laplacian spectral radius ρQ(G), we determine the extremal graphs with the minimum ρQ(G) among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.     

符号圈拼接图的零度

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.     

最大度与最小度相差不超过2的图的平衡judicious划分

图G的顶点集V(G)的一个二部划分V_1和V_2叫做平衡二部划分,如果||V_1|-|V_2||≤1成立.Bollobas和Scott猜想:每一个有m条边且最小度不小于2的图,都存在一个平衡二部划分V_1,V_2,使得max{e(V_1),e(V_2)}≤m/3,此处e(V_i)表示两顶点都在V_i(i=1,2)中的边的条数.他们证明了这个猜想对正则图(即△(G)=δ(G))成立.颜娟和许宝刚证明了每个(k,k-1)-双正则图(即△(G)-δ(G)≤1)存在一个平衡二部划分V_1,V_2,使得每一顶点集的导出子图包含大约m/4条边.这里把该结论推广到最大度和最小度相差不超过2的图G.     

几类新的上可嵌入图

讨论了几类上可嵌入的边连通简单图,得到了如下结果:若G为简单连通图,且满足以下条件1)-3)之一:1)G为1-边连通的,且不含完全图K_3,α(G)≤3,2)G为2-边连通的,且不含完全图K_3,α(G)≤5,3)G为3-边连通的,且不含完全图K_3,α(G)≤10,则G是上可嵌入的,且在上述相应条件下,独立数上界都分别是最好的.     

任意长圈覆盖指定的独立的点

An invariant σ2（G） of a graph is defined as follows： σ2（G） ：= min{d（u） ＋ d（v）｜u, v ∈V（G）,uv ∈ E（G）,u ≠ v} is the minimum degree sum of nonadjacent vertices （when G is a complete graph, we define σ2（G） = ∞）. Let k, s be integers with k ≥ 2 and s ≥ 4, G be a graph of order n sufficiently large compared with s and k. We show that if σ2（G） ≥ n ＋ k- 1, then for any set of k independent vertices v1,..., vk, G has k vertex-disjoint cycles C1,..., Ck such that ｜Ci｜ ≤ s and vi ∈ V（Ci） for all 1 ≤ i ≤ k.
The condition of degree sum σs（G） ≥ n ＋ k - 1 is sharp.     

半无爪可迹图的度和条件

可迹图即为一个含有Hamilton路的图.令$N[v]=N(v)\cup\{v\}$, $J(u,v)=\{w\in N(u)\cap N(v):N(w)\subseteq N[u]\cup N[v]\}$.若图中任意距离为2的两点$u,v$满足$J(u,v)\neq \emptyset$,则称该图为半无爪图.令$\sigma_{k}(G)=\min\{\sum_{v\in S}d(v):S$为$G$中含有$k$个点的独立集\},其中$d(v)$表示图$G$中顶点$v$的度.本论文证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{3}(G)\geq {n-2}$,则图$G$为可迹图; 文中给出一个图例,说明上述结果中的界是下确界; 此外,我们证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{2}(G)\geq \frac{2({n-2})}{3}$,则该图为可迹图.     

Dirac K4细分定理的一种推广

1960年, Dirac证明了对一个阶为$n\geq 4$的图$G$,如果$G$的边数大于$2n-3$,那么$G$一定包含一个$K_4$的细分. 作者证明了对一个阶为$n\geq 4$的图$G$和$k\geq 2$,如果$G$的边数至少为$kn-\frac{(k-1)(k+2)}{2}$, 那么$G$一定包含一个$W_{k+1}$的细分,从而推广了Dirac的结果.另外,作者利用范更华提出的边切换的方法,给出了Dirac结果的另一种证明.