On the spectral spread of bicyclic graphs with given girth

The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and |E(G)| = |V(G)|+ 1. Let B(n, g) be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B(n, g) is determined.     

双圈图的零度与其匹配数之间的关系

Let G be a graph with n(G) vertices and m(G) be its matching number.The nullity of G,denoted by η(G),is the multiplicity of the eigenvalue zero of adjacency matrix of G.It is well known that if G is a tree,then η(G) = n(G)-2m(G).Guo et al.[Jiming GUO,Weigen YAN,Yeongnan YEH.On the nullity and the matching number of unicyclic graphs.Linear Alg.Appl.,2009,431:1293 1301]proved that if G is a unicyclic graph,then η(G)equals n(G)-2m(G)-1,n(G)-2m(G),or n(G)-2m(G) +2.In this paper,we prove that if G is a bicyclic graph,then η(G) equals n(G)-2m(G),n(G)-2m(G)±1,n(G)-2m(G)±2or n(G)-2m(G) + 4.We also give a characterization of these six types of bicyclic graphs corresponding to each nullity.     

零维数等于1的单圈图

The nullity of a graph G is defined to be the multiplicity of the eigenvalue zero in its spectrum. In this paper we characterize the unicyclic graphs with nullity one in aspect of its graphical construction.     

给定悬挂点数的双圈图中具有最小Hosoya指数的图

Let G be a graph. The Hosoya index Z（G） of a graph G is defined to be the total number of its matchings. In this paper, we characterize the graph with the smallest Hosoya index of bicyclic graphs with given pendent vertices. Finally, we present a new proof about the smallest Hosoya index of bicyclic graphs.     

双圈图的无符号拉普拉斯谱半径和谱展

The signless Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the smallest eigenvalue of its signless Laplacian matrix. In this paper, we determine the first to llth largest signless Laplacian spectral radii in the class of bicyclic graphs with n vertices. Moreover, the unique bicyclic graph with the largest or the second largest signless Laplacian spread among the class of connected bicyclic graphs of order n is determined, respectively.     

Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels

A signed(res. signed total) Roman dominating function, SRDF(res.STRDF) for short, of a graph G =(V, E) is a function f : V → {-1, 1, 2} satisfying the conditions that(i)∑v∈N[v]f(v) ≥ 1(res.∑v∈N(v)f(v) ≥ 1) for any v ∈ V, where N [v] is the closed neighborhood and N(v) is the neighborhood of v, and(ii) every vertex v for which f(v) =-1 is adjacent to a vertex u for which f(u) = 2. The weight of a SRDF(res. STRDF) is the sum of its function values over all vertices.The signed(res. signed total) Roman domination number of G is the minimum weight among all signed(res. signed total) Roman dominating functions of G. In this paper,we compute the exact values of the signed(res. signed total) Roman domination numbers of complete bipartite graphs and wheels.     

符号混合控制问题的某些NP-完全性结果

Let G =(V, E) be a simple graph with vertex set V and edge set E. A signed mixed dominating function of G is a function f:V∪E→ {-1, 1} such that ∑_(y∈N_m(x)∪{x})f(y)≥ 1for every element x∈V∪E, where N_m(x) is the set of elements of V∪E adjacent or incident to x. The weight of f is w(f) =∑_(x∈V∪E)f(x). The signed mixed domination problem is to find a minimum-weight signed mixed dominating function of a graph. In this paper we study the computational complexity of signed mixed domination problem. We prove that the signed mixed domination problem is NP-complete for bipartite graphs, chordal graphs, even for planar bipartite graphs.     

双圈图的Laplace谱跨度

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.     

含有割点的图的最小特征值

Let ψ be a certain set of graphs.A graph is called a minimizing graph in the set ψ if its least eigenvalue attains the minimum among all graphs in ψ.In this paper,we determine the unique minimizing graph in ψn,where ψn denotes the set of connected graphs of order n with cut vertices.     

Orientably-regular embeddings of a class of multipartite graphs

The orientably-regular embeddings of complete multipartite graphs have been determined by the contributions of several papers. After that, a natural question can be asked: How about the regular embeddings of the multipartite graphs with m parts, while each part contains n vertices(not necessarily complete multipartite). In this paper, we classify all the orientably-regular embeddings of these graphs when m is a prime q and n is a prime power pe.     

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.     

On Proximal Relations in Transformation Semigroups Arising from Generalized Shifts

For a finite discrete topological space $X$ with at least two elements, a nonempty set $\Gamma$, and a map $\varphi:\Gamma \to \Gamma$, $\sigma_{\varphi}:X^{\Gamma} \to X^{\Gamma}$with $\sigma_{\varphi}((x_{\alpha})_{\alpha \in \Gamma})=(x_{\varphi(\alpha)})_{\alpha \in \Gamma}$ (for $(x_{\alpha})_{\alpha \in \Gamma} \in X^{\Gamma}$) is a generalized shift. In this text for $\mathcal{S} = \{\sigma_{\varphi}:\varphi \in \Gamma^{\Gamma}\}$ and $\mathcal{H}=\{\sigma_{\varphi}:\Gamma \xrightarrow{\varphi} \Gamma$ is bijective$\}$ we study proximal relations of transformation semigroups $(\mathcal{S}, X^{\Gamma})$ and $(\mathcal{H}, X^{\Gamma})$. Regarding proximal relation we prove: $$P(\mathcal{S}, X^{\Gamma}) = \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \exists \beta \in \Gamma (x_{\beta} = y_{\beta})\}$$and $P(\mathcal{H}, X^{\Gamma} ) \subseteq \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \{\beta \in \Gamma : x_{\beta} = y_{\beta}\}$ is infinite$\}$ $\cup\{($ $x,x) : x \in \mathcal{X}\}$. Moreover, for infinite $\Gamma$, both transformation semigroups $(\mathcal{S}, X^{\Gamma})$ and $(\mathcal{H}, X^{\Gamma})$ are regionally proximal, i.e., $Q(\mathcal{S}, X^{\Gamma}) = Q(\mathcal{H}, X^{\Gamma} ) = X^{\Gamma} \times X^{\Gamma}$, also for sydetically proximal relation we have $L(\mathcal{H}, X^{\Gamma}) = \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \{\gamma ∈ \Gamma :$ $x_{\gamma} \neq y_{\gamma}\}$ is finite$\}$.     

关于$3\times 3$阶三角矩阵环上模的研究

文章对$3\times 3$阶三角矩阵环$$\Gamma = \left(\begin{array}{ccc}T & 0 & 0 \\M & U & 0\\{N \otimes _U M} & N & V \\\end{array}\right)$$上的模作了研究,其中T,U,V均是环, M,N分别是U-T, V-U双模.通过用一个五元组$(A,B,C;f,g)$来描述一个左$\Gamma$-模 (其中$A \in \mod T, B\in {\rm mod} U, C \in {\rm mod} V$, $f:M \otimes _T A \to B \in {\rm mod} U, g:N \otimes _U B \to C \in {\rm mod} V$), 文章分别刻画了$\Gamma$上的一致模、空的模、有限嵌入模,并且确定了${ }_\Gamma (A \oplus B \oplus C)$的根和基座．     

森田六元组的几乎分裂序列

令ΛA_1,Λ_2为两个环,M是(A_2-Λ_1)-双模,且N是(Λ_1-Λ_2)-双模.六元组Γ=(Λ_1,Λ_2,N,M,ψ,φ)是一个森田六元组.对于Γ的表示,确定其几乎分裂序列(也称AR-序列)是非常重要的.通过modΛ_1和modΛ_2的右(左)几乎分裂同态、既约同态构造Γ上的相应同态,并进一步确定它的几乎分裂序列.     

强连通双圈有向图的无符号拉普拉斯谱刻画

设$\overrightarrow{G}$ 是一个强连通双圈有向图, $A(\overrightarrow{G})$是其邻接矩阵.设$D(\overrightarrow{G})$ 是$\overrightarrow{G}$的顶点出度的对角矩阵, $Q(\overrightarrow{G})=D(\overrightarrow{G})+A(\overrightarrow{G})$是$\overrightarrow{G}$ 的无符号拉普拉斯矩阵. $Q(\overrightarrow{G})$的谱半径称为$\overrightarrow{G}$的无符号拉普拉斯谱半径.在这篇文章中, 确定了在所有强连通双圈有向图中达到最大或最小无符号拉普拉斯谱半径的唯一有向图. 此外,还证明了任意一个强连通双圈有向图是由它的无符号拉普拉斯谱所确定的.     

Boundary -algebras for acylindrical groups

Let be an infinite, locally finite tree with more than two ends. Let be an acylindrical uniform lattice. Then the boundary algebra is a simple Cuntz-Krieger algebra whose K-theory is determined explicitly.

     

The unit ball of the Hilbert space in its weak topology

We show that the unit ball of in its weak topology is a continuous image of , and we deduce some combinatorial properties of its lattice of open sets which are not shared by the balls of other equivalent norms when is uncountable.

     

Functorial equations for lexicographic products

We generalize the main result of an earlier paper by the authors (Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997), 3177-3183) concerning the convex embeddings of a chain in a lexicographic power . For a fixed non-empty chain , we derive necessary and sufficient conditions for the existence of non-empty solutions to each of the lexicographic functorial equations

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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.