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.     

完全图删边子图的子刻画

In this paper, we show that some edges-deleted subgraphs of complete graph are determined by their spectrum with respect to the adjacency matrix as well as the Laplacian matrix.     

单圈图的零度

图的零度是指在图的谱中特征值0的重数.在文献[2]中作者给出了刻画非奇异单圈图的充分条件,并提出了一个问题,即这个条件是否也是必要的.在本文中,我们先对这个问题作出肯定回答,然后介绍一个新的概念:保留点,最后通过最大匹配数给出公式计算单圈图的零度.     

通过增强型萨格勒布指数对图排序

Recently, Furtula et al. proposed a valuable predictive index in the study of the heat of formation in octanes and heptanes, the augmented Zagreb index(AZI index) of a graph G, which is defined as AZI(G) =∑uv∈E(G)( d_u d_v/d_u + d_v-2)~3,where E(G) is the edge set of G, d u and d v are the degrees of the terminal vertices u and v of edge uv, respectively. In this paper, we obtain the first five largest(resp., the first two smallest) AZI indices of connected graphs with n vertices. Moreover, we determine the trees of order n with the first three smallest AZI indices, the unicyclic graphs of order n with the minimum, the second minimum AZI indices, and the bicyclic graphs of order n with the minimum AZI index, respectively.     

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.     

Some results on strongly regular graphs from unions of cyclotomic classes

In this paper, we construct some families of strongly regular graphs on finite fields by using unions of cyclotomic classes and index 2 Gauss sums. New infinite families of strongly regular graphs are found.     

Cycles Containing a Subset of a Given Set of Elements in Cubic Graphs

The technique of contractions and the known results in the study of cycles in $3$-connected cubic graphs are applied to obtain the following result. Let $G$ be a $3$-connected cubic graph, $X\subseteq V(G)$ with $|X| = 16$ and $e\in E(G)$. Then either for every $8$-subset $A$ of $X$, $A\cup\{e\}$ is cyclable or for some $14$-subset $A$ of $X$, $A\cup\{e\}$ is cyclable.     

序半群序同余的一个注记

序半群S的什么子集可以作为S的同余类是一个重要的问题. 在文[8]中,作者证明了如果序半群S的 理想$C$是$S$的某个同余类, 则$C$是凸的; 而且当$C$是强凸理想时，逆命题成立. 在本文中, 我们给出了序半群同余的一个新的构造，并证明了序半群$S$的理想$B$是$S$的某个同余类的充要条件是$B$是凸的.     

关于多项式映射自同构恒等集的一个注记

本文给出了判定一个仿射代数集是否是一个自同构恒等集的充分条件.作为一个推论,我们给出了Mckay-Wang的一个问题的一个新证明.我们也给出了一些具体的例子来说明主要的定理.     

关于FP-内射维数的一个注记

Let R and S be a left coherent ring and a right coherent ring respectively,RωS be a faithfully balanced self-orthogonal bimodule.We give a sufficient condition to show that l.FP-idR（ω） ∞ implies G-dimω（M） ∞,where M ∈ modR.This result generalizes the result by Huang and Tang about the relationship between the FP-injective dimension and the generalized Gorenstein dimension in 2001.In addition,we get that the left orthogonal dimension is equal to the generalized Gorenstein dimension when G-dimω（M） is finite.     

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.     

一类单圈图的拉普拉斯谱刻画

Let H（n; q, n1, n2, n3, n4） be a unicyclic graph with n vertices containing a cycle Cq and four hanging paths Ph1＋1, Pn2＋1, Pn3＋1 and Pn4＋1 attached at the same vertex of the cycle. In this paper, it is proved that all unicyclic graphs H （n; q, n1, n2, n3, n4） are determined by their Laplacian spectra.     

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.     

单圈图的极小能量

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Undenote the set of all connected unicyclic graphs with order n, and Ur n= {G ∈ Un| d(x) = r for any vertex x ∈ V(Cl)}, where r ≥ 2 and Cl is the unique cycle in G. Every unicyclic graph in Ur nis said to be a cycle-r-regular graph.In this paper, we completely characterize that C39(2, 2, 2) ο Sn-8is the unique graph having minimal energy in U4 n. Moreover, the graph with minimal energy is uniquely determined in Ur nfor r = 3, 4.     

强正则自补图的一个注记

Kotzig put forward a question on strongly-regular self-complementary graphs, that is, for any natural number k, whether there exists a strongly-regular self- complementary graph whose order is 4k 1, where 4k 1=x2 y2, x and y are positive integers; what is the minimum number that made there exist at least two non-isomorphic strongly-regular self-complementary graphs. In this paper, we use two famous lemmas to generalize the existential conditions for strongly-regular self-complementary circular graphs with 4k 1 orders.     

A Note on Degenerate and Spectrally Degenerate Graphs

A graph G is called spectrally d‐degenerate if the largest eigenvalue of each subgraph of it with maximum degree D is at most . We prove that for every constant M there is a graph with minimum degree M, which is spectrally 50‐degenerate. This settles a problem of Dvo?ák and Mohar (Spectrally degenerate graphs: Hereditary case, arXiv: 1010.3367).     

单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.     

Degree Condition for Subdivisions of Unicyclic Graphs

If H is any graph of order n with k non-trivial components, each of which contains at most one cycle, then every graph of order at least n and minimum degree at least n − k contains a subdivision of H such that only edges contained in a cycle in H are subdivided.