Erds-Ko-Rado Theorem for Ladder Graphs

For a graph G and an integer r≥1,G is r-EKR if no intersecting family of independent r-sets of G is larger than the largest star(a family of independent r-sets containing some fixed vertex in G),and G is strictly r-EKR if every extremal intersecting family of independent r-sets is a star.Recently,Hurlbert and Kamat gave a preliminary result about EKR property of ladder graphs.They showed that a ladder graph with n rungs is 3-EKR for all n≥3.The present paper proves that this graph is r-EKR for all 1≤r≤n,and strictly r-EKR except for r=n-1.     

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.     

5连通图中的可去边

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.     

Bounds on the Aα-Spectral Radius of a C3-Free Graph

Let G be a simple undirected graph.For any real numberα∈[0,1],Nikiforov defined the A_α-matrix of G as A_α(G)=αD(G)+(1-α)A(G)in 2017,where A(G)and D(G)are the adjacency matrix and the degree diagonal matrix of G,respectively.In this paper,we obtain a lower bound on the A_α-spectral radius of a C₃-free graph forα∈[0,1)and a sharp upper bound on the Aα-spectral radius of a C₃-free k-cycle graph forα∈[1/2,1).     

A result on quasi k-connected graphs

Let G be a k-connected graph, and T be a subset of V(G)If G- T is not connected,then T is said to be a cut-set of GA k-cut-set T of G is a cut-set of G with |T | = kLet T be a k-cut-set of a k-connected graph GIf G- T can be partitioned into subgraphs G1 and G2 such that |G1| ≥ 2, |G2| ≥ 2, then we call T a nontrivial k-cut-set of GSuppose that G is a(k- 1)-connected graph without nontrivial(k- 1)-cut-setThen we call G a quasi k-connected graphIn this paper, we prove that for any integer k ≥ 5, if G is a k-connected graph without K-4, then every vertex of G is incident with an edge whose contraction yields a quasi k-connected graph, and so there are at least|V(G)|2edges of G such that the contraction of every member of them results in a quasi k-connected graph.     

On Edge Singularity and Eigenvectors of Mixed Graphs

Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.     

（g,f）-FACTORS WITH SPECIAL PROPERTIES IN BIPARTITE （mg,mf）-GRAPHS

Let G be a bipartite graph and g and f be two positive integer-valued functions defined on vertex set V(G) of G such that g(x)≤f(x). In this paper,some sufficient conditions related to the connectivity and edge-connectivity for a bipartite (mg,mf)-graph to have a (g,f)-factor with special properties are obtained and some previous results are generalized.Furthermore,the new results are proved to be the best possible.     

Continuous Forcing Spectra of Even Polygonal Chains

Let G be a graph that admits a perfect matching M.A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G.The cardinality of a forcing set of M with the smallest size is called the forcing number of M,denoted by f(G,M).The forcing spectrum of G is defined as:Spec(G)={f(G,M)|M is a perfect matching of G}.In this paper,by applying the Ztransformation graph(resonance graph)we show that for any polyomino with perfect matchings and any even polygonal chain,their forcing spectra are integral intervals.Further we obtain some sharp bounds on maximum and minimum forcing numbers of hexagonal chains with given number of kinks.Forcing spectra of two extremal chains are determined.     

Vertex partitions of r-edge-colored graphs

Let G be an edge-colored graph.The monochromatic tree partition problem is to find the minimum number of vertex disjoint monochromatic trees to cover the all vertices of G.In the authors' previous work,it has been proved that the problem is NP-complete and there does not exist any constant factor approximation algorithm for it unless P=NP.In this paper the authors show that for any fixed integer r≥5,if the edges of a graph G are colored by r colors,called an r-edge-colored graph,the problem remains NP-complete.Similar result holds for the monochromatic path(cycle)partition problem.Therefore,to find some classes of interesting graphs for which the problem can be solved in polynomial time seems interesting. A linear time algorithm for the monochromatic path partition problem for edge-colored trees is given.     

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.     

以完全二部图作为星补的正则图的刻画

设$G$是一个$n$阶图, $\mu$是$G$的一个$(k\ge 1)$重邻接特征值. 图$G$中关于$\mu$的星补$H$是指$G$的不含特征值$\mu$的$n-k$阶诱导子图,且顶点集$X=V(G-H)$称为图$G$中关于$\mu$的星集.星补技术提供了利用部分子结构来重建满足特定性质的整个图的谱工具. 本文我们研究了关于特征值$\mu$的以$K_{t,s}~(s\ge t\ge 2)$作为是补的正则图, 特别地, 我们完全刻画了$t=3$的情形, 获得了当$t=s$时的一些性质, 并提出了有待进一步研究的问题.     

由圈长分布确定的二部图~$K_{n,n r}-A\ (|A|\leq 3)$

The cycle length distribution of a graph G of order n is a sequence （c1 （G）,…, cn （G））, where ci （G） is the number of cycles of length i in G. In general, the graphs with cycle length distribution （c1（G） ,…,cn（G）） are not unique. A graph G is determined by its cycle length distribution if the graph with cycle length distribution （c1 （G）,…, cn （G）） is unique. Let Kn,n＋r be a complete bipartite graph and A lohtaib in E（Kn,n＋r）. In this paper, we obtain： Let s 〉 1 be an integer. （1） If r = 2s, n 〉 s（s - 1） ＋ 2｜A｜, then Kn,n＋r - A （A lohtain in E（Kn,n＋r）,｜A｜ ≤ 3） is determined by its cycle length distribution; （2） If r = 2s ＋ 1,n 〉 s^2 ＋ 2｜A｜, Kn,n＋r - A （A lohtain in E（Kn,n＋r）, ｜A｜ ≤3） is determined by its cycle length distribution.     

树和单圈图的斜谱矩

给定图$G$,对图$G$的每条边确定一个方向,称为$G$的定向图$G^\sigma$, $G$称为$G^\sigma$的基础图. $G^\sigma$的斜邻接矩阵$S(G^\sigma)$是反对称矩阵,其特征值是0或纯虚数. $S(G^\sigma)$所有特征值的$k$次幂之和称为$G^\sigma$的$k$阶斜谱矩,其中$k$是非负整数.斜谱矩序列可用于对图进行排序.本文主要研究定向树和定向单圈图的斜谱矩,并对这两类图的斜谱矩序列依照字典序进行排序.首先确定了直径为$d$的树作为基础图的所有定向树中,斜谱矩序最大的$2\lfloor\frac{d}{4}\rfloor$个图; 然后确定以围长为$g$的单圈图作为基础图的所有定向单圈图中, 斜谱矩序最大的$2\lfloor\frac{g}{4}\rfloor+1$个图.     

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.     

一类几乎唯一泛圈图

设G是阶为n的简单Hamilton图．若存在m(3(?)m相似文献   

有约束条件的图的(g,f)-因子

设G是一个图,用V(G)和E(G)表示它的顶点集和边集,并设g和f是定义在V(G)上的两个整数值函数且g＜f.图G的一个(g,f)-因子是G的一个支撑子图F使对任意的x∈V(G)有g(x)≤dF(x)≤f(x).如果过图G的任意k条边都有一个(g,f)-因子,则称图G是一个(g,f)-k-覆盖图.如果图G的任意k条边不属于它的一个(g,f)-因子,则称图G是一个(g,f)-k-消去图.作者分别给出了一个图是(g,f)-k-覆盖图和(g,f)-k-消去图的充分条件.     

不含三角形图的$A_\alpha$-谱半径的界

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.     

二部图的E-H-不可收缩性

By End（G） and hEnd（G） we denote the set of endomorphisms and half-strong endomorphisms of a graph G respectively. A graph G is said to be E-H-unretractive if End（G） = hEnd（G）. A general characterization of an E-H-unretractive graph seems to be difficult. In this paper, bipartite graphs with E-H-unretractivity are characterized explicitly.     

2-连通半无爪图的最长圈

A graph G is called quasi-claw-free if it satisfies the property:d(x,y)=2 there exists a vertex u∈N(x)∩N(y)such that N[u]■N[x]∪N[y].In this paper,we show that every 2-connected quasi-claw-free graph of order n with G■F contains a cycle of length at least min{3δ+2,n},where F is a family of graphs.