A New Construction for Cohen–Macaulay Graphs

Let G be a finite simple graph on a vertex set V(G) = {x ₁₁,…, x _n1}. Also let m ₁,…, m _n  ≥ 2 be integers and G ₁,…, G _n be connected simple graphs on the vertex sets V(G _i ) = {x _i1,…, x _{im i} }. In this article, we provide necessary and sufficient conditions on G ₁,…, G _n for which the graph obtained by attaching the G _i to G is unmixed or vertex decomposable. Then we characterize Cohen–Macaulay and sequentially Cohen–Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to arbitrary graphs.     

给定边连通度的图的最小距离谱半径(英文)

本文研究了边连通度为r的n阶连通图中距离谱半径最小的极图问题,利用组合的方法,确定了K(n-1,r)为唯一的极图,其中K(n-1,r)是由完全图K_(n-1)添加一个顶点v以及连接v与K_(n-1)中r个顶点的边所构成.上述结论推广了极图理论中的相关结果.     

Distance-transitive dihedrants

The main result of this article is a classification of distance-transitive Cayley graphs on dihedral groups. We show that a Cayley graph X on a dihedral group is distance-transitive if and only if X is isomorphic to one of the following graphs: the complete graph K _2n ; a complete multipartite graph K _t×m with t anticliques of size m, where t m is even; the complete bipartite graph without 1-factor K _n,n − nK ₂; the cycle C _2n ; the incidence or the non-incidence graph of the projective geometry PG _d-1(d,q), d ≥ 2; the incidence or the non-incidence graph of a symmetric design on 11 vertices.     

Cubic one-regular graphs of order twice a square-free integer

A graph is one-regular if its automorphism group acts regularly on the set of its arcs.Let n be a square-free integer.In this paper,we show that a cubic one-regular graph of order 2n exists if and only if n=3~tp1p2…p_s≥13,where t≤1,s≥1 and p_i's are distinct primes such that 3|(P_i—1). For such an integer n,there are 2~(s-1) non-isomorphic cubic one-regular graphs of order 2n,which are all Cayley graphs on the dihedral group of order 2n.As a result,no cubic one-regular graphs of order 4 times an odd square-free integer exist.     

On quadrilaterals in layers of the cube and extremal problems for directed and oriented graphs

Erdős has conjectured that every subgraph of the n‐cube Q_n having more than (1/2 + o(1))e(Q_n) edges will contain a 4‐cycle. In this note we consider ‘layer’ graphs, namely, subgraphs of the cube spanned by the subsets of sizes k − 1, k and k + 1, where we are thinking of the vertices of Q_n as being the power set of {1,…, n}. Observe that every 4‐cycle in Q_n lies in some layer graph. We investigate the maximum density of 4‐cycle free subgraphs of layer graphs, principally the case k = 2. The questions that arise in this case are equivalent to natural questions in the extremal theory of directed and undirected graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 66–82, 2000     

Bounding the Clique‐Width of H‐Free Chordal Graphs

A graph is H‐free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique‐width. Brandstädt, Le, and Mosca erroneously claimed that the gem and co‐gem are the only two 1‐vertex P₄‐extensions H for which the class of H‐free chordal graphs has bounded clique‐width. In fact we prove that bull‐free chordal and co‐chair‐free chordal graphs have clique‐width at most 3 and 4, respectively. In particular, we find four new classes of H‐free chordal graphs of bounded clique‐width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H‐free chordal graphs has bounded clique‐width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of ‐free graphs has bounded clique‐width via a reduction to K₄‐free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique‐width of H‐free weakly chordal graphs.     

Hamilton cycle rich two‐factorizations of complete graphs

For all integers n ≥ 5, it is shown that the graph obtained from the n‐cycle by joining vertices at distance 2 has a 2‐factorization is which one 2‐factor is a Hamilton cycle, and the other is isomorphic to any given 2‐regular graph of order n. This result is used to prove several results on 2‐factorizations of the complete graph K_n of order n. For example, it is shown that for all odd n ≥ 11, K_n has a 2‐factorization in which three of the 2‐factors are isomorphic to any three given 2‐regular graphs of order n, and the remaining 2‐factors are Hamilton cycles. For any two given 2‐regular graphs of even order n, the corresponding result is proved for the graph K_n ‐ I obtained from the complete graph by removing the edges of a 1‐factor. © 2004 Wiley Periodicals, Inc.     

On graphs determining links with maximal number of components via medial construction

Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and μ(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that μ(D(G))≤n(G)+1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with μ(D(G))=n(G)+1. An algorithm is given firstly to judge whether a graph is extremal or not, then we prove that all extremal graphs can be obtained from K₁ by applying two graph operations repeatedly. We also present a dual characterization of extremal graphs and finally we provide a simple criterion on structures of bridgeless extremal graphs.     

Zero-Divisor Semigroups and Some Simple Graphs

In this article, we study commutative zero-divisor semigroups determined by graphs. We prove that for all n ≥ 4, the complete graph K _n together with two end vertices has a unique corresponding zero-divisor semigroup, while the complete graph K _n together with three end vertices has no corresponding semigroups. We determine all the twenty zero-divisor semigroups whose zero-divisor graphs are the complete graph K ₃ together with an end vertex.     

Pairs of Chromatically Equivalent Graphs

Two graphs are said to be chromatically equivalent if they have the same chromatic polynomial. In this paper we give the means to construct infinitely many pairs of chromatically equivalent graphs where one graph in the pair is clique-separable, that is, can be obtained by identifying an r-clique in some graph H ₁ with an r-clique in some graph H ₂, and the other graph is non-clique-separable. There are known methods for finding pairs of chromatically equivalent graphs where both graphs are clique-separable or both graphs are non-clique-separable. Although examples of pairs of chromatically equivalent graphs where only one of the graphs is clique-separable are known, a method for the construction of infinitely many such pairs was not known. Our method constructs such pairs of graphs with odd order n ≥ 9.     

EMBEDDINGS OF ONE KIND OF GRAPHS

The purpose of this paper is to display a new kind of simple graphs which belong to B. inwhich any graph has its orientable genus n,n≥3. Furthermore, for any integer k,1≤k≤n,there exists a graph B^kn of B. such that the non-orientable genus of B^kn is k.     

On graphs with the largest Laplacian index

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results. The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006). The work was done when Z. Chen was on sabbatical in China.     

Commuting projections on graphs

Motivated by the increasing importance of large‐scale networks typically modeled by graphs, this paper is concerned with the development of mathematical tools for solving problems associated with the popular graph Laplacian. We exploit its mixed formulation based on its natural factorization as product of two operators. The goal is to construct a coarse version of the mixed graph Laplacian operator with the purpose to construct two‐level, and by recursion, a multilevel hierarchy of graphs and associated operators. In many situations in practice, having a coarse (i.e., reduced dimension) model that maintains some inherent features of the original large‐scale graph and respective graph Laplacian offers potential to develop efficient algorithms to analyze the underlined network modeled by this large‐scale graph. One possible application of such a hierarchy is to develop multilevel methods that have the potential to be of optimal complexity. In this paper, we consider general (connected) graphs and function spaces defined on its edges and its vertices. These two spaces are related by a discrete gradient operator, ‘Grad’ and its adjoint, ‘ ? Div’, referred to as (negative) discrete divergence. We also consider a coarse graph obtained by aggregation of vertices of the original one. Then, a coarse vertex space is identified with the subspace of piecewise constant functions over the aggregates. We consider the ?₂‐projection Q_H onto the space of these piecewise constants. In the present paper, our main result is the construction of a projection π_H from the original edge‐space onto a properly constructed coarse edge‐space associated with the edges of the coarse graph. The projections π_H and Q_H commute with the discrete divergence operator, that is, we have Div π_H = Q_H div. The respective pair of coarse edge‐space and coarse vertex‐space offer the potential to construct two‐level, and by recursion, multilevel methods for the mixed formulation of the graph Laplacian, which utilizes the discrete divergence operator. The performance of one two‐level method with overlapping Schwarz smoothing and correction based on the constructed coarse spaces for solving such mixed graph Laplacian systems is illustrated on a number of graph examples. Copyright © 2013 John Wiley & Sons, Ltd.     

K3,3-free intersection graphs of finite groups

The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if H∩K≠1 where 1 denotes the trivial subgroup of G. In this paper we classify all finite groups whose intersection graphs are K_3,3-free.     

The Number of Spanning Trees in <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-Complements of Quasi-Threshold Graphs

In this paper we examine the classes of graphs whose K_n-complements are trees or quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of K_n, the K_n-complement of H is the graph K_n–H which is obtained from K_n by removing the edges of H. Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of the graph. Our results generalize previous results and extend the family of graphs of the form K_n–H admitting formulas for the number of their spanning trees.Final version received: March 18, 2004     

Maximal σ‐polynomials of connected 3‐chromatic graphs

In the set of graphs of order n and chromatic number k the following partial order relation is defined. One says that a graph G is less than a graph H if c_i(G) ≤ c_i(H) holds for every i, k ≤ i ≤ n and at least one inequality is strict, where c_i(G) denotes the number of i‐color partitions of G. In this paper the first ? n/2 ? levels of the diagram of the partially ordered set of connected 3‐chromatic graphs of order n are described. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 210–222, 2003     

The Crossing Number of Cartesian Products of Complete Bipartite Graphs K 2, m with Paths P n

Most results on the crossing number of a graph focus on the special graphs, such as Cartesian products of small graphs with paths P_n, cycles C_n or stars S_n. In this paper, we extend the results to Cartesian products of complete bipartite graphs K_2,m with paths P_n for arbitrary m ≥ 2 and n ≥ 1. Supported by the NSFC (No. 10771062) and the program for New Century Excellent Talents in University.     

Almost regular edge colorings and regular decompositions of complete graphs

For k = 1 and k = 2, we prove that the obvious necessary numerical conditions for packing t pairwise edge‐disjoint k‐regular subgraphs of specified orders m₁,m₂,… ,m_t in the complete graph of order n are also sufficient. To do so, we present an edge‐coloring technique which also yields new proofs of various known results on graph factorizations. For example, a new construction for Hamilton cycle decompositions of complete graphs is given. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 499–506, 2008