关于单圈图与无交双圈图的邻接矩阵的行列式

研究图的邻接矩阵的行列式主要是为了研究图的零特征值的重数，而零特征值的重数在化学分子结构图的稳定性问题中有广泛的应用．本文给出了单圈图及无交双圈图的邻接矩阵的行列式分类．     

Cycle embedding in star graphs with more conditional faulty edges

The star graph is one of the most attractive interconnection networks. The cycle embedding problem is widely discussed in many networks, and edge fault tolerance is an important issue for networks since edge failures may occur when a network is put into use. In this paper, we investigate the cycle embedding problem in star graphs with conditional faulty edges. We show that there exist fault-free cycles of all even lengths from 6 to n! in any n-dimensional star graph S_n (n ? 4) with ?3n − 10 faulty edges in which each node is incident with at least two fault-free edges. Our result not only improves the previously best known result where the number of tolerable faulty edges is up to 2n − 7, but also extends the result that there exists a fault-free Hamiltonian cycle under the same condition.     

On the adjacency matrix of a threshold graph

A threshold graph on n   vertices is coded by a binary string of length n−1$n-1$. We obtain a formula for the inertia of (the adjacency matrix of) a threshold graph in terms of the code of the graph. It is shown that the number of negative eigenvalues of the adjacency matrix of a threshold graph is the number of ones in the code, whereas the nullity is given by the number of zeros in the code that are preceded by either a zero or a blank. A formula for the determinant of the adjacency matrix of a generalized threshold graph and the inverse, when it exists, of the adjacency matrix of a threshold graph are obtained. Results for antiregular graphs follow as special cases.     

The competition number of a graph and the dimension of its hole space

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of the important research problems in the study of competition graphs to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph has been studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is not smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.     

Short Cycle Structure of Graphs on Surfaces （1） -the Uniqueness Theorems

In this paper we investigate cycle base structures of a （weighted） graph and show that much information of short cycles is contained in a MCB （minimum cycle base）. After setting up a Hall type theorem for base-transformation, we give a sufficient and necessary condition for a cycle base to be a MCB. Further more, we show that the structure of MCB in a （weighted） graph is unique. In the case of nonnegative weight, every pair of MCB have the same number of k-cycles for each integer k ≥ 3. The property is also true for those having longest length （although much work has been down in evaluating MCB, little is known for those having longest length）.     

The Ramsey number for a cycle of length six versus a clique of order eight

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C_m denote a cycle of length m and K_n a complete graph of order n. In this paper, it is shown that R(C₆,K₈)=36.     

Fundamental group and cycle space of dual graphs and applications

In this work, we study the fundamental group of dual graph of a planar graph. Moreover, we show that a planar graph G has no cut vertex if and only if N(Π(D(G))) = N(Π(D(G − v))) − 1 for any v ∈ V(G). Some applications relevant to quantum space time are indicated. Our results generalize and extend results in paper [1] [S.I. Nada, E.H. Hamouda, Fundamental group of dual graphs and applications to quantum space time, Chaos Soliton Fractals 42 (2009) 500-503].     

On even cycle decompositions of line graphs of cubic graphs

An even cycle decomposition of a graph is a partition of its edges into cycles of even length. In 2012, Markström conjectured that the line graph of every 2-connected cubic graph has an even cycle decomposition and proved this conjecture for cubic graphs with oddness at most 2. However, for 2-connected cubic graphs with oddness 2, Markström only considered these graphs with a chordless 2-factor. (A chordless 2-factor of a graph is a 2-factor consisting of only induced cycles.) In this paper, we first construct an infinite family of 2-connected cubic graphs with oddness 2 and without chordless 2-factors. We then give a complete proof of Markström’s result and further prove this conjecture for cubic graphs with oddness 4.     

CYCLE SPACES OF GRAPHS ON THE SPHERE AND THE PROJECTIVE PLANE

Cycle base theory of a graph has been well studied in abstract mathematical field such matroid theory as Whitney and Tutte did and found many applications in pratical uses such as electric circuit theory and structure analysis, etc. In this paper graph embedding theory is used to investigate cycle base structures of a 2-(edge)-connected graph on the sphere and the projective plane and it is shown that short cycles do generate the cycle spaces in the case of ““““small face-embeddings““““. As applications the authors find the exact formulae for the minimum lengthes of cycle bases of some types of graphs and present several known results. Infinite examples shows that the conditions in their main results are best possible and there are many 3-connected planar graphs whose minimum cycle bases can not be determined by the planar formulae but may be located by re-embedding them into the projective plane.     

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   

Dynamic feedback control for cycle slipping in a phase-controlled system

This paper concerns the problem of cycle slipping for continuous phase-controlled systems with periodic nonlinearity. The number of slipped cycles is an important property in the transient mode of such nonlinear systems. On the basis of the Yakubovich–Kalman lemma, linear matrix inequality (LMI) characterizations are derived for the number of slipped cycles of such systems and an efficient way of estimating the number is proposed by solving a generalized eigenvalue minimization problem. Furthermore, by virtue of these results, a dynamic output feedback controller is designed to guarantee the nonexistence of cycle slipping. As a result, the transient performance of phase-controlled system is improved. A concrete application to the phase-locked loop shows the applicability and validity of the proposed approach.     

Diameter graphs of polygons and the proof of a conjecture of Graham

We show that for an n-gon with unit diameter to have maximum area, its diameter graph must contain a cycle, and we derive an isodiametric theorem for such n-gons in terms of the length of the cycle. We then apply this theorem to prove Graham's 1975 conjecture that the diameter graph of a maximal 2m-gon (m?3) must be a cycle of length 2m−1 with one additional edge attached to it.     

Path and cycle decompositions of complete equipartite graphs: Four parts

We show that a complete equipartite graph with four partite sets has an edge-disjoint decomposition into cycles of length k if and only if k≥3, the partite set size is even, k divides the number of edges in the equipartite graph and the total number of vertices in the graph is at least k. We also show that a complete equipartite graph with four even partite sets has an edge-disjoint decomposition into paths with k edges if and only if k divides the number of edges in the equipartite graph and the total number of vertices in the graph is at least k+1.     

The structure of graphs with given lengths of cycles

The cycle spectrum of a given graph G is the lengths of cycles in G. In this paper, we introduce the following problem: determining the maximum number of edges of an n-vertex graph with given cycle spectrum. In particular, we determine the maximum number of edges of an n-vertex graph without containing cycles of lengths three and at least k. This can be viewed as an extension of a well-known result of Erd?s and Gallai concerning the maximum number of edges of an n-vertex graph without containing cycles of lengths at least k. We also determine the maximum number of edges of an n-vertex graph whose cycle spectrum is a subset of two positive integers.     

A bound on the total size of a cut cover

A cycle cover (cut cover) of a graph G is a collection of cycles (cuts) of G that covers every edge of G at least once. The total size of a cycle cover (cut cover) is the sum of the number of edges of the cycles (cuts) in the cover.We discuss several results for cycle covers and the corresponding results for cut covers. Our main result is that every connected graph on n vertices and e edges has a cut cover of total size at most 2e-n+1 with equality precisely when every block of the graph is an odd cycle or a complete graph (other than K₄ or K₈). This corresponds to the result of Fan [J. Combin. Theory Ser. B 74 (1998) 353-367] that every graph without cut-edges has a cycle cover of total size at most e+n-1.     

A contribution to queens graphs: A substitution method

A graph G is a queens graph if the vertices of G can be mapped to queens on the chessboard such that two vertices are adjacent if and only if the corresponding queens attack each other, i.e. they are in horizontal, vertical or diagonal position.We prove a conjecture of Beineke, Broere and Henning that the Cartesian product of an odd cycle and a path is a queens graph. We show that the same does not hold for two odd cycles. The representation of the Cartesian product of an odd cycle and an even cycle remains an open problem.We also prove constructively that any finite subgraph of the rectangular grid or the hexagonal grid is a queens graph.Using a small computer search we solve another conjecture of the authors mentioned above, saying that K_3,4 minus an edge is a minimal non-queens graph.     

On mixed block graphs

A mixed complete graph is obtained from a directed cycle of length at least three by adding all the possible arcs between any non-adjacent vertices of the underlying cycle. A mixed block graph is a strongly connected directed graph whose blocks are mixed complete graphs. In this paper, we give the inverse of the distance matrix of the mixed block graph.     

Hamilton cycles in graph bundles over a cycle with tree as a fibre

A sufficient and necessary condition for the existence of a Hamilton cycle in a graph bundle with a cycle as a base and a tree as a fibre is obtained.     

Ore-type conditions for bipartite graphs containing hexagons

Let G=(V₁,V₂;E) be a bipartite graph with |V₁|=|V₂|=3k, where k>0. In this paper it is proved that if d(x)+d(y)≥4k−1 for every pair of nonadjacent vertices x∈V₁, y∈V₂, then G contains k−1 independent cycles of order 6 and a path of order 6 such that all of them are independent. Furthermore, if d(x)+d(y)≥4k for every pair of nonadjacent vertices x∈V₁, y∈V₂ and k>2, then G contains k−2 independent cycles of order 6 and a cycle of order 12 such that all of them are independent.     

Embedding cycles of various lengths into star graphs with both edge and vertex faults

The n-dimensional star graph S_n is an attractive alternative to the hypercube graph and is a bipartite graph with two partite sets of equal size. Let F_v and F_e be the sets of faulty vertices and faulty edges of S_n, respectively. We prove that S_n − F_v − F_e contains a fault-free cycle of every even length from 6 to n! − 2∣F_v∣ with ∣F_v∣ + ∣F_e∣ ? n − 3 for every n ? 4. We also show that S_n − F_v − F_e contains a fault-free path of length n! − 2∣F_v∣ − 1 (respectively, n! − 2∣F_v∣ − 2) between two arbitrary vertices of S_n in different partite sets (respectively, the same partite set) with ∣F_v∣ + ∣F_e∣ ? n − 3 for every n ? 4.