The join of split graphs whose half-strong endomorphisms form a monoid

In this paper, the half-strong endomorphisms of the join of split graphs are investigated. We give the conditions under which the half-strong endomorphisms of the join of split graphs form a monoid.     

Unretractive and S-unretractive joins and lexicographic products of graphs

Graphs without proper endomorphisms are the subject of this article. It is shown that the join of two graphs has this property if and only if both summands have it, and that the lexicographic product of a complete graph or an odd circuit as first factors has this property if and only if the second factor has it. A somewhat stronger theorem is proved if the lexicographic product has no proper strong endomorphism. The corresponding result for the join is the same as for usual endomorphisms.     

一族具有正则绐幺半群的图

本文给出了一族具有正则自同态么半群的图及相应的自同态计数公式．     

Graphs whose endomorphism monoids are regular

In this paper, we give several approaches to construct new End-regular (-orthodox) graphs by means of the join and the lexicographic product of two graphs with certain conditions. In particular, the join of two connected bipartite graphs with a regular (orthodox) endomorphism monoid is explicitly described.     

广义联图的正则性

讨论了两个图的广义联图的End-正则性,给出了当图X、Y的广义联图G（y1,…ym）End-正则时,图X也End-正则应满足的条件。     

Endomorphisms of graphs II. Various unretractive graphs

In this part of the article we investigate graphs for which different endomorphism monoids coincide. We consider endomorphisms, strong endomorphisms and automorphisms. Coincidences are investigated for joins of graphs and some lexicographic products. In an additional section the graphs with the respective properties are listed up to 8 vertices in two cases and up to 5 vertices in the remaining case.     

Green's relations on the strong endomorphism monoid of a graph

Combinatorial characteristics of Green's relations on the monoid of strong endomorphisms of graphs are explored. As an application, a short proof of the fact thats End(G) is regular is given. The author would like to thank Professor Dr. U. Knauer and Dr. E. Wilkeit for valuable advice and Professor T. E. Hall for helpful suggestions and corrections made to an earlier version of this paper.     

On betweenness-uniform graphs

The betweenness centrality of a vertex of a graph is the fraction of shortest paths between all pairs of vertices passing through that vertex. In this paper, we study properties and constructions of graphs whose vertices have the same value of betweenness centrality (betweenness-uniform graphs); we show that this property holds for distanceregular graphs (which include strongly regular graphs) and various graphs obtained by graph cloning and local join operation. In addition, we show that, for sufficiently large n, there are superpolynomially many betweenness-uniform graphs on n vertices, and explore the structure of betweenness-uniform graphs having a universal or sub-universal vertex.     

A generalization of de Bruijn graphs and classification of endomorphisms of Cuntz algebras by graph invariants

The classification problem of endomorphisms of the Cuntz algebra O_N\mathcal{O}_{N} is solved by using graph theory. We introduce permutative de Bruijn graphs as generalizations of de Bruijn graphs. Branching laws for a permutative endomorphism ρ of O_N\mathcal{O}_{N} are computed by using the permutative de Bruijn graph associated with ρ. According to this correspondence between endomorphisms and graphs, we classify permutative endomorphisms of O_N\mathcal{O}_{N} by graph invariants concretely.     

Spectra of graphs obtained by a generalization of the join graph operation

Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.     

Decomposition of perfect graphs

We describe composition and decomposition schemes for perfect graphs, which generalize all recent results in this area, e.g., the amalgam and the 2-amalgam split. Our approach is based on the consideration of induced cycles and their complements in perfect graphs (as opposed to the consideration of cycles for defining biconnected or 3-connected graphs). Our notion of 1-inseparable graphs is “parallel” to that of biconnected graphs in that different edges in different inseparable components of a graph are not contained in any induced cycle or any complement of an induced cycle. Furthermore, in a special case which generalizes the join operation, this definition is self-complementary in a natural fashion. Our 2-composition operation, which only creates even induced cycles in the composed graphs, is based on two perfection-preserving vertex merge operations on perfect graphs. As a by-product, some new properties of minimally imperfect graphs are presented.     

Split graphs whose half-strong endomorphisms form a monoid

In this paper,the half-strong,the locally strong and the quasi-strong endomorphisms of a split graph are investigated.Let X be a split graph and let End(X),hEnd(X),lEnd(X) and qEnd(X) be the endomorphism monoid,the set of all half-strong endomorphisms,the set of all locally strong endomorphisms and the set of all quasi-strong endomorphisms of X,respectively.The conditions under which hEnd(X) forms a submonoid of End(X) are given.It is shown that lEnd(X) = qEnd(X) for any split graph X.The conditions under which lEnd(X)(resp.qEnd(X)) forms a submonoid of End(X) are also given.In particular,if hEnd(X) forms a monoid,then lEnd(X)(resp.qEnd(X)) forms a monoid too.     

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.     

Endomorphism Spectra of Bipartite Graphs with Diameter Three and Girth Six

There are different endomorphisms for a graph. For a more systematic treatment of these endomorphisms the endomorphism spectrum and the endomorphism type of a graph are defined. Knauer characterized trees using their endomorphism types. The endomorphism type of bipartite graphs with diameter three and girth six is given in this paper.AMS Subject Classification (1991): 05C25 20M20Supported by the National Natural Science Foundation of China (19901012)     

Minimizing cyclic cutwidth of graphs using a memetic algorithm

Cyclic cutwidth minimization problem (CCMP) consists of embedding a graph onto a circle such that the maximum cutwidth in a region is minimized. It is an NP-complete problem and for some classes of graphs, exact results of cyclic cutwidth have been proved in literature. However, no algorithm has been reported for general graphs. In this paper, a memetic algorithm is proposed for CCMP in which we have designed six construction heuristics in order to generate a good initial population and also local search is employed to improve the solutions in each generational phase. The algorithm achieves optimal results for the classes of graphs with known exact results. Extensive experiments have also been conducted on some classes of graphs for which exact results are not known such as the complete split graph, join of hypercubes, toroidal meshes, cone graph and some instances of Harwell-Boeing graphs which are a subset of public domain Matrix Market library. Trends observed in the experimental results for some of the classes of graphs have led to conjectures for cyclic cutwidth.     

Finite Invariant Sets for a Family of Endomorphisms in Compact Graphs

We prove several results concerning the existence of common invariant finite sets of vertices or of common invariant finite connected subgraphs for some families of endomorphisms of graphs whose ends are all dominated.     

Circuit decompositions of join‐covered graphs

In this paper, we focus our attention on join‐covered graphs, that is, ±1‐weighted graphs, without negative circuits, in which every edge lies in a zero‐weight circuit. Join covered graphs are a natural generalization of matching‐covered graphs. Many important properties of matching covered graphs, such as the existence of a canonical partition, tight cut decomposition and ear decomposition, have been generalized to join covered graphs by A. Seb? [5]. In this paper we prove that any two edges of a join‐covered graph lie on a zero‐weight circuit (under an equivalent weighting), generalize this statement to an arbitrary number of edges, and characterize minimal bipartite join‐covered graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 220–233, 2009     

Ear decompositions of join covered graphs

Join covered graphs are ±1-weighted graphs, without negative circuits, in which every edge lies in a zero-weight circuit. Join covered graphs are a natural generalization of matching covered graphs. Many important properties of matching covered graphs have been generalized to join covered graphs. In this paper, we generalize Lovász and Plummerʼs ear decomposition theorem of matching covered graphs to join covered graphs.     

Spectral characterization of multicone graphs

A multicone graph is defined to be the join of a clique and a regular graph. Based on Zhou and Cho’s result [B. Zhou, H.H. Cho, Remarks on spectral radius and Laplacian eigenvalues of a graph, Czech. Math. J. 55 (130) (2005), 781–790], the spectral characterization of multicone graphs is investigated. Particularly, we determine a necessary and sufficient condition for two multicone graphs to be cospectral graphs and investigate the structures of graphs cospectral to a multicone graph. Additionally, lower and upper bounds for the largest eigenvalue of a multicone graph are given.