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.     

Almost self-centered graphs

The center of a graph is the set of vertices with minimum eccentricity. Graphs in which all vertices are central are called self-centered graphs. In this paper almost self-centered (ASC) graphs are introduced as the graphs with exactly two non-central vertices. The block structure of these graphs is described and constructions for generating such graphs are proposed. Embeddings of arbitrary graphs into ASC graphs are studied. In particular it is shown that any graph can be embedded into an ASC graph of prescribed radius. Embeddings into ASC graphs of radius two are studied in more detail. ASC index of a graph G is introduced as the smallest number of vertices needed to add to G such that G is an induced subgraph of an ASC graph.     

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.     

Fast generation of regular graphs and construction of cages

The construction of complete lists of regular graphs up to isomorphism is one of the oldest problems in constructive combinatorics. In this article an efficient algorithm to generate regular graphs with a given number of vertices and vertex degree is introduced. The method is based on orderly generation refined by criteria to avoid isomorphism checking and combined with a fast test for canonicity. The implementation allows computing even large classes of graphs, like construction of the 4‐regular graphs on 18 vertices and, for the first time, the 5‐regular graphs on 16 vertices. Also in cases with given girth, some remarkable results are obtained. For instance, the 5‐regular graphs with girth 5 and minimal number of vertices were generated in less than 1 h. There exist exactly four (5, 5)‐cages. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 137–146, 1999     

Almost all graphs are rigid—revisited

It is shown that almost all graphs are unretractive, i.e. have no endomorphisms other than their automorphisms. A more general result has already been published in [V. Koubek, V. Rödl, On the minimum order of graphs with given semigroup, J. Combin. Theory Ser. B 36 (1984) 135–155]. In the paper at hand, a different proof is presented, following an approach of P. Erdős and A. Rényi that was used in their proof [P. Erdős, A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hungar. 14 (1963) 295–315] that almost all graphs are asymmetric (have a trivial automorphism group). The approach is modified using an algebraically motivated reduction to idempotent endomorphisms. These take the role of the automorphisms in the proof of Erdős and Rényi. A bound of is provided for the ratio of retractive graphs among all graphs with n vertices, confirming an earlier statement by Babai [L. Babai, Automorphism groups, isomorphism, reconstruction, in: R.L. Graham, M. Grötschel, L. Lovász (Eds.), in: Handbook of Combinatorics, vol. 2, Elsevier, Amsterdam, 1995, pp. 1447–1540]. The fact that almost all graphs are unretractive and asymmetric can be summarized in the statement that almost all graphs are rigid (have a trivial endomorphism monoid), and the same bound can be obtained for corresponding ratios of nonrigid graphs.     

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.     

Enumeration of point-determining graphs

Point-determining graphs are graphs in which no two vertices have the same neighborhoods, co-point-determining graphs are those whose complements are point-determining, and bi-point-determining graphs are those both point-determining and co-point-determining. Bicolored point-determining graphs are point-determining graphs whose vertices are properly colored with white and black. We use the combinatorial theory of species to enumerate these graphs as well as the connected cases.     

Walks and regular integral graphs

We establish a useful correspondence between the closed walks in regular graphs and the walks in infinite regular trees, which, after counting the walks of a given length between vertices at a given distance in an infinite regular tree, provides a lower bound on the number of closed walks in regular graphs. This lower bound is then applied to reduce the number of the feasible spectra of the 4-regular bipartite integral graphs by more than a half.Next, we give the details of the exhaustive computer search on all 4-regular bipartite graphs with up to 24 vertices, which yields a total of 47 integral graphs.     

A solitaire game played on 2-colored graphs

We introduce a solitaire game played on a graph. Initially one disk is placed at each vertex, one face green and the other red, oriented with either color facing up. Each move of the game consists of selecting a vertex whose disk shows green, flipping over the disks at neighboring vertices, and deleting the selected vertex. The game is won if all vertices are eliminated. We derive a simple parity-based necessary condition for winnability of a given game instance. By studying graph operations that construct new graphs from old ones, we obtain broad classes of graphs where this condition also suffices, thus characterizing the winnable games on such graphs. Concerning two familiar (but narrow) classes of graphs, we show that for trees a game is winnable if and only if the number of green vertices is odd, and for n-cubes a game is winnable if and only if the number of green vertices is even and not all vertices have the same color. We provide a linear-time algorithm for deciding winnability for games on maximal outerplanar graphs. We reduce the decision problem for winnability of a game on an arbitrary graph G to winnability of games on its blocks, and to winnability on homeomorphic images of G obtained by contracting edges at 2-valent vertices.     

Weakly triangulated graphs

A graph is triangulated if it has no chordless cycle with four or more vertices. It follows that the complement of a triangulated graph cannot contain a chordless cycle with five or more vertices. We introduce a class of graphs (namely, weakly triangulated graphs) which includes both triangulated graphs and complements of triangulated graphs (we define a graph as weakly triangulated if neither it nor its complement contains a chordless cycle with five or more vertices). Our main result is a structural theorem which leads to a proof that weakly triangulated graphs are perfect.     

三圈图的Harary指数

图G的Harary指数是指图G中所有顶点对间的距离倒数之和. 三圈图是指边数等于顶点数加2的连通图. 研究了三圈图的Harary数, 给出了所有三圈图中具有极大Harary指数的图的结构以及含有三个圈的三圈图中具有次大Harary指数的图的结构.     

On extensions of exceptional strongly regular graphs with eigenvalue 3

The study of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with eigenvalue 3 was initiated by Makhnev. In particular, he reduced these graphs to graphs in which neighborhoods of vertices are exceptional graphs or pseudogeometric graphs for pG _s?3(s, t). Makhnev and Paduchikh found parameters of exceptional graphs (see the proposition). In the present paper, we study amply regular graphs in which neighborhoods of vertices are exceptional strongly regular graphs with nonprincipal eigenvalue 3 and parameters from conditions 3–6 of the Proposition.     

Crossing minimization in extended level drawings of graphs

The most popular method of drawing directed graphs is to place vertices on a set of horizontal or concentric levels, known as level drawings. Level drawings are well studied in Graph Drawing due to their strong application for the visualization of hierarchy in graphs. There are two drawing conventions: Horizontal drawings use a set of parallel lines and radial drawings use a set of concentric circles.In level drawings, edges are only allowed between vertices on different levels. However, many real world graphs exhibit hierarchies with edges between vertices on the same level. In this paper, we initiate the new problem of extended level drawings of graphs, which was addressed as one of the open problems in social network visualization, in particular, displaying centrality values of actors. More specifically, we study minimizing the number of edge crossings in extended level drawings of graphs. The main problem can be formulated as the extended one-sided crossing minimization problem between two adjacent levels, as it is folklore with the one-sided crossing minimization problem in horizontal drawings.We first show that the extended one-sided crossing minimization problem is NP-hard for both horizontal and radial drawings, and then present efficient heuristics for minimizing edge crossings in extended level drawings. Our extensive experimental results show that our new methods reduce up to 30% of edge crossings.     

Infinite quantum graphs

Infinite quantum graphs with δ-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a quantum graph and a certain discrete Laplacian given on a graph with infinitely many vertices and edges is established. In particular, it is shown that these operators are self-adjoint, lower semibounded, nonnegative, discrete, etc. only simultaneously.     

Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

A chain graph is a digraph whose strong components are undirected graphs and a directed acyclic graph (ADG or DAG) G is essential if the Markov equivalence class of G consists of only one element. We provide recurrence relations for counting labelled chain graphs by the number of chain components and vertices; labelled essential DAGs by the number of vertices. The second one is a lower bound for the number of labelled essential graphs. The formula for labelled chain graphs can be extended in such a way, that allows us to count digraphs with two additional properties, which essential graphs have.     

Complement reducible graphs

In this paper we study the family of graphs which can be reduced to single vertices by recursively complementing all connected subgraphs. It is shown that these graphs can be uniquely represented by a tree where the leaves of the tree correspond to the vertices of the graph. From this tree representation we derive many new structural and algorithmic properties. Furthermore, it is shown that these graphs have arisen independently in various diverse areas of mathematics.     

More on betweenness-uniform graphs

We study graphs whose vertices possess the same value of betweenness centrality (which is defined as the sum of relative numbers of shortest paths passing through a given vertex). Extending previously known results of S. Gago, J. Hurajová, T. Madaras (2013), we show that, apart of cycles, such graphs cannot contain 2-valent vertices and, moreover, are 3-connected if their diameter is 2. In addition, we prove that the betweenness uniformity is satisfied in a wide graph family of semi-symmetric graphs, which enables us to construct a variety of nontrivial cubic betweenness-uniform graphs.     

Regular graphs with high edge degree

For any vertex x of a graph G let Δ(x) denote the set of vertices adjacent to x. We seek to describe the connected graphs G which are regular of valence n and in which for all adjacent vertices x and y |Δ(x) ∩ Δ(y)| = n ? 1 ? s. It is known that the complete graphs are the graphs for which s = 0. For any s, any complete many-partite graph, each part containing s + 1 vertices, is such a graph. We show that these are the only such graphs for which the valence exceeds 2s² ? s + 1. The graphs satisfying these conditions for s = 1 or 2 are characterized (up to the class of trivalent triangle-free graphs.)     

On the structure of clique‐free graphs

Using a clever inductive counting argument Erd?s, Kleitman and Rothschild showed in 1976 that almost all triangle‐free graphs are bipartite, i.e., that the cardinality of the two graph classes is asymptotically equal. In this paper we investigate the structure of the “few” triangle‐free graphs which are not bipartite. As it turns out, with high probability, these graphs are bipartite up to a few vertices. More precisely, almost all of them can be made bipartite by removing just one vertex. Almost all others can be made bipartite by removing two vertices, and then three vertices and so on. We also show that similar results hold if we replace “triangle‐free” by K_??+1‐free and “bipartite” by ??‐partite. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 37–53, 2001     

Disconnecting strongly regular graphs

In this paper, we show that the minimum number of vertices whose removal disconnects a connected strongly regular graph into non-singleton components equals the size of the neighborhood of an edge for many graphs. These include block graphs of Steiner 2-designs, many Latin square graphs and strongly regular graphs whose intersection parameters are at most a quarter of their valency.