Homogeneous Graphs Research Articles

Edge partitions of the countable triangle free homogeneous graph

In this paper we investigate edge partition problems of the countable triangle free homogeneous graph. As consequences of the main result, we obtain the following theorems. For every coloring of the edges of the countable triangle free homogeneous graph U with finitely many colors there exists a copy of U in U whose edges are colored with at most two of the colors. The countable triangle free homogeneous graph U is weakly edge indivisible, that is, for every coloring of the edges of U with two colors, say red and blue, the following holds: If there is a finite triangle free graph G so that every copy of G in U contains a red edge, then there is a copy of U in U which has red edges only.

The second lowest extremal invariant measure of the contact process

We study the ergodic behavior of the contact process on infinite connected graphs of bounded degree. We show that the fundamental notion of complete convergence is not as well behaved as it was thought to be. In particular there are graphs for which complete convergence holds in any number of separated intervals of values of the infection parameter and fails for the other values of this parameter. We then introduce a basic invariant probability measure related to the recurrence properties of the process, and an associated notion of convergence that we call “partial convergence.” This notion is shown to be better behaved than complete convergence, and to hold in certain cases in which complete convergence fails. Relations between partial and complete convergence are presented, as well as tools to verify when these properties hold. For homogeneous graphs we show that whenever recurrence takes place (i.e., whenever local survival occurs) there are exactly two extremal invariant measures.

Homogeneously orderable graphs

In this paper we introduce homogeneously orderable graphs which are a common generalization of distance-hereditary graphs, dually chordal graphs and homogeneous graphs. We present a characterization of the new class in terms of a tree structure of the closed neighborhoods of homogeneous sets in 2-graphs which is closely related to the defining elimination ordering. Moreover, we characterize the hereditary homogeneously orderable graphs by forbidden induced subgraphs as the house-hole-domino-sun-free graphs. The local structure of homogeneously orderable graphs implies a simple polynomial-time recognition algorithm for these graphs. Finally, we give a polynomial-time solution for the Steiner tree problem on homogeneously orderable graphs which extends the efficient solutions of that problem on distance-hereditary graphs, dually chordal graphs and homogeneous graphs.

Edge partitions of the Rado graph

We will prove that for every colouring of the edges of the Rado graph,ℛ (that is the countable homogeneous graph), with finitely many coulours, it contains an isomorphic copy whose edges are coloured with at most two of the colours. It was known [4] that there need not be a copy whose edges are coloured with only one of the colours. For the proof we use the lexicographical order on the vertices of the Rado graph, defined by Erdos, Hajnal and Posa. Using the result we are able to describe a “Ramsey basis” for the class of Rado graphs whose edges are coloured with at most a finite number,r, of colours. This answers an old question of M. Pouzet.

The spectrum of the averaging operator for a finite homogeneous graph

We give an estimate for the spectrum of the averaging operator T1(Γ, 1) over the radius 1 for the finite (q+1)-homogeneous quotient graph Γ/X, where X is an infinite (q+1)-homogeneous tree associated with the free group G over a finite set of generators S={x1 ..., xp} (2p=q+1), and Γ, a subgroup of finite index in G. T1(Γ, 1) is defined on the subspace L2(Γ/G, 1) ⊖ Eex, where Eex is the subspace of eigenfunctions of T1(Γ, 1) with eigenvalue λ such that |λ|=q+1. We present a construction of some finite homogeneous graphs such that the spectrum of their adjacency matrices can be calculated explicitly. Bibliography: 11 titles.

Homological identities for a finite homogeneous graph

An application of the Atiyah-Bott trace identity to the study of the spectral characteristics of a finite (q+1)-homogeneous factorgraph Y=Γ/X is given (X is an infinite (q+1)-homogeneous tree, Γ a free group of isometries of X). Bibliography: 9 titles.

Infinite homogeneous bipartite graphs with unequal sides

We call a bipartite graph homogeneous if every finite partial automorphism which respects left and right can be extended to a total automorphism. A (κ, λ) bipartite graph is a bipartite graph with left side of size κ and right side of size λ. We show that there is a homogeneous ▪ 0,2 ▪0 bipartite graph of girth 4 (thus answering negatively a question by Kupitz and Perles), and that depending on the underlying set theory all homogeneous ▪ 0, ▪ 1)bipartite graphs may be isomorphic, or there may be 2 ▪1 many isomorphism types of ( ▪ 0, ▪ 1 homogeneous graphs.

Expansions of Ultrahomogeneous Graphs

Lachlan and Woodrow have completly classified the countable ultrahomogeneous graphs. We expand the language of graphs to include a new unary predicate. In this expanded language, ultrahomogeneous vertex 2-colorings of ultrahomogeneous graphs are classified.

Convergence in homogeneous random graphs

AbstractFor a sequence p̄ = (p(1), p(2), ‥ .), let G(n, p̄) denote the random graph with vertex set {1, 2, ‥ ., n} in which two vertices i, j are adjacent with probability p(|i − j|), independently for each pair. We study how the convergence of probabilities of first order properties of G(n, p̄), can be affected by the behaviour of p̄ and the strength of the language we use.

THE MEAN-FIELD THEORY OF DIRECTED POLYMERS IN RANDOM MEDIA AND SPIN GLASS MODELS

We give a unifying framework for the mean-field theory for models of spin glasses and directed polymers in a random medium defined on homogeneous graphs. Their phase diagram is studied in the complex plane of temperature.

Covering projections of graphs preserving links of vertices and edges

The link of a vertex υ of a graph G is the subgraph induced by the set of vertices of G adjacent to υ. If all the links of G are isomorphic to a given graph H, then G is called locally H, or locally homogeneous. We deal with the problem of characterization of all connected locally graphs for H fixed. It turns out that the problem is closely related to the theory of covering spaces. This fundamental observation is generalized and developed in Section 1 of the paper. Theoretical results from Section 1 are used for deriving of some new results on locally homogeneous graphs and reproving of some old ones. For instance, Theorem 2.7 establishes that if a graph H has a limited number of edges then either there is no locally H graph, or there are infinitely many finite connected locally H graphs.

A Construction of Locally Homogeneous Graphs

A Construction of Locally Homogeneous Graphs

Eliminating the boundary effect of a large-scale personal communication service network simulation

Eliminating the boundary effects is an important issue for a large-scale personal communication service (PCS) network simulation. A PCS network is often modeled by a network of hexagonal cells. The boundary may significantly bias the ouput statistics if the number of hexagonal cells is small in a PCS network simulation. On the other hand, if the simulation is to be completed within a reasonable time on the available computing resources, the number of cells in the simulation cannot be too large. To avoid the inaccuracy caused by the boundary effect for a PCS network simulation with limited computing resources, we propose wrapping the hexagonal mesh into a homogeneous graph (i.e., all nodes in the graph are topologically identical). We show that by using the wrapped hexagonal mesh, the inaccuracy of the output measures can be limited even though the number of cells in the simulation is small. We can thus obtain the same statistical accuracy while using significantly less computation power than required for a simulation without cell wrapping.

Homogeneous Graphs and Regular Near Polygons

We consider a distance-regular graph having homogeneous edge patterns in each entry of its intersection diagram with respect to an edge. We call such graphs homogeneous graphs. We study elementary properties of homogeneous graphs, and we show these graphs are related deeply with regular near polygons.

RPCT algorithm and its VLSI implementation

This paper presents the regional projection contour transformation (RPCT) which transforms a compound pattern or multicontour pattern into a unique outer contour. Two RPCT's, (1) diagonal-diagonal regional projection contour transformation and (2) horizontal-vertical regional projection contour transformation, are presented. They are applicable to a wide range of areas such as image analysis, pattern recognition, etc. A very large scale integration (VLSI) architecture to implement the RPCT has also been designed based on a canonical methodology which maps homogeneous dependence graphs into processor arrays. In this paper, a linear array has been designed, where an N/2-element vector is used to process a pattern with a size of N/spl times/N. It can speed up the recognition process considerably with a time complexity of O(N) compared with O(N/sup 2/) when a uniprocessor is used. >

Covering spaces of locally homogeneous graphs

A graph G is called locally homogeneous, or locally G 0, if for each vertex u of G the subgraph induced on the set of vertices adjacent with u is isomorphic to some graph G 0. In this paper we use the concept of covering spaces for deriving various results on the set of all connected locally G 0 graphs (for given G 0). For instance, we prove that if e( G 0) is small and there exists a locally G 0 graph, then there are infinitely many finite connected locally G 0 graphs. Further, a sufficient condition for existence of an infinite locally G 0 graph given in terms of minors of G is presented. As a by-product, we obtain a characterization of contraction minimal locally cyclic triangulations of the projective plane, which is also interesting for its own sake.

On the Divisibility of Homogeneous Directed Graphs

AbstractLet be a finite set of finite tournaments. We will give a necessary and sufficient condition for the -free homogeneous directed graph to be divisible. That is, that there is a partition of into two classes such that neither of them contains an isomorphic copy of .

Laplacian and vibrational spectra for homogeneous graphs

AbstractA homogeneous graph is a graph togerther with a group that acts transitively on vertices as symmertries of the graph. We consider Laplacians of homogeneous graphs and generalizations of Laplacians whose eigenvalues can be associated with various equilibria of forces in molecules (such as vibrational modes of buckyballs). Methods are given for calculating such eigenvalues by combining concepts and techniques in group representation theory, gauge theory and graph theory.

Inclusive fitness in a homogeneous environment

A general formulation of inclusive fitness is proposed which specifically accounts for competitive effects between relatives. As an example, for an asexual population in a homogeneous inelastic environment, such as is found in a stepping-stone model of dispersal, the inclusive fitness of a breeding female, under weak selection, is independent of her direct effect on the fitness of other individuals in the population. More precisely, suppose a female acts in a way that changes not only her own fitness but the fitness of several other females in the population who may be relatives (i.e. she changes the number of their offspring). I call these changes the direct effects of the action. There will also be indirect effects on the fitness of these and other females which come from the competitive impact of the extra offspring in the next generation. The principal result is that these indirect effects exactly cancel all the direct effects except the direct effect of the actor on herself.

Distanced graphs

This paper studies model theoretic conditions that arise in the study of distanced graphs. These are graphs with additional relation symbols for finite distances. Every embedding of distanced graphs is an isometric embedding. The conditions we study are the version of homogeneity, injectivity and universality for distanced graphs. There is a countable graph which isometrically embeds every countable graph. There is a unique countable distace homogeneous graph with this universal property. Its first-order theory is the model completion of the theory of distanced graphs. It has a unique countable connected model, but it has more than one connected model of each uncountable cardinality.