Semigroups of mappings on graphs

At present, there are quite a few investigations in the theory of semigroups devoted to semigroups of mappings on graphs. Up to now, endomorphism semigroups of graphs, extensive transformation semigroups of graphs, coloring semigroups of graphs and other semigroups of special mappings on graphs have been studied. The results obtained show the way graphs are determined by the above-mentioned semigroups. They also show the structure of semigroups of mappings and interrelations between properties of graphs and corresponding properties of semigroups associated with the graphs. This paper gives a survey of the main results in this field.     

Random perfect graphs

We investigate the asymptotic structure of a random perfect graph P_n sampled uniformly from the set of perfect graphs on vertex set . Our approach is based on the result of Prömel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number and clique number is close to a concentrated distribution L(n) which plays an important role in our generation method. We also prove that the probability that P_n contains any given graph H as an induced subgraph is asymptotically 0 or or 1. Further we show that almost all perfect graphs are 2‐clique‐colorable, improving a result of Bacsó et al. from 2004; they are almost all Hamiltonian; they almost all have connectivity equal to their minimum degree; they are almost all in class one (edge‐colorable using Δ colors, where Δ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon .     

Random I-colorable graphs

In this article we investigate properties of the class of all l-colorable graphs on n vertices, where l = l(n) may depend on n. Let G^l_n denote a uniformly chosen element of this class, i.e., a random l-colorable graph. For a random graph G^l_n we study in particular the property of being uniquely l-colorable. We show that not only does there exist a threshold function l = l(n) for this property, but this threshold corresponds to the chromatic number of a random graph. We also prove similar results for the class of all l-colorable graphs on n vertices with m = m(n) edges.     

Random interval graphs

We consider models for random interval graphs that are based on stochastic service systems, with vertices corresponding to customers and edges corresponding to pairs of customers that are in the system simultaneously. The number N of vertices in a connected component thus corresponds to the number of customers arriving during a busy period, while the size K of the largest clique (which for interval graphs is equal to the chromatic number) corresponds to the maximum number of customers in the system during a busy period. We obtain the following results for both the M/D/∞ and the M/M/∞ models, with arrival rate λ per mean service time. The expected number of vertices is e^λ, and the distribution of the N/e^λ converges pointwise to an exponential distribution with mean 1 as λ tends to infinity. This implies that the distribution of log N−λ converges pointwise to a distribution with mean −γ (where γ is Euler's constant) and variance π²/6. The size K of the largest clique falls in the interval [eλ−2 log λ, eλ+1] with probability tending to 1 as λ tends to infinity. Thus the distribution of the ratio K/log N converges pointwise to that of the constant e, in contrast to the situation for random graphs generated by unbiased coin flips, in which the distribution of K/log N converges pointwise to that of the constant 2/log 2. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 361–380, 1998     

Random interval graphs

In this paper we introduce a notion ofrandom interval graphs: the intersection graphs of real, compact intervals whose end points are chosen at random. We establish results about the number of edges, degrees, Hamiltonicity, chromatic number and independence number of almost all interval graphs.     

Random graphs and covering graphs of posets

For a graph G, we define c(G) to be the minimal number of edges we must delete in order to make G into a covering graph of some poset. We prove that, if p=n ^-1+(n) ,where (n) is bounded away from 0, then there is a constant k ₀>0 such that, for a.e. G _p , c(G _p )k ₀ n ¹⁺⁽ⁿ⁾ .In other words, to make G _p into a covering graph, we must almost surely delete a positive constant proportion of the edges. On the other hand, if p=n ^-1+(n) , where (n)0, thenc(G _p )=o(n ¹⁺⁽ⁿ⁾ ), almost surely.Partially supported by MCS Grant 8104854.     

On set-valued mappings with starlike graphs

The paper studies some topological properties of starlike bodies. It is proved that the boundary of a starlike body is a Lipschitz surface. A separability theorem for starlike bodies is proved. It is shown that under some additional assumptions the starlike property of the graph provides the local Lipschitz property of the set-valued mapping itself. It is shown that F. Clark’s contingent and tangential cones are Boltyansky tents. On the base of these results, some lower and upper differentials for set-valued mappings with starlike graphs are constructed. Some theorems on fixed points of set-valued mappings with starlike values are proved.     

Random walks on graphs

In this paper the following Markov chains are considered: the state space is the set of vertices of a connected graph, and for each vertex the transition is always to an adjacent vertex, such that each of the adjacent vertices has the same probability. Detailed results are given on the expectation of recurrence times, of first-entrance times, and of symmetrized first-entrance times (called commuting times). The problem of characterizing all connected graphs for which the commuting time is constant over all pairs of adjacent vertices is solved almost completely.     

Random Latin square graphs

In this paper we introduce new models of random graphs, arising from Latin squares which include random Cayley graphs as a special case. We investigate some properties of these graphs including their clique, independence and chromatic numbers, their expansion properties as well as their connectivity and Hamiltonicity. The results obtained are compared with other models of random graphs and several similarities and differences are pointed out. For many properties our results for the general case are as strong as the known results for random Cayley graphs and sometimes improve the previously best results for the Cayley case. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Random graphs on surfaces

Counting labelled planar graphs, and typical properties of random labelled planar graphs, have received much attention recently. We start the process here of extending these investigations to graphs embeddable on any fixed surface S. In particular we show that the labelled graphs embeddable on S have the same growth constant as for planar graphs, and the same holds for unlabelled graphs. Also, if we pick a graph uniformly at random from the graphs embeddable on S which have vertex set {1,…,n}, then with probability tending to 1 as n→∞, this random graph either is connected or consists of one giant component together with a few nodes in small planar components.     

Random walks generated by affine mappings

This article is concerned with Markov chains on ^m constructed by randomly choosing an affine map at each stage, and then making the transition from the current point to its image under this map. The distribution of the random affine map can depend on the current point (i.e., state of the chain). Sufficient conditions are given under which this chain is ergodic.     

Vertex transitive graphs from injective linear mappings

Based on a motivation coming from the study of the metric structure of the category of finite dimensional vector spaces over a finite field \(\mathbb {F}\) , we examine a family of graphs, defined for each pair of integers \(1 \le k \le n\) , with vertex set formed by all injective linear transformations \(\mathbb {F}^k \rightarrow \mathbb {F}^n\) and edges corresponding to pairs of mappings, \(f\) and \(g\) , with \(\lambda (f,g)= \dim \mathrm{Im }(f-g)=1 \) . For \(\mathbb {F}\cong \mathrm{GF }(q)\) , this graph will be denoted by \(\mathrm{INJ }_q(k,n)\) . We show that all such graphs are vertex transitive and Hamiltonian and describe the full automorphism group of each \(\mathrm{INJ }_q (k,n)\) for \(k . Using the properties of line-transitive groups, we completely determine which of the graphs \(\mathrm{INJ }_q (k,n)\) are Cayley and which are not. The Cayley ones consist of three infinite families, corresponding to pairs \((1,n),\,(n-1,n)\) , and \((n,n)\) , with \(n\) and \(q\) arbitrary, and of two sporadic examples \(\mathrm{INJ }_{2} (2,5)\) and \(\mathrm{INJ }_{2}(3,5)\) . Hence, the overwhelming majority of our graphs is not Cayley.     

Analytic sets extending the graphs of holomorphic mappings

Let D, D′ ⊂ ℂⁿ be bounded domains with smooth real analytic boundaries and ƒ: D → D′ be a proper holomorphic map. Our main result implies that if the graph of ƒ extends as an analytic set to a neighborhood of a poìnt (a, a′) ∈ ∂D × 3D′ with a′ ∈ cl_ƒ(a), then ƒ extends holomorphically to a neighborhood of a.     

Random walks on colored graphs

We initiate a study of random walks on undirected graphs with colored edges. In our model, a sequence of colors is specified before the walk begins, and it dictates the color of edge to be followed at each step. We give tight upper and lower bounds on the expected cover time of a random walk on an undirected graph with colored edges. We show that, in general, graphs with two colors have exponential expected cover time, and graphs with three or more colors have doubly-exponential expected cover time. We also give polynomial bounds on the expected cover time in a number of interesting special cases. We described applications of our results to understanding the dominant eigenvectors of products and weighted averages of stochastic matrices, and to problems on time-inhomogeneous Markov chains. © 1994 John Wiley & Sons, Inc.