Some results on the ordering of the Laplacian spectral radii of unicyclic graphs

A unicyclic graph is a graph whose number of edges is equal to the number of vertices. Guo Shu-Guang [S.G. Guo, The largest Laplacian spectral radius of unicyclic graph, Appl. Math. J. Chinese Univ. Ser. A. 16 (2) (2001) 131–135] determined the first four largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices. In this paper, we extend this ordering by determining the fifth to the ninth largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices.     

The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real‐world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007     

Appearance of complex components in a random bigraph

This is an investigation into the limiting distribution of the sparse connected components in an n₁ × n₂ bipartite multigraph with approximately ½n edges, where n₁, n₂ ≈? ½n. We will show that the probability of finding no complex components in a random n₁ × n₂ bigraph, with approximately ½n edges, is asymptotically the same as for random graphs with n vertices and approximately ½n edges, namely √2/3. In addition, we will show that, for n₁ × n₂ multi-bigraphs, the probability distribution for finitely many bicyclic components, tricyclic components, etc., with no components having more than a given number of cycles, asymptotically equals the corresponding distribution for random multigraphs with n vertices and approximately half as many edges. As an application of this analysis we present a method for estimating the efficiency of memory access in a distributed, replicated data base.     

Connectivity threshold of Bluetooth graphs

We study the connectivity properties of random Bluetooth graphs that model certain “ad hoc” wireless networks. The graphs are obtained as “irrigation subgraphs” of the well‐known random geometric graph model. There are two parameters that control the model: the radius r that determines the “visible neighbors” of each vertex and the number of edges c that each vertex is allowed to send to these. The randomness comes from the underlying distribution of vertices in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters r, c and completely characterize the connectivity threshold (in c) for values of r close the critical value for connectivity in the underlying random geometric graph.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 45–66, 2014     

Sparse graphs: Metrics and random models

Recently, Bollobás, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with n vertices and Θ(n) edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function κ: [0, 1]² → [0, ∞). To understand these models, we should like to know when different kernels κ give rise to “similar” graphs, and, given a real‐world network, how “similar” is it to a typical graph G(n, κ) derived from a given kernel κ. The analogous questions for dense graphs, with Θ(n²) edges, are answered by recent results of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in [0, 1]. Possible generalizations of these results to graphs with o(n²) but ω(n) edges are discussed in a companion article [Bollobás and Riordan, London Math Soc Lecture Note Series 365 (2009), 211–287]; here we focus only on graphs with Θ(n) edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models and vice versa. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 1‐38, 2011     

Automorphisms of random graphs with specified vertices

Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1. These conditions involve the probability that such a graph has a given subgraph. One implication is that the probability that a random unlabelledk-regular simple graph onn vertices has only the trivial group of automorphisms is asymptotic to 1 asn → ∞ with 3≦k=O(n ^1/2−c). In combination with previously known results, this produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymptotic results on the probable connectivity and girth of such graphs. Corresponding results for graphs with more arbitrary degree sequences are obtained. The main results apply equally well to graphs in which multiple edges and loops are permitted, and also to bicoloured graphs. Research of the second author supported by U. S. National Science Foundation Grant MCS-8101555, and by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowships Scheme. Current address: Mathematics Department, University of Auckland, Auckland, New Zealand.     

The Bohman‐Frieze process near criticality

The Erd?s‐Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erd?s and Rényi states that the Erd?s‐Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erd?s and Rényi, various random graph models have been introduced and studied. In this paper we study the Bohman‐Frieze process, a simple modification of the Erd?s‐Rényi process. The Bohman‐Frieze process also begins with an empty graph on n vertices. At each step two random edges are presented, and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman‐Frieze process. We show that it has a qualitatively similar phase transition to the Erd?s‐Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time t_c ? ? (that is, when the number of edges are (t_c ? ?)n/2) are trees or unicyclic components and that the largest component is of size Ω(?^‐2log n). Further, at t_c + ?, all components apart from the giant component are trees or unicyclic and the size of the second‐largest component is Θ(?^‐2log n). Each of these results corresponds to an analogous well‐known result for the Erd?s‐Rényi process. Our proof techniques include combinatorial arguments, the differential equation method for random processes, and the singularity analysis of the moment generating function for the susceptibility, which satisfies a quasi‐linear partial differential equation. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

The random planar graph process

We consider the following variant of the classical random graph process introduced by Erd?s and Rényi. Starting with an empty graph on n vertices, choose the next edge uniformly at random among all edges not yet considered, but only insert it if the graph remains planar. We show that for all ε > 0, with high probability, θ(n²) edges have to be tested before the number of edges in the graph reaches (1 + ε)n. At this point, the graph is connected with high probability and contains a linear number of induced copies of any fixed connected planar graph, the first property being in contrast and the second one in accordance with the uniform random planar graph model. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

On the least eigenvalue of a unicyclic mixed graph

Using the result on Fiedler vectors of a simple graph, we obtain a property on the structure of the eigenvectors of a nonsingular unicyclic mixed graph corresponding to its least eigenvalue. With the property, we get some results on minimizing and maximizing the least eigenvalue over all nonsingular unicyclic mixed graphs on n vertices with fixed girth. In particular, the graphs which minimize and maximize, respectively, the least eigenvalue are given over all such graphs with girth 3.     

Phase transitions of the Moran process and algorithmic consequences

The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches “fixation,” where all vertices are mutants, or “extinction,” where none are. Our main result is an almost‐tight upper bound on expected absorption time. For all ?>0, we show that the expected absorption time on an n‐vertex graph is o(n^3+?). Specifically, it is at most , and there is a family of graphs where it is Ω(n³). In proving this, we establish a phase transition in the probability of fixation, depending on the mutants' fitness r. We show that no similar phase transition occurs for digraphs, where it is already known that the expected absorption time can be exponential. Finally, we give an improved fully polynomial randomized approximation scheme (FPRAS) for approximating the probability of fixation. On degree‐bounded graphs where some basic properties are given, its running time is independent of the number of vertices.     

On the Maximum Number of Cliques in a Graph

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K₅-minor or no K_3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2). Research supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.     

Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvátal, Trotter and the two senior authors proved that for any Δ there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph KN with N?Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Δ. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and edges, with N=⌈B^′n⌉ for some constant B^′ that depends only on Δ. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Δ is . Our approach is based on random graphs; in fact, we show that the classical Erd?s–Rényi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Δ.The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed.     

单圈混合图的极大谱半径(英文)

设U*为一个未定向的n个顶点上的单圈混合图,它是由一个三角形在其某个顶点上附加n-3个悬挂边而获得.在文[Largest eigenvalue of a unicyclic mixed graph,Applied Mathematics A Journal of Chinese Universities(Ser.B),2004,19(2):140-148]中,作者证明了:在相差符号同构意下,在所有n个顶点上的单圈混合图中,U*是唯一的达到最大Laplace谱半径的混合图.本文应用非负矩阵的Perron向量,给出上述结论的一个简单的证明.     

On vertex symmetric digraphs

The problem of how “near” we can come to a n-coloring of a given graph is investigated. I.e., what is the minimum possible number of edges joining equicolored vertices if we color the vertices of a given graph with n colors. In its generality the problem of finding such an optimal color assignment to the vertices (given the graph and the number of colors) is NP-complete. For each graph G, however, colors can be assigned to the vertices in such a way that the number of offending edges is less than the total number of edges divided by the number of colors. Furthermore, an Ω(epn) deterministic algorithm for finding such an n-color assignment is exhibited where e is the number of edges and p is the number of vertices of the graph (e?p?n). A priori solutions for the minimal number of offending edges are given for complete graphs; similarly for equicolored K_m in K_p and equicolored graphs in K_p.     

An edge-grafting theorem on Laplacian spectra of graphs and its application

Let 𝒰(n,?d) be the class of unicyclic graphs on n vertices with diameter d. This article presents an edge-grafting theorem on Laplacian spectra of graphs. By applying this theorem, we determine the unique graph with the maximum Laplacian spectral radius in 𝒰(n,?d). This extremal graph is different from that for the corresponding problem on the adjacency spectral radius as done by Liu et al. [Q. Liu, M. Lu, and F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007), 449–457].     

A strong law of large numbers for random biased connected graphs

We consider a class of random connected graphs with random vertices and random edges in which the randomness of the vertices is determined by a continuous probability distribution and the randomness of the edges is determined by a connection function. We derive a strong law of large numbers on the total lengths of all random edges for a random biased connected graph that is a generalization of a directed k-nearest-neighbor graph.     

Algebraic Connectivity of Connected Graphs with Fixed Number of Pendant Vertices

In this paper, we consider the following problem. Over the class of all simple connected graphs of order n with k pendant vertices (n, k being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.     

Sparse universal graphs for bounded‐degree graphs

Let ℋ︁ be a family of graphs. A graph T is ℋ︁‐universal if it contains a copy of each H ∈ℋ︁ as a subgraph. Let ℋ︁(k,n) denote the family of graphs on n vertices with maximum degree at most k. For all positive integers k and n, we construct an ℋ︁(k,n)‐universal graph T with edges and exactly n vertices. The number of edges is almost as small as possible, as Ω(n^2‐2/k) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic and is based on a novel graph decomposition result and on the properties of random walks on expanders. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Probabilities of Sentences about Very Sparse Random Graphs

We consider random graphs with edge probability βn^?α, where n is the number of vertices of the graph, β > 0 is fixed, and α = 1 or α = (l + 1) /l for some fixed positive integer l. We prove that for every first-order sentence, the probability that the sentence is true for the random graph has an asymptotic limit.     

单圈混合图的极大谱半径

设U＊为一个未定向的n个顶点上的单圈混合图，它是由一个三角形在其某个顶点上附加”一3个悬挂边而获得．在文[Largest eigenvalue of aunicyclic mixed graph，Applied Mathematics A Journal of Chinese Universities （Ser．B），2004，19（2）：140-J48]中，作者证明了：在相差符号同构意下，在所有n个顶点上的单圈混合图中，U＊是唯一的达到最大Laplace谱半径的混合图．本文应用非负矩阵的Perron向量，给出上述结论的一个简单的证明．