Rumor spreading in random evolving graphs

We analyze randomized broadcast in dynamic networks modeled as edge‐Markovian evolving graphs. The most realistic range of edge‐Markovian parameters yields sparse and disconnected graphs. We prove that, in this setting, the “push” protocol completes with high probability in optimal logarithmic time. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 290–312, 2016     

Modularity in random regular graphs and lattices

Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity of G is the maximum modularity of a partition.We give an upper bound on the modularity of r-regular graphs as a function of the edge expansion (or isoperimetric number) under the restriction that each part in our partition has a sub-linear numbers of vertices. This leads to results for random r-regular graphs. In particular we show the modularity of a random cubic graph partitioned into sub-linear parts is almost surely in the interval (0.66, 0.88).The modularity of a complete rectangular section of the integer lattice in a fixed dimension was estimated in Guimer et. al. [R. Guimerà, M. Sales-Pardo and L.A. Amaral, Modularity from fluctuations in random graphs and complex networks, Phys. Rev. E 70 (2) (2004) 025101]. We extend this result to any subgraph of such a lattice, and indeed to more general graphs.     

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u, v) is at least d_C(u, v)?e(n). Let ω(n) be any function tending to infinity with n. We consider a random d‐regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n) = log_d?1log_d?1n+ ω(n) and |C| = 2log_d?1n+ O(ω(n)). Along the way, we obtain results on near‐geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. Copyright © 2010 John Wiley & Sons, Ltd. J Graph Theory 66:115‐136, 2011     

Distribution of subgraphs of random regular graphs

We obtain the asymptotic distribution of the number of copies of a fixed subgraph H in a random d‐regular graph, provided H is strictly balanced and d = d(n) is chosen so that the expected number of copies of H tends to infinity (but not too quickly), and the expected number of copies sharing edges with two other copies is bounded. The proof of asymptotic normality of the distribution uses a method of factorial moments for variables with unbounded means that was recently derived by the authors. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Random walks on random simple graphs

This paper looks at random regular simple graphs and considers nearest neighbor random walks on such graphs. This paper considers walks where the degree d of each vertex is around (log n)^a where a is a constant which is at least 2 and where n is the number of vertices. By extending techniques of Dou, this paper shows that for most such graphs, the position of the random walk becomes close to uniformly distributed after slightly more than log n/log d steps. This paper also gets similar results for the random graph G(n, p), where p = d/(n − 1). © 1996 John Wiley & Sons, Inc.     

The generalized acyclic edge chromatic number of random regular graphs

The r‐acyclic edge chromatic number of a graph is defined to be the minimum number of colors required to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min(|C|, r) colors. We show that (r ? 2)d is asymptotically almost surely (a.a.s.) an upper bound on the r‐acyclic edge chromatic number of a random d‐regular graph, for all constants r ≥ 4 and d ≥ 2. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 101–125, 2006     

Critical percolation on random regular graphs

The behavior of the random graph G(n,p) around the critical probability p_c = is well understood. When p = (1 + O(n^1/3))p_c the components are roughly of size n^2/3 and converge, when scaled by n^?2/3, to excursion lengths of a Brownian motion with parabolic drift. In particular, in this regime, they are not concentrated. When p = (1 ‐ ?(n))p_c with ?(n)n^1/3 →∞ (the subcritical regime) the largest component is concentrated around 2?^?2 log(?³n). When p = (1 + ?(n))p_c with ?(n)n^1/3 →∞ (the supercritical regime), the largest component is concentrated around 2?n and a duality principle holds: other component sizes are distributed as in the subcritical regime. Itai Benjamini asked whether the same phenomenon occurs in a random d‐regular graph. Some results in this direction were obtained by (Pittel, Ann probab 36 (2008) 1359–1389). In this work, we give a complete affirmative answer, showing that the same limiting behavior (with suitable d dependent factors in the non‐critical regimes) extends to random d‐regular graphs. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Small subgraphs of random regular graphs

The main aim of this short paper is to answer the following question. Given a fixed graph H, for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one?     

Packing vertices and edges in random regular graphs

In this paper we consider the problem of finding large collections of vertices and edges satisfying particular separation properties in random regular graphs of degree r, for each fixed r ≥ 3. We prove both constructive lower bounds and combinatorial upper bounds on the maximal sizes of these sets. The lower bounds are proved by analyzing a class of algorithms that return feasible solutions for the given problems. The analysis uses the differential equation method proposed by Wormald [Lectures on Approximation and Randomized Algorithms, PWN, Wassaw, 1999, pp. 239–298]. The upper bounds are proved by direct combinatorial means. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Maximum acyclic and fragmented sets in regular graphs

We show that a typical d‐regular graph G of order n does not contain an induced forest with around vertices, when n ? d ? 1, this bound being best possible because of a result of Frieze and ?uczak [6]. We then deduce an affirmative answer to an open question of Edwards and Farr (see [4]) about fragmentability, which concerns large subgraphs with components of bounded size. An alternative, direct answer to the question is also given. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 149–156, 2008     

Expansion properties of random Cayley graphs and vertex transitive graphs via matrix martingales

The Alon–Roichman theorem states that for every ε> 0 there is a constant c(ε), such that the Cayley graph of a finite group G with respect to c(ε)log ∣G∣ elements of G, chosen independently and uniformly at random, has expected second largest eigenvalue less than ε. In particular, such a graph is an expander with high probability. Landau and Russell, and independently Loh and Schulman, improved the bounds of the theorem. Following Landau and Russell we give a new proof of the result, improving the bounds even further. When considered for a general group G, our bounds are in a sense best possible. We also give a generalization of the Alon–Roichman theorem to random coset graphs. Our proof uses a Hoeffding‐type result for operator valued random variables, which we believe can be of independent interest. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

A note on regular Ramsey graphs

We prove that there is an absolute constant C>0 so that for every natural n there exists a triangle‐free regular graph with no independent set of size at least \({{C}}\sqrt{{{n}}\log{{n}}}\). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 244–249, 2010     

5-chromatic strongly regular graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic. In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.     

Sparse random graphs: Eigenvalues and eigenvectors

In this paper, we prove the semi‐circular law for the eigenvalues of regular random graph G_n,d in the case d →∞, complementing a previous result of McKay for fixed d. We also obtain a upper bound on the infinity norm of eigenvectors of Erd?s–Rényi random graph G(n,p), answering a question raised by Dekel–Lee–Linial. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Retarded regular graphs are regular or semiregular

A graph G is said to be retarded regular if there is a positive integral number s such that the number of walks of length s starting at vertices of G is a constant function. Regular and semiregular graphs are retarded regular with s?=?1 and s\!≤ \!2, respectively. We prove that any retarded regular connected graph is either regular or semiregular.     

Embedding arbitrary finite simple graphs into small strongly regular graphs

It is well known that any finite simple graph Γ is an induced subgraph of some exponentially larger strongly regular graph Γ (e.g., [2, 8]). No general polynomial‐size construction has been known. For a given finite simple graph Γ on υ vertices, we present a construction of a strongly regular graph Γ on O(υ⁴) vertices that contains Γ as its induced subgraph. A discussion is included of the size of the smallest possible strongly regular graph with this property. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 1–8, 2000     

Randomly coloring random graphs

We consider the problem of generating a coloring of the random graph ??_n,p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are βΔ colors available, where Δ is the maximum degree of the graph, and we wish to determine the least β = β(p) such that the distribution is close to uniform in O(n log n) steps of the chain. This problem has been previously studied for ??_n,p in cases where np is relatively small. Here we consider the “dense” cases, where np ε [ω ln n, n] and ω = ω(n) → ∞. Our methods are closely tailored to the random graph setting, but we obtain considerably better bounds on β(p) than can be achieved using more general techniques. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009