Pseudo-random Graphs Research Articles

Pseudo-Wigner Matrices

We consider the problem of generating pseudo-random matrices based on the similarity of their spectra to Wigner’s semicircular law. We introduce the notion of an $r$ -independent pseudo-Wigner matrix ensemble and prove the closeness of the spectra of its matrices to the semicircular density in the Kolmogorov distance. We give an explicit construction of a family of $N \times N$ pseudo-Wigner ensembles using dual BCH codes and show that the Kolmogorov complexity of the obtained matrices is of the order of ${\mathrm{log}}(N)$ bits for a fixed designed Kolmogorov distance precision. We compare our construction with the quasi-random graphs introduced by Chung et al. and demonstrate that the pseudo-Wigner matrices pass stronger randomness tests than the adjacency matrices of these graphs (lifted by the mapping $0 \rightarrow 1$ and $1 \rightarrow -1$ ) do. Finally, we provide numerical simulations verifying our theoretical results.

Counting designs

We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an expository treatment of our recently developed method of Randomised Algebraic Construction: we give a simpler proof of a special case of our result on clique decompositions of hypergraphs, namely triangle decompositions of quasirandom graphs.

Quasirandom geometric networks from low-discrepancy sequences.

We define quasirandom geometric networks using low-discrepancy sequences, such as Halton, Sobol, and Niederreiter. The networks are built in d dimensions by considering the d-tuples of digits generated by these sequences as the coordinates of the vertices of the networks in a d-dimensional I^{d} unit hypercube. Then, two vertices are connected by an edge if they are at a distance smaller than a connection radius. We investigate computationally 11 network-theoretic properties of two-dimensional quasirandom networks and compare them with analogous random geometric networks. We also study their degree distribution and their spectral density distributions. We conclude from this intensive computational study that in terms of the uniformity of the distribution of the vertices in the unit square, the quasirandom networks look more random than the random geometric networks. We include an analysis of potential strategies for generating higher-dimensional quasirandom networks, where it is know that some of the low-discrepancy sequences are highly correlated. In this respect, we conclude that up to dimension 20, the use of scrambling, skipping and leaping strategies generate quasirandom networks with the desired properties of uniformity. Finally, we consider a diffusive process taking place on the nodes and edges of the quasirandom and random geometric graphs. We show that the diffusion time is shorter in the quasirandom graphs as a consequence of their larger structural homogeneity. In the random geometric graphs the diffusion produces clusters of concentration that make the process more slow. Such clusters are a direct consequence of the heterogeneous and irregular distribution of the nodes in the unit square in which the generation of random geometric graphs is based on.

Quasirandomness in hypergraphs

A graph G is called quasirandom if it possesses typical properties of the corresponding random graph G(n,p) with the same edge density as G. A well-known theorem of Chung, Graham and Wilson states that, in fact, many such ‘typical’ properties are asymptotically equivalent and, thus, a graph G possessing one property immediately satisfies the others.In recent years, more quasirandom graph properties have been found and extensions to hypergraphs have been explored. For the latter, however, there exist several distinct notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. The purpose of this paper is to give short purely combinatorial proofs of most of Towsner's results.

On the Local Approach to Sidorenko's Conjecture

A well-known conjecture of Erdős-Simonovits and Sidorenko states that, if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same number of vertices and edge density. A strengthening known as the forcing conjecture states that, if H is a bipartite graph with at least one cycle, then quasirandom graphs of density p are the only graphs of density p that asymptotically minimize the number of copies of H. Lovász proved a local version of Sidorenko's conjecture. We characterize those graphs for which Sidorenko's conjecture holds locally. Namely, it holds locally for H if and only if H has even girth or is a forest. Furthermore, a local version of the forcing conjecture holds precisely for graphs of even girth. As a corollary, we prove that for such H there is δH>0 such that Sidorenko's conjecture and the forcing conjecture holds for all p>1−δH.

Powers of Hamilton cycles in pseudorandom graphs

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (e,p,k,l)-pseudorandom if for all disjoint X, Y ⊂ V(G) with |X| ≥ ep k n and |Y| ≥ ep l n we have e(X,Y) = (1±e)p|X||Y|. We prove that for all β > 0 there is an e > 0 such that an (e,p,1,2)-pseudorandom graph on n vertices with minimum degree at least βpn contains the square of a Hamilton cycle. In particular, this implies that (n,d,λ)-graphs with λ ≪ d 5/2 n − 3/2 contain the square of a Hamilton cycle, and thus a triangle factor if n is a multiple of 3. This improves on a result of Krivelevich, Sudakov and Szabo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403–426]. We also obtain results for higher powers of Hamilton cycles and establish corresponding counting versions. Our proofs are constructive, and yield deterministic polynomial time algorithms.

Квазислучайные графы и структурная устойчивость сложных дискретных систем

Квазислучайные графы и структурная устойчивость сложных дискретных систем

Discrepancy and eigenvalues of Cayley graphs

We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This positively answers a question of Chung and Graham ["Sparse quasi-random graphs", Combinatorica 22 (2002), no. 2, 217-244] for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.

Optimal path and cycle decompositions of dense quasirandom graphs

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0<p<1 be constant and let G∼Gn,p. Let odd(G) be the number of odd degree vertices in G. Then a.a.s. the following hold:(i)G can be decomposed into ⌊Δ(G)/2⌋ cycles and a matching of size odd(G)/2.(ii)G can be decomposed into max⁡{odd(G)/2,⌈Δ(G)/2⌉} paths.(iii)G can be decomposed into ⌈Δ(G)/2⌉ linear forests. Each of these bounds is best possible. We actually derive (i)–(iii) from ‘quasirandom’ versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by Kühn and Osthus.

The Phase Transition in Site Percolation on Pseudo-Random Graphs

We establish the existence of the phase transition in site percolation on pseudo-random $d$-regular graphs. Let $G=(V,E)$ be an $(n,d,\lambda)$-graph, that is, a $d$-regular graph on $n$ vertices in which all eigenvalues of the adjacency matrix, but the first one, are at most $\lambda$ in their absolute values. Form a random subset $R$ of $V$ by putting every vertex $v\in V$ into $R$ independently with probability $p$. Then for any small enough constant $\epsilon&gt;0$, if $p=\frac{1-\epsilon}{d}$, then with high probability all connected components of the subgraph of $G$ induced by $R$ are of size at most logarithmic in $n$, while for $p=\frac{1+\epsilon}{d}$, if the eigenvalue ratio $\lambda/d$ is small enough as a function of $\epsilon$, then typically $R$ contains a connected component of size at least $\frac{\epsilon n}{d}$ and a path of length proportional to $\frac{\epsilon^2n}{d}$.

Matrix and discrepancy view of generalized random and quasirandom graphs

Abstract We will discuss how graph based matrices are capable to find classification of the graph vertices with small within- and between-cluster discrepancies. The structural eigenvalues together with the corresponding spectral subspaces of the normalized modularity matrix are used to find a block-structure in the graph. The notions are extended to rectangular arrays of nonnegative entries and to directed graphs. We also investigate relations between spectral properties, multiway discrepancies, and degree distribution of generalized random graphs. These properties are regarded as generalized quasirandom properties, and we conjecture and partly prove that they are also equivalent for certain deterministic graph sequences, irrespective of stochastic models.

On the graph limit question of Vera T. Sós

In the dense graph limit theory, the topology of the set of graphs is defined by the distribution of the subgraphs spanned by finite number of random vertices. Vera T. Sós proposed a question that if we consider only the number of edges in the spanned subgraphs, then whether it provides an equivalent definition. We show that the answer is positive on quasirandom graphs.

Optimal path and cycle decompositions of dense quasirandom graphs

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0<p<1 be constant and let G∼Gn,p. Let odd(G) be the number of odd degree vertices in G. Then a.a.s. the following hold:(i)G can be decomposed into ⌊Δ(G)/2⌋ cycles and a matching of size odd(G)/2.(ii)G can be decomposed into max⁡{odd(G)/2,⌈Δ(G)/2⌉} paths.(iii)G can be decomposed into ⌈Δ(G)/2⌉ linear forests. Each of these bounds is best possible. We actually derive (i)–(iii) from ‘quasi-random’ versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by Kühn and Osthus.

Perfect Packings in Quasirandom Hypergraphs II

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ⩾ 4 and 0 &lt; p &lt; 1. Suppose that H is an n-vertex triple system with r|n and the following two properties: •for every graph G with V(G) = V(H), at least p proportion of the triangles in G are also edges of H,•for every vertex x of H, the link graph of x is a quasirandom graph with density at least p. Then H has a perfect Kr(3)-packing. Moreover, we show that neither of the hypotheses above can be weakened, so in this sense our result is tight. A similar conclusion for this special case can be proved by Keevash's Hypergraph Blow-up Lemma, with a slightly stronger hypothesis on H.

On the graph limit question of Vera T. Sós

In the dense graph limit theory, the topology of the set of graphs is defined by the distribution of the subgraphs spanned by finite number of random vertices. Vera T. Sós proposed a question that if we consider only the number of edges in the spanned subgraphs, then whether it provides an equivalent definition. We show that the answer is positive on quasirandom graphs, and we prove a generalization of the statement.

Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

The Kuramoto model (KM) of coupled phase oscillators on complete, Paley, and Erdos-Renyi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the KM on these graphs can be qualitatively different. Specifically, we identify twisted states, steady state solutions of the KM on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the IVPs for the KM on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the KM on Cayley and random graphs.

More on quasi-random graphs, subgraph counts and graph limits

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It has been shown by several authors that several such conditions are quasi-random, but that there are exceptions. In order to understand this better, we investigate some new properties of this type. We show that these properties too are quasi-random, at least in some cases; however, there are also cases that are left as open problems, and we discuss why the proofs fail in these cases.The proofs are based on the theory of graph limits; and on the method and results developed by Janson (2011), this translates the combinatorial problem to an analytic problem, which then is translated to an algebraic problem.

Extremal results for odd cycles in sparse pseudorandom graphs

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $\Gamma$ the generalized Tur\'an density $\pi_F(\Gamma)$ denotes the density of a maximum subgraph of $\Gamma$, which contains no copy of~$F$. Extending classical Tur\'an type results for odd cycles, we show that $\pi_{F}(\Gamma)=1/2$ provided $F$ is an odd cycle and $\Gamma$ is a sufficiently pseudorandom graph. In particular, for $(n,d,\lambda)$-graphs $\Gamma$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues in the interval $[-\lambda,\lambda]$, our result holds for odd cycles of length $\ell$, provided \[ \lambda^{\ell-2}\ll \frac{d^{\ell-1}}n\log(n)^{-(\ell-2)(\ell-3)}\,. \] Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szab\'o, and Vu, who addressed the case when $F$ is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free $(n,d,\lambda)$-graphs) shows that our assumption on $\Gamma$ is best possible up to the polylog-factor for every odd $\ell\geq 5$.

Optimal covers with Hamilton cycles in random graphs

A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has size \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szab\'o recently initiated research on the `dual' problem, where one asks for a set of Hamilton cycles covering all edges of G. Our main result states that for log^{117}n / n < p < 1-n^{-1/8}, a.a.s. the edges of G_n,p can be covered by \lceil Delta(G_n,p)/2 \rceil Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szab\'o, which holds for p > n^{-1+\eps}. Our proof is based on a result of Knox, K\"uhn and Osthus on packing Hamilton cycles in pseudorandom graphs.

Packing Tree Factors in Random and Pseudo-Random Graphs

For a fixed graph $H$ with $t$ vertices, an $H$-factor of a graph $G$ with $n$ vertices, where $t$ divides $n$, is a collection of vertex disjoint (not necessarily induced) copies of $H$ in $G$ covering all vertices of $G$. We prove that for a fixed tree $T$ on $t$ vertices and $\epsilon&gt;0$, the random graph $G_{n,p}$, with $n$ a multiple of $t$, with high probability contains a family of edge-disjoint $T$-factors covering all but an $\epsilon$-fraction of its edges, as long as $\epsilon^4 n p \gg \log^2 n$. Assuming stronger divisibility conditions, the edge probability can be taken down to $p&gt;\frac{C\log n}{n}$. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.