arXiv subjects

Viktor Harangi

Publications and source records attributed to Viktor Harangi.

At least 19 recordsLinked to original sources

arXiv2026

The least balanced graphs and trees

Given a connected graph, the principal eigenvector of the adjacency matrix (often called the Perron vector) can be used to assign positive weights to the vertices. A natural way to measure the homogeneousness of this vector is by considering the ratio of its $\ell^1$ and $\ell^2$ norms. It is easy to see that the most balanced graphs in this sense (i.e., the ones with the largest ratio) are the regular graphs. What can we say about the least balanced (or most centralized) graphs with the smallest ratio? It was conjectured by Rücker, Rücker and Gutman that, for any given $n \geq 6$, among $n$-vertex connected graphs the smallest ratio is achieved by the complete graph $K_4$ with a single path $P_{n-4}$ attached to one of its vertices. In this paper we confirm this conjecture. We also verify the analogous conjecture for trees: for any given $n \geq 8$, among $n$-vertex trees the smallest ratio is achieved by the star graph $S_5$ with a path $P_{n-5}$ attached to its central vertex.

math.CO↗

arXiv2026

Boosted second moment method in random regular graphs

Determining the asymptotic independence ratio of random regular graphs is a key challenge in the area of sparse random graphs. Due to the interpolation method, we have very good upper bounds at our disposal, which are actually known to be sharp for sufficiently large degrees. However, we are still in need of good explicit lower bounds for specific degrees. The classical approach by Frieze and Łuczak achieves a lower bound by first applying the second moment method to sparse Erdős--Rényi graphs, and then cleverly transitioning from that model to regular graphs. They obtain an asymptotic formula (as the degree tends to infinity) but no explicit lower bounds are derived for specific degrees. In contrast, in this paper, we apply the second moment method directly to random regular graphs. This approach has a number of advantages. First, we can numerically compute good explicit lower bounds for any given degree $d$. Moreover, we can even boost this lower bound by arguing that the obtained independent set has a certain spatial Markov property. One can then exploit this property by making local modifications to the independent set, resulting in substantial improvements, and beating the previous best bounds for any $d \geq 10$. Finally, this method gives finer asymptotics as $d \to \infty$ than the original Frieze--Łuczak approach. Moreover, these results can be useful even beyond the scope of the independence ratio due to the fact that independent sets with the Markov property may be used to construct other objects in random regular graphs. To demonstrate this, we consider the problem of decomposing random regular graphs into stars.

math.CO↗

arXiv2025

RSB bounds on the maximum cut

In the context of random regular graphs, the size of the maximum cut is probably the second most studied graph parameter after the independence ratio. Zdeborová and Boettcher used the cavity method, a non-rigorous statistical physics technique, to predict one-step replica symmetry breaking (1-RSB) formulas. Coja-Ohglan et al. confirmed these predictions as rigorous upper bounds using the interpolation method. While these upper bounds were not expected to be exact, they may be very close to the true values. In this paper, we establish 2-RSB upper bounds and fine-tune their parameters to beat the aforementioned 1-RSB bounds.

math.CO↗

arXiv2025

Star decompositions via orientations

A $k$-star decomposition of a graph is a partition of its edges into $k$-stars (i.e., $k$ edges with a common vertex). The paper studies the following problem: given $k \leq d/2$, does the random $d$-regular graph have a $k$-star decomposition (asymptotically almost surely, provided that the number of edges is divisible by $k$)? Delcourt, Greenhill, Isaev, Lidický, and Postle proved the a.a.s. existence for every odd $k$ using earlier results regarding orientations satisfying certain degree conditions modulo $k$. In this paper we give a direct, self-contained proof that works for every $d$ and every $k<d/2-1$. In fact, we prove stronger results. Let $s\geq 1$ denote the integer part of $d/(2k)$. We show that the random $d$-regular graph a.a.s. has a $k$-star decomposition such that the number of stars centered at each vertex is either $s$ or $s+1$. Moreover, if $k < d/3$ or $k \leq d/2 - 2.6 \log d$, we can even prescribe the set of vertices with $s$ stars, as long as it is of the appropriate size.

math.CO↗

arXiv2025

Star decompositions and independent sets in random regular graphs

A $k$-star decomposition of a graph is a partition of its edges into $k$-stars (i.e., $k$ edges with a common vertex). The paper studies the following problem: for what values of $k>d/2$ does the random $d$-regular graph have a $k$-star decomposition (asymptotically almost surely, provided that the number of edges is divisible by $k$)? Delcourt, Greenhill, Isaev, Lidický, and Postle proposed the following conjecture. It is easy to see that a $k$-star decomposition necessitates the existence of an independent set of density $1-d/(2k)$. So let $k^{\mathrm{ind}}_d$ be the largest $k$ for which the random $d$-regular graph a.a.s. contains an independent set of this density. Clearly, $k$-star decompositions cannot exist for $k>k^{\mathrm{ind}}_d$. The conjecture suggests that this is essentially the only restriction: there is a threshold $k^\star_d$ such that $k$-star decompositions exist if and only if $k \leq k^\star_d$, and it (basically) coincides with the other threshold, i.e., $k^\star_d \approx k^{\mathrm{ind}}_d$. We confirm this conjecture for sufficiently large $d$ by showing that a $k$-star decomposition exists if $d/2< k < k^{\mathrm{ind}}_d$. In fact, we prove the existence even if $k=k^{\mathrm{ind}}_d$ for degrees $d$ with asymptotic density $1$.

math.CO↗

arXiv2023

Conditional graph entropy as an alternating minimization problem

Conditional graph entropy is known to be the minimal rate for a natural functional compression problem with side information at the receiver. In this paper we show that it can be formulated as an alternating minimization problem, which gives rise to a simple iterative algorithm for numerically computing (conditional) graph entropy. This also leads to a new formula which shows that conditional graph entropy is part of a more general framework: the solution of an optimization problem over a convex corner. In the special case of graph entropy (i.e., unconditioned version) this was known due to Csiszár, Körner, Lovász, Marton, and Simonyi. In that case the role of the convex corner was played by the so-called vertex packing polytope. In the conditional version it is a more intricate convex body but the function to minimize is the same. Furthermore, we describe a dual problem that leads to an optimality check and an error bound for the iterative algorithm.

cs.IT↗

arXiv2023

Generalizing Körner's graph entropy to graphons

Körner introduced the notion of graph entropy in 1973 as the minimal code rate of a natural coding problem where not all pairs of letters can be distinguished in the alphabet. Later it turned out that it can be expressed as the solution of a minimization problem over the so-called vertex-packing polytope. In this paper we generalize this notion to graphons. We show that the analogous minimization problem provides an upper bound for graphon entropy. We also give a lower bound in the shape of a maximization problem. The main result of the paper is that for most graphons these two bounds actually coincide and hence precisely determine the entropy in question. Furthermore, graphon entropy has a nice connection to the fractional chromatic number and the fractional clique number.

math.CO↗

arXiv2022

Improved replica bounds for the independence ratio of random regular graphs

Studying independent sets of maximum size is equivalent to considering the hard-core model with the fugacity parameter $λ$ tending to infinity. Finding the independence ratio of random $d$-regular graphs for some fixed degree $d$ has received much attention both in random graph theory and in statistical physics. For $d \geq 20$ the problem is conjectured to exhibit 1-step replica symmetry breaking (1-RSB). The corresponding 1-RSB formula for the independence ratio was confirmed for (very) large $d$ in a breakthrough paper by Ding, Sly, and Sun. Furthermore, the so-called interpolation method shows that this 1-RSB formula is an upper bound for each $d \geq 3$. For $d \leq 19$ this bound is not tight and full-RSB is expected. In this work we use numerical optimization to find good substituting parameters for discrete $r$-RSB formulas ($r=2,3,4,5$) to obtain improved rigorous upper bounds for the independence ratio for each degree $3 \leq d \leq 19$. As $r$ grows, these formulas get increasingly complicated and it becomes challenging to compute their numerical values efficiently. Also, the functions to minimize have a large number of local minima, making global optimization a difficult task.

math.CO↗

arXiv2020

Entropy and expansion

Shearer's inequality bounds the sum of joint entropies of random variables in terms of the total joint entropy. We give another lower bound for the same sum in terms of the individual entropies when the variables are functions of independent random seeds. The inequality involves a constant characterizing the expansion properties of the system. Our results generalize to entropy inequalities used in recent work in invariant settings, including the edge-vertex inequality for factor-of-IID processes, Bowen's entropy inequalities, and Bollobás's entropy bounds in random regular graphs. The proof method yields inequalities for other measures of randomness, including covariance. As an application, we give upper bounds for independent sets in both finite and infinite graphs.

math.PR↗

arXiv2017

Entropy inequalities for factors of IID

This paper is concerned with certain invariant random processes (called factors of IID) on infinite trees. Given such a process, one can assign entropies to different finite subgraphs of the tree. There are linear inequalities between these entropies that hold for any factor of IID process (e.g. "edge versus vertex" or "star versus edge"). These inequalities turned out to be very useful: they have several applications already, the most recent one is the Backhausz-Szegedy result on the eigenvectors of random regular graphs. We present new entropy inequalities in this paper. In fact, our approach provides a general "recipe" for how to find and prove such inequalities. Our key tool is a generalization of the edge-vertex inequality for a broader class of factor processes with fewer symmetries.

math.PR↗

arXiv2017

Acute sets of exponentially optimal size

We present a simple construction of an acute set of size $2^{d-1}+1$ in $\mathbb{R}^d$ for any dimension $d$. That is, we explicitly give $2^{d-1}+1$ points in the $d$-dimensional Euclidean space with the property that any three points form an acute triangle. It is known that the maximal number of such points is less than $2^d$. Our result significantly improves upon a recent construction, due to Dmitriy Zakharov, with size of order $φ^d$ where $φ= (1+\sqrt{5})/2 \approx 1.618$ is the golden ratio.

math.MG↗

arXiv2017

Mutual information decay for factors of IID

This paper is concerned with factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We study the mutual information of the values on two given vertices. If the vertices are neighbors (i.e., their distance is $1$), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$, of order $(d-1)^{-k/2}$. Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process the rate of decay of the mutual information is much faster, essentially of order $(d-1)^{-k}$.

math.PR↗

arXiv2016

Correlation bound for distant parts of factor of IID processes

We study factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most $k(d-1) / (\sqrt{d-1})^k$, where $k$ denotes the distance of the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.

math.PR↗

arXiv2015

Independence ratio and random eigenvectors in transitive graphs

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $λ_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a $3$-regular transitive graph is at least \[q=\frac{1}{2}-\frac{3}{4π}\arccos\biggl(\frac{1-λ_{\min}}{4}\biggr).\] The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least $q-o(1)$. We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.

math.PR↗

arXiv2013

Invariant Gaussian processes and independent sets on regular graphs of large girth

We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue λ. We show that such processes can be approximated by i.i.d. factors provided that $|λ| \leq 2\sqrt{d-1}$. We then use these approximations for $λ= -2\sqrt{d-1}$ to produce factor of i.i.d. independent sets on regular trees.

math.CO↗

arXiv2012

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and $\al$, how large Hausdorff dimension can a compact set $A\su\Rn$ have if $A$ does not contain three points that form an angle $\al$? (2) Given $\al$ and $\de$, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\de$-neighborhood of $\al$? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles.

math.CA↗

arXiv2012

Intersection of continua and rectifiable curves

We prove that for any non-degenerate continuum $K \subseteq \mathbb{R}^d$ there exists a rectifiable curve such that its intersection with $K$ has Hausdorff dimension 1. This answers a question of B. Kirchheim.

math.CA↗

arXiv2011

On the density of triangles and squares in regular finite and unimodular random graphs

We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these densities. As a corollary one gets an explicit description of the possible pairs of the third and fourth moments of the spectral measure of r-regular unimodular random graphs.

math.CO↗