D-regular Graph Research Articles

An elementary approach to component sizes in critical random graphs

AbstractIn this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our method to a model of a random intersection graph, a random graph obtained through p-bond percolation on a general d-regular graph, and a model of an inhomogeneous random graph.

Freely Independent Coin Tosses, Standard Young Tableaux, and the Kesten–McKay Law

In this article, we shall start with a closed walk on a regular tree of degree d. These walks are described by the Kesten–McKay law which arises as the asymptotic distribution of a random d-regular graph on n vertices. We will show that the moments of the Kesten–McKay law are given by counting standard Young tableaux with at most two rows, and how some properties of the walk make sense even when d is not an integer. We will use free probability to instruct us about how to build an explicit model in random matrix theory.

(Dis)assortative partitions on random regular graphs

We study the problem of assortative and disassortative partitions on random d-regular graphs. Nodes in the graph are partitioned into two non-empty groups. In the assortative partition every node requires at least H of their neighbors to be in their own group. In the disassortative partition they require less than H neighbors to be in their own group. Using the cavity method based on analysis of the belief propagation algorithm we establish for which combinations of parameters (d, H) these partitions exist with high probability and for which they do not. For we establish that the structure of solutions to the assortative partition problems corresponds to the so-called frozen-one-step replica symmetry breaking. This entails a conjecture of algorithmic hardness of finding these partitions efficiently. For we argue that the assortative partition problem is algorithmically easy on average for all d. Further we provide arguments about asymptotic equivalence between the assortative partition problem and the disassortative one, going through a close relation to the problem of single-spin-flip-stable states in spin glasses. In the context of spin glasses, our results on algorithmic hardness imply a conjecture that gapped single spin flip stable states are hard to find which may be a universal reason behind the observation that physical dynamics in glassy systems display convergence to marginal stability.

Evaluations of Tutte polynomials of regular graphs

Let TG(x,y) be the Tutte polynomial of a graph G. In this paper we show that if (Gn)n is a sequence of d-regular graphs with girth g(Gn)→∞, then for x≥1 and 0≤y≤1 we havelimn→∞⁡TGn(x,y)1/v(Gn)=td(x,y), wheretd(x,y)={(d−1)((d−1)2(d−1)2−x)d/2−1ifx≤d−1and0≤y≤1,x(1+1x−1)d/2−1ifx>d−1and0≤y≤1. If (Gn)n is a sequence of random d-regular graphs, then the same statement holds true asymptotically almost surely.This theorem generalizes results of McKay (x=1,y=1, spanning trees of random d-regular graphs) and Lyons (x=1,y=1, spanning trees of large-girth d-regular graphs). Interesting special cases are TG(2,1) counting the number of spanning forests, and TG(2,0) counting the number of acyclic orientations.

Locating-dominating sets: From graphs to oriented graphs

A locating-dominating set of an undirected graph is a subset of vertices S such that S is dominating and for every u,v∉S, the neighbourhood of u and v on S are distinct (i.e. N(u)∩S≠N(v)∩S). Locating-dominating sets have received a considerable attention in the last decades. In this paper, we consider the oriented version of the problem. A locating-dominating set in an oriented graph is a set S such that for each w∈V∖S, N−(w)∩S≠∅ and for each pair of distinct vertices u,v∈V∖S, N−(u)∩S≠N−(v)∩S. We consider the following two parameters. Given an undirected graph G, we look for γ→LD(G) (Γ→LD(G)) which is the size of the smallest (largest) optimal locating-dominating set over all orientations of G. In particular, if D is an orientation of G, then γ→LD(G)≤γLD(D)≤Γ→LD(G) where γLD(D) is the minimum size of a locating-dominating set of D.For the best orientation, we prove that, for every twin-free graph G on n vertices, γ→LD(G)≤n/2 which proves a “directed version” of a widely studied conjecture on the location-domination number. As a side result we obtain a new improved upper bound for the location-domination number in undirected trees. Moreover, we give some bounds for γ→LD(G) on many graph classes and drastically improve the value n/2 for (almost) d-regular graphs by showing that γ→LD(G)∈O(log⁡d/d⋅n) using a probabilistic argument.While γ→LD(G)≤γLD(G) holds for every graph G, we give some graph classes such as outerplanar graphs for which Γ→LD(G)≥γLD(G) and some for which Γ→LD(G)≤γLD(G) such as complete graphs. We also give general bounds for Γ→LD(G) such as Γ→LD(G)≥α(G). Finally, we show that for many graph classes Γ→LD(G) is polynomial on n but we leave open the question whether there exist graphs with Γ→LD(G)∈O(log⁡n).

Sandwiching dense random regular graphs between binomial random graphs

Kim and Vu made the following conjecture (Advances in Mathematics, 2004): if dgg log n, then the random d-regular graph {mathscr {G}}(n,d) can asymptotically almost surely be “sandwiched” between {mathscr {G}}(n,p_1) and {mathscr {G}}(n,p_2) where p_1 and p_2 are both (1+o(1))d/n. They proved this conjecture for log nll dleqslant n^{1/3-o(1)}, with a defect in the sandwiching: {mathscr {G}}(n,d) contains {mathscr {G}}(n,p_1) perfectly, but is not completely contained in {mathscr {G}}(n,p_2). The embedding {mathscr {G}}(n,p_1) subseteq {mathscr {G}}(n,d) was improved by Dudek, Frieze, Ruciński and Šileikis to d=o(n). In this paper, we prove Kim–Vu’s sandwich conjecture, with perfect containment on both sides, for all d where min {d, n-d}gg n/sqrt{log n}. The sandwich theorem allows translation of many results from {mathscr {G}}(n,p) to {mathscr {G}}(n,d) such as Hamiltonicity, the chromatic number, the diameter, etc. It also allows translation of threshold functions of phase transitions from {mathscr {G}}(n,p) to bond percolation of {mathscr {G}}(n,d). In addition to sandwiching regular graphs, our results cover graphs whose degrees are asymptotically equal. The proofs rely on estimates for the probability of small subgraph appearances in a random factor of a pseudorandom graph, which is of independent interest.

On the Second Eigenvalue of Random Bipartite Biregular Graphs

We consider the spectral gap of a uniformly chosen random $$(d_1,d_2)$$ -biregular bipartite graph G with $$|V_1|=n, |V_2|=m$$ , where $$d_1,d_2$$ could possibly grow with n and m. Let A be the adjacency matrix of G. Under the assumption that $$d_1\ge d_2$$ and $$d_2=O(n^{2/3}),$$ we show that $$\lambda _2(A)=O(\sqrt{d_1})$$ with high probability. As a corollary, combining the results from (Tikhomirov and Yousse in Ann Probab 47(1):362–419, 2019), we show that the second singular value of a uniform random d-regular digraph is $$O(\sqrt{d})$$ for $$1\le d\le n/2$$ with high probability. Assuming $$d_2$$ is fixed and $$d_1=O(n^2)$$ , we further prove that for a random $$(d_1,d_2)$$ -biregular bipartite graph, $$|\lambda _i^2(A)-d_1|=O(\sqrt{d_1})$$ for all $$2\le i\le n+m-1$$ with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook et al. (Ann Probab 46(1):72–125, 2018) for random d-regular graphs and several new switching operations we define for random bipartite biregular graphs.

On 12-regular nut graphs

A nut graph is a simple graph whose adjacency matrix is singular with 1-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each d ∈ {3, 4, …, 11} all values n such that there exists a d-regular nut graph of order n. In the present paper, we resolve the first open case d = 12, i.e. we show that there exists a 12-regular nut graph of order n if and only if n ≥ 16. We also present a result by which there are infinitely many circulant nut graphs of degree d ≡ 0 (mod 4) and no circulant nut graphs of degree d ≡ 2 (mod 4). The former result partially resolves a question by Fowler et al. on existence of vertex-transitive nut graphs of order n and degree d. We conclude the paper with problems, conjectures and ideas for further work.

Star colouring of bounded degree graphs and regular graphs

A k-star colouring of a graph G is a function f:V(G)→{0,1,…,k−1} such that f(u)≠f(v) for every edge uv of G, and every bicoloured connected subgraph of G is a star. The star chromatic number of G, χs(G), is the least integer k such that G is k-star colourable. We prove that χs(G)≥⌈(d+4)/2⌉ for every d-regular graph G with d≥3. We reveal the structure and properties of even-degree regular graphs G that attain this lower bound. The structure of such graphs G is linked with a certain type of Eulerian orientations of G. Moreover, this structure can be expressed in the LC-VSP framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by an FPT algorithm with the parameter either treewidth, cliquewidth, or rankwidth. We prove that for p≥2, a 2p-regular graph G is (p+2)-star colourable only if n:=|V(G)| is divisible by (p+1)(p+2). For each p≥2 and n divisible by (p+1)(p+2), we construct a 2p-regular Hamiltonian graph on n vertices which is (p+2)-star colourable.The problem k-Star Colourability takes a graph G as input and asks whether G is k-star colourable. We prove that 3-Star Colourability is NP-complete for planar bipartite graphs of maximum degree three and arbitrarily large girth. Besides, it is coNP-hard to test whether a bipartite graph of maximum degree eight has a unique 3-star colouring up to colour swaps. For k≥3, k-Star Colourability of bipartite graphs of maximum degree k is NP-complete, and does not even admit a 2o(n)-time algorithm unless ETH fails.

Cutoff for random lifts of weighted graphs

We prove the cutoff phenomenon for the random walk on random n-lifts of finite weighted graphs, even when the random walk on the base graph G of the lift is not reversible. The mixing time is w.h.p. tmix=h−1logn, where h is a constant associated to G, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all n-lifts of G. In the particular case where the base graph is a vertex with d/2 loops, d even, we obtain a cutoff for a d-regular random graph, as did Lubetzky and Sly in (Duke Math. J. 153 (2010) 475–510) (with a slightly different distribution on d-regular graphs, but the mixing time is the same).

Invertibility of adjacency matrices for random d-regular graphs

Let d≥3 be a fixed integer, and let A be the adjacency matrix of a random d-regular directed or undirected graph on n vertices. We show that there exists a constant d>0 such that P(A is singular in R)≤n−d, for n sufficiently large. This answers an open problem by Frieze and Vu. The key idea is to study the singularity probability over a finite field Fp. The proof combines a local central limit theorem and a large deviation estimate.

On the bipartiteness constant and expansion of Cayley graphs

Let G be a finite, undirected, d-regular graph and A(G) its normalized adjacency matrix, with eigenvalues 1=λ1(A)≥⋯≥λn≥−1. It is a classical fact that λn=−1 if and only if G is bipartite. Our main result provides a quantitative separation of λn from −1 in the case of Cayley graphs, in terms of their expansion. Denoting hout by the (outer boundary) vertex expansion of G, we show that if G is a non-bipartite Cayley graph (constructed using a group and a symmetric generating set of size d) then λn≥−1+chout2d2, for c an absolute constant. We exhibit graphs for which this result is tight up to a factor depending on d. This improves upon a recent result by Biswas and Saha (2021) who showed λn≥−1+hout429d8. We also note that such a result could not be true for general non-bipartite graphs.

Fast Algorithms for General Spin Systems on Bipartite Expanders

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ( n 2 ) time, where n is the size of the graph.

Joint large deviation principle for some empirical measures of the d-regular random graphs

In this paper, we define a d–regular random model by perfect matching of vertices or paring of vertices. For each vertex, we assign a q–state spin. From this d–regular graph model, we define the empirical co-operate measure, which enumerates the number of co-operation between a given couple of spins, and empirical spin measure, which enumerates the number of sites having a given spin on the d–regular random graph model. For these empirical measures, we obtain large deviation principle(LDP) in the weak topology.

A proof of the upper matching conjecture for large graphs

We prove that the ‘Upper Matching Conjecture’ of Friedland, Krop, and Markström and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every d and every large enough n divisible by 2d, a union of n/(2d) copies of the complete d-regular bipartite graph maximizes the number of independent sets and matchings of size k for each k over all d-regular graphs on n vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth.

On the Number of Forests and Connected Spanning Subgraphs

Let F(G) be the number of forests of a graph G. Similarly let C(G) be the number of connected spanning subgraphs of a connected graph G. We bound F(G) and C(G) for regular graphs and for graphs with a fixed average degree. Among many other things we study f_d=sup _{Gin {mathcal {G}}_d}F(G)^{1/v(G)}, where {mathcal {G}}_d is the family of d-regular graphs, and v(G) denotes the number of vertices of a graph G. We show that f_3=2^{3/2}, and if (G_n)_n is a sequence of 3-regular graphs with the length of the shortest cycle tending to infinity, then lim _{nrightarrow infty }F(G_n)^{1/v(G_n)}=2^{3/2}. We also improve on the previous best bounds on f_d for 4le dle 9.

A Cheeger type inequality in finite Cayley sum graphs

Let G be a finite group with |G|⩾4 and S be a subset of G with |S|=d such that the Cayley sum graph C Σ (G,S) is undirected and connected. We show that the non-trivial spectrum of the normalised adjacency operator of C Σ (G,S) is controlled by its Cheeger constant and its degree. We establish an explicit lower bound for the non-trivial spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval - 1 + h Σ (G) 4 η , 1 - h Σ (G) 2 2d 2 , where h Σ (G) denotes the vertex Cheeger constant of the d-regular graph C Σ (G,S) and η=2 9 d 8 . Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the Cayley graph of finite groups.

Spectral Threshold for Extremal Cyclic Edge-Connectivity

The universal cyclic edge-connectivity of a graph G is the least k such that there exists a set of k edges whose removal disconnects G into components where every component contains a cycle. We show that for graphs of minimum degree at least 3 and girth g at least 4, the universal cyclic edge-connectivity is bounded above by \((\Delta -2)g\) where \(\Delta \) is the maximum degree. We then prove that if the second eigenvalue of the adjacency matrix of a d-regular graph of girth \(g\ge 4\) is sufficiently small, then the universal cyclic edge-connectivity is \((d-2)g\), providing a spectral condition for when this upper bound on universal cyclic edge-connectivity is tight.

Identity-aware Graph Neural Networks

Message passing Graph Neural Networks (GNNs) provide a powerful modeling framework for relational data. However, the expressive power of existing GNNs is upper-bounded by the 1-Weisfeiler-Lehman (1-WL) graph isomorphism test, which means GNNs that are not able to predict node clustering coefficients and shortest path distances, and cannot differentiate between different d-regular graphs. Here we develop a class of message passing GNNs, named Identity-aware Graph Neural Networks (ID-GNNs), with greater expressive power than the 1-WL test. ID-GNN offers a minimal but powerful solution to limitations of existing GNNs. ID-GNN extends existing GNN architectures by inductively considering nodes’ identities during message passing. To embed a given node, ID-GNN first extracts the ego network centered at the node, then conducts rounds of heterogeneous message passing, where different sets of parameters are applied to the center node than to other surrounding nodes in the ego network. We further propose a simplified but faster version of ID-GNN that injects node identity information as augmented node features. Alto- gether, both versions of ID-GNN represent general extensions of message passing GNNs, where experiments show that transforming existing GNNs to ID-GNNs yields on average 40% accuracy improvement on challenging node, edge, and graph property prediction tasks; 3% accuracy improvement on node and graph classification benchmarks; and 15% ROC AUC improvement on real-world link prediction tasks. Additionally, ID-GNNs demonstrate improved or comparable performance over other task-specific graph networks.

Local classical MAX-CUT algorithm outperforms p=2 QAOA on high-girth regular graphs

The p-stage Quantum Approximate Optimization Algorithm (QAOAp) is a promising approach for combinatorial optimization on noisy intermediate-scale quantum (NISQ) devices, but its theoretical behavior is not well understood beyond p=1. We analyze QAOA2 for the maximum cut problem (MAX-CUT), deriving a graph-size-independent expression for the expected cut fraction on any D-regular graph of girth >5 (i.e. without triangles, squares, or pentagons).We show that for all degrees D≥2 and every D-regular graph G of girth >5, QAOA2 has a larger expected cut fraction than QAOA1 on G. However, we also show that there exists a 2-local randomized classical algorithm A such that A has a larger expected cut fraction than QAOA2 on all G. This supports our conjecture that for every constant p, there exists a local classical MAX-CUT algorithm that performs as well as QAOAp on all graphs.