Finite Regular Graphs Research Articles

Degrees in Link Graphs of Regular Graphs

We analyse an extremal question on the degrees of the link graphs of a finite regular graph, that is, the subgraphs induced by non-trivial spheres. We show that if $G$ is $d$-regular and connected but not complete then some link graph of $G$ has minimum degree at most $\lfloor{2d/3}\rfloor-1$, and if $G$ is sufficiently large in terms of $d$ then some link graph has minimum degree at most $\lfloor{d/2}\rfloor-1$; both bounds are best possible. We also give the corresponding best-possible result for the corresponding problem where subgraphs induced by balls, rather than spheres, are considered.
 We motivate these questions by posing a conjecture concerning expansion of link graphs in large bounded-degree graphs, together with a heuristic justification thereof.

APPROXIMATIONS OF REGULAR GRAPHS

The paper [11] raised the question of describing the cardinality and types of approximations for natural families of theories. In the present paper, a partial answer to this question is given, and the study of approximation and topological properties of natural classes of theories is also continued. We consider a cycle graph consisting of one cycle or, in other words, a certain number of vertices (at least 3 if the graph is simple) connected into a closed chain. It is shown that an infinite cycle graph is approximated by finite cycle graphs. Approximations of regular graphs by finite regular graphs are considered. On the other hand, approximations of acyclic regular graphs by finite regular graphs are considered. It is proved that any infinite regular graph is pseudofinite. And also, for any k, any k-regular graph is homogeneous and pseudofinite.Examples of pseudofinite 3-regular and 4-regular graphs are given.

Monotonic Normalized Heat Diffusion for Regular Bipartite Graphs with Four Eigenvalues

Let $X=(V, E)$ be a finite regular graph and $H_t(u, v), \, u, v \in V$, the heat kernel on $X$. We prove that, if the graph $X$ is bipartite and has four distinct Laplacian eigenvalues, the ratio $H_t(u, v)/H_t(u, u), \, u, v \in V,$ is monotonically non-decreasing as a function of $t$. The key to the proof is the fact that such a graph is an incidence graph of a symmetric 2-design.

Quantum ergodicity for large equilateral quantum graphs

Consider a sequence of finite regular graphs (GN) converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero coupling constant α) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum, the eigenfunctions of the converging quantum graphs satisfy a quantum ergodicity theorem. In case α = 0 and U = 0, the limit measure is the uniform measure on the edges. In general, it has an explicit analytic density. We finally prove a stronger quantum ergodicity theorem involving integral operators, the purpose of which is to study eigenfunction correlations.

Moments of the inverse participation ratio for the Laplacian on finite regular graphs

We investigate the first and second moments of the inverse participation ratio (IPR) for all eigenvectors of the Laplacian on finite random regular graphs with n vertices and degree z. By exactly diagonalizing a large set of z-regular graphs, we find that as n becomes large, the mean of the inverse participation ratio on each graph, when averaged over a large ensemble of graphs, approaches the numerical value 3. This universal number is understood as the large-n limit of the average of the quartic polynomial corresponding to the IPR over an appropriate -dimensional hypersphere of . For a large, but not exhaustive ensemble of graphs, the mean variance of the inverse participation ratio for all graph Laplacian eigenvectors deviates from its continuous hypersphere average due to large graph-to-graph fluctuations that arise from the existence of highly localized modes.

Derangement action digraphs and graphs

A derangement of a set X is a fixed-point-free permutation of X. Derangement action digraphs are closely related to group action digraphs introduced by Annexstein, Baumslag and Rosenberg in 1990. For a non-empty set X and a non-empty subset S of derangements of X, the derangement action digraph DA⃗(X,S) has vertex set X, and an arc from x to y if and only if y is the image of x under the action of some element of S, so by definition it is a simple digraph. In common with Cayley graphs and Cayley digraphs, derangement action digraphs may be useful to model networks since the same routing and communication schemes can be implemented at each vertex. We prove that the family of derangement action digraphs contains all Cayley digraphs, all finite vertex-transitive simple graphs, and all finite regular simple graphs of even valency. We determine necessary and sufficient conditions on S under which DA⃗(X,S) may be viewed as a simple undirected graph of valency |S|. We investigate structural and symmetry properties of these digraphs and graphs, pose several open problems, and give many examples.

Invariant random perfect matchings in Cayley graphs

We prove that every non-amenable Cayley graph admits a factor of IID perfect matching. We also show that any connected d -regular vertex transitive graph admits a perfect matching. The two results together imply that every Cayley graph admits an invariant random perfect matching. A key step in the proof is a result on graphings that also applies to finite graphs. The finite version says that for any partial matching of a finite regular graph that is a good expander, one can always find an augmenting path whose length is poly-logarithmic in one over the ratio of unmatched vertices.

Quantum automorphism group of the lexicographic product of finite regular graphs

We study the quantum automorphism group of the lexicographic product of two finite regular graphs, providing a quantum generalization of Sabidussi's structure theorem on the automorphism group of such a graph.

On Quantum Percolation in Finite Regular Graphs

The aim of this paper is twofold. First, we study eigenvalues and eigenvectors of the adjacency matrix of a bond percolation graph when the base graph is finite and well approximated locally by an infinite regular graph. We relate quantitatively the empirical measure of the eigenvalues and the delocalization of the eigenvectors to the spectrum of the adjacency operator of the percolation on the infinite graph. Secondly, we prove that percolation on an infinite regular tree with degree at least three preserves the existence of an absolutely continuous spectrum if the removal probability is small enough. These two results are notably relevant for bond percolation on a uniformly sampled regular graph or a Cayley graph with large girth.

Poisson Eigenvalue Statistics for Random Schrödinger Operators on Regular Graphs

For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular, localization and pure point spectrum in infinite volume is expected to correspond to Poisson eigenvalue statistics. Motivated by results on the Anderson model on the infinite tree we consider random Schrodinger operators on finite regular graphs. We study local spectral statistics: we analyze the number of eigenvalues in intervals with length comparable to the inverse of the number of vertices of the graph, in the limit where this number tends to infinity. We show that the random point process generated by the rescaled eigenvalues converges in certain spectral regimes of localization to a Poisson process. The corresponding result on the lattice was proved by Minami. However, due to the geometric structure of regular graphs the known methods turn out to be difficult to adapt. Therefore, we develop a new approach based on direct comparison of eigenvectors.

The independent set sequence of regular bipartite graphs

Let it(G) be the number of independent sets of size t in a graph G. Alavi, Erdős, Malde and Schwenk made the conjecture that if G is a tree then the independent set sequence {it(G)}t≥0 of G is unimodal; Levit and Mandrescu further conjectured that this should hold for all bipartite G.We consider the independent set sequence of finite regular bipartite graphs, and graphs obtained from these by percolation (independent deletion of edges). Using bounds on the independent set polynomial P(G,λ)≔∑t≥0it(G)λt for these graphs, we obtain partial unimodality results in these cases.We then focus on the discrete hypercube Qd, the graph on vertex set {0,1}d with two strings adjacent if they differ on exactly one coordinate. We obtain asymptotically tight estimates for it(d)(Qd) in the range t(d)/2d−1>1−1/2, and nearly matching upper and lower bounds otherwise. We use these estimates to obtain a stronger partial unimodality result for the independent set sequence of Qd.

On induced matchings as star complements in regular graphs

We determine all the finite regular graphs which have an induced matching or a cocktail party graph as a star complement.

Dynamical properties of profinite actions

AbstractWe study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

Star complements in regular graphs: Old and new results

We survey results concerning star complements in finite regular graphs, and note the connection with designs and strongly regular graphs in certain cases. We include improved proofs along with new results on stars and windmills as star complements.

Orbits in finite regular graphs

Given three integers k , ν and ϵ , we prove that there exists a finite k -regular graph whose automorphism group has exactly ν orbits on the set of vertices and ϵ orbits on the set of edges if and only if { ( ν , ϵ ) = ( 1 , 0 ) when k = 0 ( ν , ϵ ) = ( 1 , 1 ) when k = 1 ν = ϵ ≥ 1 when k = 2 1 ≤ ν ≤ 2 ϵ ≤ 2 k ν when k ≥ 3 . Given an arbitrary odd prime p , we construct countably many pairwise non-isomorphic p -regular graphs which are edge-transitive but not vertex-transitive.

The second eigenvalue of regular graphs of given girth

Lower bounds on the subdominant eigenvalue of regular graphs of given girth are derived. Our approach is to approximate the discrete spectrum of a finite regular graph by the continuous spectrum of an infinite regular tree. We interpret these spectra as probability distributions and the girth condition as equalities between the moments of these distributions. Then the associated orthogonal polynomials coincide up to a degree equal to half the girth, and their extremal zeroes provide bounds on the supports of these distributions.

Groups of automorphisms of finite regular cubic graphs

Groups of automorphisms of finite regular cubic graphs

Arc transitive covering digraphs and their eigenvalues

AbstractWe prove that every finite regular digraph has an arc‐transitive covering digraph (whose arcs are equivalent under automorphisms) and every finite regular graph has a 2‐arc‐transitive covering graph. As a corollary, we sharpen C. D. Godsil's results on eigenvalues and minimum polynomials of vertex‐transitive graphs and digraphs. Using Godsil's results, we prove, that given an integral matrix A there exists an arc‐transitive digraph X such that the minimum polynomial of A divides that of X. It follows that there exist arc‐transitive digraphs with nondiagonalizable adjacency matrices, answering a problem by P. J. Cameron. For symmetric matrices A, we construct a 2‐arc‐transitive graphs X.

Finite common coverings of pairs of regular graphs

The graph G is a covering of the graph H if there exists a (projection) map p from the vertex set of G to the vertex set of H which induces a one-to-one correspondence between the vertices adjacent to v in G and the vertices adjacent to p( v) in H, for every vertex v of G. We show that for any two finite regular graphs G and H of the same degree, there exists a finite graph K that is simultaneously a covering both of G and H. The proof uses only Hall's theorem on 1-factors in regular bipartite graphs.

Rozklad konečného pravidelného grafu nepárneho stupňa na dva faktory

Rozklad konečného pravidelného grafu nepárneho stupňa na dva faktory