Expander graphs have many interesting applications in communication networks and other areas, and thus these graphs have been extensively studied in theoretic computer sciences and in applied mathematics. In this paper, we use reversible difference sets and generalized difference sets to construct more expander graphs, some of them are Ramanujan graphs. Three classes of elementary constructions of infinite families of Ramanujan graphs are provided. It is proved that for every even integer $k>4$, if $2(k+2)=rs$ for two even numbers $r$ and $s$ with $s\geq 4$ and $2s>r\geq s$, or $r\geq 4$ and $2r>s\geq r$, then there exists an $k$-regular Ramanujan graph. As a consequence, there exists an $k$-regular Ramanujan graph with $k=2t^2-2$ for every integer $t>2$. It is also proved that for every odd integer $m$, there is an $(2^{2m-2}+2^{m-1})$-regular Ramanujan graph. These results partially solved the long hanging open question for the existence of $k$-regular Ramanujan graphs for every positive integer $k$.
For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a family of expanders.
Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of $r$-uniform hypergraph expanders for all $r \geq 3$.
In this paper, we study new Cayley graphs over the additive group of Galois rings. First we prove that they are expander graphs by using a Weil-Carlitz-Uchiyama type estimation of character sums for Galois rings. We also show that Cayley graphs from Galois rings of characteristic 4 form a new infinite family of Ramanujan graphs by an elementary eigenvalue estimation. Moreover some other spectral properties of our graphs are also discussed.
The Cayley graphs of finite groups are known to provide several examples of families of expanders, and some of them are Ramanujan graphs. Babai studied isospectral non-isomorphic Cayley graphs of the dihedral groups. Lubotzky, Samuels and Vishne proved that there are isospectral non-isomorphic Cayley graphs of $\mathrm{PSL}_d(\mathbb F_q)$ for every $d\geq 5$ ($d \neq 6$) and prime power $q> 2$. In this article, we focus on three variants of Cayley graphs, viz., the Cayley sum graphs, the twisted Cayley graphs, and the twisted Cayley sum graphs. We prove the existence of non-isomorphic expande…
We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.
A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of Chung.
We construct a new family of trivalent expanders tessellating hyperbolic surfaces with large isometry groups. These graphs are obtained from a family of Cayley graphs of nilpotent groups via $(Δ-Y)$-transformations. We compare this family with Platonic graphs and their associated hyperbolic surfaces and see that they are generally very different with only one hyperbolic surface in the intersection. Moreover, we study combinatorial, topological and spectral properties of our trivalent graphs and their associated hyperbolic surfaces.
We present a construction of expander graphs obtained from Cayley graphs of narrow ray class groups, whose eigenvalue bounds follow from the Generalized Riemann Hypothesis. Our result implies that the Cayley graph of (Z/qZ)* with respect to small prime generators is an expander. As another application, we show that the graph of small prime degree isogenies between ordinary elliptic curves achieves non-negligible eigenvalue separation, and explain the relationship between the expansion properties of these graphs and the security of the elliptic curve discrete logarithm problem.
We provide an explicit construction of finite 4-regular graphs $(Γ_k)_{k\in \mathbb N}$ with ${girth Γ_k\to\infty}$ as $k\to\infty$ and $\frac{diam Γ_k}{girth Γ_k}\leqslant D$ for some $D>0$ and all $k\in\mathbb{N}$. For each fixed dimension $n\geqslant 2,$ we find a pair of matrices in $SL_{n}(\mathbb{Z})$ such that (i) they generate a free subgroup, (ii)~their reductions $\bmod\, p$ generate $SL_{n}(\mathbb{F}_{p})$ for all sufficiently large primes $p$, (iii) the corresponding Cayley graphs of $SL_{n}(\mathbb{F}_{p})$ have girth at least $c_n\log p$ for some $c_n>0$. Relying on growth resul…
We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of random walks on groups, card shuffling and other areas.
We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon--Roichman about expanding properties of random Cayley graphs. Applying the general setting we show e.g., that with high probability (in a strong exp…
We show that for integers k > 1 and n > 2, the diameter of the Cayley graph of SL_n(Z/kZ) associated to a standard two-element generating set, is at most a constant times n^2 ln k. This answers a question of A. Lubotzky concerning SL_n(F_p) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing elements of SL_n(Z/kZ).
We prove that any non-amenable Cayley graph admits a factor of IID perfect matching. We also show that any connected d-regular vertex tran- sitive 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.
Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon--Roichman. His pr…
A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths. For minimally rigid graphs, the double-exponential Gröbner Bases algorithm with an elimination order can be applied, in theory, but it is impractical even for small instances. By relating the problem to the computation of circuit polynomials in the Cayley-Menger ideal, we recently proposed an algebraic-combinatorial approach and an elimination algorithm for c…
Let the group $G$ act transitively on the finite set $Ω$, and let $S \subseteq G$ be closed under taking inverses. The Schreier graph $Sch(G \circlearrowleft Ω,S)$ is the graph with vertex set $Ω$ and edge set $\{ (ω,ω^s) : ω\in Ω, s \in S \}$. In this paper, we show that random Schreier graphs on $C \log|Ω|$ elements exhibit a (two-sided) spectral gap with high probability, magnifying a well known theorem of Alon and Roichman for Cayley graphs. On the other hand, depending on the particular action of $G$ on $Ω$, we give a lower bound on the number of elements which are necessary to provide a …
We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.
We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree.
For n at least 2, the concept of n-way expanders was defined by various researchers. Bigger n gives a weaker notion in general, and 2-way expanders coincide with expanders in usual sense. Koji Fujiwara asked whether these concepts are equivalent to that of ordinary expanders for all n for a sequence of Cayley graphs. In this paper, we answer his question in the affirmative. Furthermore, we obtain universal inequalities on multi-way isoperimetric constants on any finite connected vertex-transitive graph, and show that gaps between these constants imply the imprimitivity of the group action on t…
The main contribution of this work is a new type of graph product, which we call the {\it zig-zag product}. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of arbitrary size, starting from one constant-size expander. Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as ``entropy wave" propagat…
Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_Σ(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}η, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_Σ(G,S)$ and $η= 2…
Commutative Distance-Regularity: Algebraic Hierarchies and Cayley Graph Constructions
Centro de Investigaciones en Ciencias Exactas y Naturales (CIEM), Universidad Nacional de Córdoba（科尔多瓦国立大学精确与自然科学研究中心）；Instituto de Matemática Aplicada San Luis, Universidad Nacional de San Luis and CONICET（圣路易斯应用数学研究所，圣路易斯国立大学与阿根廷国家科学研究委员会）；Departamento de Matemática, Universidad Nacional de San Luis（圣路易斯国立大学数学系）
We prove Bowen's conjecture that every sequence of finite graphs that locally converges to the Cayley graph of a countably infinite group with Kazhdan Property (T) is essentially a vertex-disjoint union of expander graphs. We characterize graph sequences that are essentially a vertex-disjoint union of expander graphs in terms of the Markov operator.
A two-dimensional simplicial complex is called $d$-{\em regular} if every edge of it is contained in exactly $d$ distinct triangles. It is called $ε$-expanding if its up-down two-dimensional random walk has a normalized maximal eigenvalue which is at most $1-ε$. In this work, we present a class of bounded degree 2-dimensional expanders, which is the result of a small 2-complex action on a vertex set. The resulted complexes are fully transitive, meaning the automorphism group acts transitively on their faces. Such two-dimensional expanders are rare! Known constructions of such bounded degree tw…
A balanced graph is a bipartite graph with no induced circuit of length 2 mod 4. These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented.
A fundamental fact about bounded-degree graph expanders is that three notions of expansion -- vertex expansion, edge expansion, and spectral expansion -- are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion. There are two well-studied notions of linear-algebraic expansion, namely dimension expansion (defined in analogy to graph vertex expansion) and quantum expansion (defined in analogy to graph spectral expansion). Lubotzky and Zelmanov proved that the latter implies the former. We prove that the converse is false: there…
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper, establishing strongly dense subgroups in simple algebraic groups.
We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, "(3,6)-fullerenes", have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form $\{λ,-λ\}$ except for the four eigenvalues $\{3,-1,-1,-1\}$. We exhibit other families of graphs which are "spectrally nearly bipartite" in this sense. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum grap…
In 1988, Duval introduced the concept of directed strongly regular graphs, which can be viewed as a directed graph version of strongly regular graphs. Such directed graphs have similar structural and algebraic properties to strongly regular graphs. In the past three decades, it was found that Cayley graphs, especially those over dihedral groups, play a key role in the construction of directed strongly regular graphs. In this paper, we focus on the characterization of directed strongly regular Cayley graphs over more general groups. Let $G$ be a non-abelian group with an abelian subgroup of ind…
The concept of directed strongly regular graphs (DSRG) was introduced by Duval in "A Directed Graph Version of Strongly Regular Graphs" [Journal of Combinatorial Theory, Series A 47(1988)71-100]. Duval also provided several construction methods for directed strongly regular graphs. In this paper, We construct several new classes of directed strongly regular graphs which are obtained by using Kronecker matrix product, Semidirect product and Cayley coset graph. At the same time, using group representation, for two special cases, we give some other sufficient and necessary conditions of Cayley gr…
This paper deals with some of the algebraic properties of Sierpiński graphs and a family of regular generalized Sierpiński graphs. For the family of regular generalized Sierpiński graphs, we obtain their spectrum and characterize those graphs that are Cayley graphs. As a by-product, a new family of non-Cayley vertex-transitive graphs, and consequently, a new set of non-Cayley numbers are introduced. We also obtain the Laplacian spectrum of Sierpiński graphs in some particular cases, and make a conjecture on the general case.
In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower bounds on the chromatic number of these graphs. This provides a lower bound for the chromatic number of Cayley graphs of the regular graphs associated to the ring of $n\times n$ matrices over finite fields. Using Weil's bound for Kloosterman sums we will also prove an analogous result for $\mathrm{SL}_2$ over finite rings.
The splitting field $\mathbb{SF}(Γ)$ of a mixed graph $Γ$ is the smallest field extension of $\mathbb{Q}$ which contains all eigenvalues of the Hermitian adjacency matrix of $Γ$. The extension degree $[\mathbb{SF}(Γ):\mathbb{Q}]$ is called the algebraic degree of $Γ$. In this paper, we determine the splitting fields and algebraic degrees of mixed Cayley graphs over abelian groups. This generalizes the main results of [K. Mönius, Splitting fields of spectra of circulant graphs, J. Algebra 594(15) (2022) 154--169] and [M. Kadyan, B. Bhattacharjya, Integral mixed Cayley graphs over abelian groups…
The classic Cayley identity states that \det(\partial) (\det X)^s = s(s+1)...(s+n-1) (\det X)^{s-1} where X=(x_{ij}) is an n-by-n matrix of indeterminates and \partial=(\partial/\partial x_{ij}) is the corresponding matrix of partial derivatives. In this paper we present straightforward combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann-Berezin integration. Among the new identities proven here are a pair of "diagonal-parametrized" Cayley identities, a pair of "Laplacian-par…
We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: - We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu's Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. - We construct the first examples of finitely generated groups, with expanders …
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more standard approach to Cayley's formula. Moreover, this algebraic structure sheds light on the problem of counting the unlabeled trees. In particular, it indicates how counting the number of unlabeled trees on $n$ vertices is connected to finding the number of partitions of $n-2$
