In this paper we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let $A$ be an abelian group and let $ι$ be the automorphism of $A$ defined by $a^ι=a^{-1}$, for every $a\in A$. A Cayley graph $\Cay(A, S)$ is said to have an automorphism group as small as possible if $\Aut(\Cay(A,S)) = \langle A,ι\rangle$. In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley digraphs.
We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group $\C{A}$. Using these characterizations, we find enumeration formulae of the equivalence classes and weak equivalence classes of Cayley graphs. As an application, we find the number of weak equivalence classes of circulant graphs.
Cayley's 1875 enumerations of centered and bicentered alkanes (unlabeled trees of valency at most 4) are corrected and extended -- possibly for the first time in 124 years.
Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified. A nature question is to classify regular Cayley maps of elementary abelian $p$-groups $Z_p^n$. In this paper, a complete classification of regular Cayley maps of $Z_p^n$ is given and moreover, the number of these maps and their genera are enumerated.
Let $p$ be an odd prime, and $D_{2p}=\langle a,b\mid a^p=b^2=1,bab=a^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we completely classify the cubic Cayley graphs on $D_{2p}$ up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on $D_{2p}$ are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on $D_{2p}$ by using Gauss' celebrated law of quadratic reciprocity.
We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group -- the part of this bipartite Cayley graph which contains the identity element -- and $\mathbb{Z}_{2}$. We apply this to the case of bipartite distance-regular Cayley graphs with diameter three, and consider cases where the sufficient conditions are not satisfied for some specific groups such as the dihedral group. We also extend a result by Miklavič and Potočnik that relates difference sets to …
In this paper we show that two distinct conjectures, the first proposed by Babai and Godsil in $1982$ and the second proposed by Xu in $1998$, concerning the asymptotic enumeration of Cayley graphs are in fact equivalent. This result follows from a more general theorem concerning the asymptotic enumeration of a certain family of Cayley graphs.
Using the theory of combinatorial species, we compute the cycle index for bipartite graphs, which we use to count unlabeled bipartite graphs and bipartite blocks.
In this paper, we show that every bipartite distance-regular Cayley graph with diameter $3$ can be constructed on the semidirect product of a group and $\mathbb{Z}_{2}$, except possibly for one case.
A graph $Γ$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $Γ$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $Γ$ be a connected bipartite bi-Cayley graph over the group $G$. In this paper, we prove that $G$ is normal in the full automorphism group ${\rm Aut}(Γ)$ of $Γ$ when $G$ is a Sylow $p$-subgroup of ${\rm Aut}(Γ)$. As an application, we classify half-arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $2p$. Furthermore, it is shown that there are no semisymmetric and no …
In this paper we show that almost all Cayley digraphs have automorphism group as small as possible; that is, they are digraphical regular representations (DRRs). More precisely, we show that as $r$ tends to infinity, for every finite group $R$ of order $r$, out of all possible Cayley digraphs on $R$ the proportion whose automorphism group is as small as possible tends to $1$. This proves a natural conjecture first proposed in $1982$ by Babai and Godsil.
Let $C_{d,k}$ be the largest number of vertices in a Cayley digraph of degree $d$ and diameter $k$, and let $BC_{d,k}$ be the largest order of a bipartite Cayley digraph for given $d$ and $k$. For every degree $d\geq2$ and for every odd $k$ we construct Cayley digraphs of order $2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ and diameter at most $k$, where $k\ge 3$, and bipartite Cayley digraphs of order $2(k-1)\left(\lfloor\frac{d}{2}\rfloor\right)^{k-1}$ and diameter at most $k$, where $k\ge 5$. These constructions yield the bounds $C_{d,k} \ge 2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ for od…
In \cite{Chan95}, the authors classified the 2-extendable abelian Cayley graphs and posed the problem of characterizing all 2-extendable Cayley graphs. We first show that a connected bipartite Cayley (vertex-transitive) graph is 2-extendable if and only if it is not a cycle. It is known that a non-bipartite Cayley (vertex-transitive) graph is 2-extendable when it is of minimum degree at least five \cite{sun}. We next classify all 2-extendable cubic non-bipartite Cayley graphs and obtain that: a cubic non-bipartite Cayley graph with girth $g$ is 2-extendable if and only if $g\geq 4$ and it does…
A binary Cayley graph is a Cayley graph based on a binary group. In 1982, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd-cycle, implying that such a graph cannot have chromatic number 3. We strengthen this result first by proving that any non-bipartite binary Cayley graph must contain a projective cube as a subgraph. We further conjecture that any homo- morphism of a non-bipartite binary Cayley graph to a projective cube must be surjective and we prove some special case of this conjecture.
We give a simple formula for the number of hypertrees with $k$ hyperedges of given sizes and $n+1$ labelled vertices with prescribed degrees. A slight generalization of this formula counts labelled bipartite trees with prescribed degrees in each class of vertices.
In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group $R$ has a nontrivial proper n…
Let $G$ be a group and let $S$ be an inverse-closed and identity-free generating set of $G$. The \emph{Cayley graph} $\Cay(G,S)$ has vertex-set $G$ and two vertices $u$ and $v$ are adjacent if and only if $uv^{-1}\in S$. Let $CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order at most $n$. We show that $\log(CAY_d(n))\inΘ(d(\log n)^2)$, as $n\to\infty$. We also obtain some stronger results in the case $d=3$.
Let $G$ be a finite transitive group on a set $Ω$, let $α\in Ω$ and let $G_α$ be the stabilizer of the point $α$ in $G$. In this paper, we are interested in the proportion $$\frac{|\{ω\in Ω\mid ω\textrm{ lies in a }G_α\textrm{-orbit of cardinality at most two}\}|}{|Ω|},$$ that is, the proportion of elements of $Ω$ lying in a suborbit of cardinality at most two. We show that, if this proportion is greater than $5/6$, then each element of $Ω$ lies in a suborbit of cardinality at most two and hence $G$ is classified by a result of Bergman and Lenstra. We also classify the permutation groups attai…
A map is a connected topological graph $Γ$ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are presented. By determining automorphisms of maps of Cayley graph $Γ={\rm Cay}(G:S)$ with ${\rm Aut} Γ\cong G\times H$ on locally, orientable and non-orientable surfaces, formulae for the number of non-equivalent maps of $Γ$ on surfaces (orientable, non-orientable or locally orientable) are obtained . Meanwhile, using reseults on GRR graph for finite groups, we enumerat…
Let $p$ be an odd prime, and $D_{2p}=\langle τ,σ\mid τ^p=σ^2=e,στσ=τ^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we provide the number of (connected) Cayley (di-)graphs on $D_{2p}$ up to isomorphism by using the Pólya enumeration theorem. In the process, we also enumerate (connected) Cayley digraphs on $D_{2p}$ of out-degree $k$ up to isomorphism for each $k$.
This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players …
We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation capturing exactly these groups. Extending this in a companion paper, we find group presentations capturing the planar finitely generated Cayley graphs. Thus we obtain an effective enumeration of these Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al.
In this paper we introduce and study a type of Cayley graph -- subnormal Cayley graph. We prove that a subnormal 2-arc transitive Cayley graph is a normal Cayley graph or a normal cover of a complete bipartite graph $K_{p^d,p^d}$ with $p$ prime. Then we obtain a generic method for constructing half-symmetric (namely edge transitive but not arc transitive) Cayley graphs.
Let $G$ be a finite, undirected $d$-regular graph and $A(G)$ its normalized adjacency matrix, with eigenvalues $1 = λ_1(A)\geq \dots \ge λ_n \ge -1$. It is a classical fact that $λ_n = -1$ if and only if $G$ is bipartite. Our main result provides a quantitative separation of $λ_n$ from $-1$ in the case of Cayley graphs, in terms of their expansion. Denoting $h_{out}$ 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 $λ_n \ge -1 + ch_{out}^2/d^2\,,$ for $c$ an absolute …
The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in the cases where $n$ is a prime power, obtained in collaboration with Du, Kwak, Nedela and Škoviera, together with results of Itô, Hall, Huppert and Wielandt on factorisable groups and on finite solvable groups.
The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley graphs on dicyclic groups is obtained. More specifically, it is shown that every distance-regular Cayley graph on a dicyclic group is a complete graph, a complete multipartite graph, or a non-antipodal bipartite distance-regular graph with diameter $3$ satisfying some additional conditions.
The theory of Γ-species is developed to allow species-theoretic study of quotient structures in a categorically rigorous fashion. This new approach is then applied to two graph-enumeration problems which were previously unsolved in the unlabeled case-bipartite blocks and general k-trees.
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.
We give a generating function for the number of graphs with given numerical properties and prescribed weighted number of connected components. As an application, we give a generating function for the number of bipartite graphs of given order, size and number of connected components.
A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main resu…
A Cayley graph of a group $H$ is a finite simple graph $Γ$ such that ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the bipartite sets of $Σ$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner…
We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequences, for a wide range of parameters. Our results cover medium range densities and close the gaps between the results known for the sparse and dense ranges. In the case of bipartite graphs, these results were proved by Greenhill, McKay and Wang in 2006 and by Canfield, Greenhill and McKay in 2008, respectively. Our method also essentially covers the sparse range, for which much less was known in the case of loopless digraphs. For the range o…
Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high-dimensional complex integrals. Motivated by the need for a systematic study of the asymptotic behaviour of such integrals, we establish explicit bounds on the exponentials of complex martingales. Those bounds applied to the case of truncated normal distributions are precise enough to include and extend many enumerative results of Barvinok, Canfield, Gao, Greenhill, Hartigan, Isaev, McKay, Wang, Wormald, and others. Our method applies to sums as well as inte…
We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group $\C{A}$. Using these characterizations, we find degree distribution polynomials for weak equivalence of some graphs including 1) circulant graphs of prime power order, 2) circulant graphs of order $4p$, 3) circulant graphs of square free order and 4) Cayley graphs of order $p$ or $2p$. As an application, we find an enumeration formula for the number of weak equivalence classes of circulant graphs of prime power order, order $4p$ and square free order and Cayley graphs of order $p$ or $2p$.
The Ferrers bound conjecture is a natural graph-theoretic extension of the enumeration of spanning trees for Ferrers graphs. We document the current status of the conjecture and provide a further conjecture which implies it.
We show that a group admits a planar, finitely generated Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a consequence, we obtain an effective enumeration of the planar Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al. asking whether the planar groups can be effectively enumerated.
A poset is (3+1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets play a central role in the (3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have enumerated (3+1)-free posets in the graded case by decomposing them into bipartite graphs, but until now the general enumeration problem has remained open. We give a finer decomposition into bipartite graphs which applies to all (3+1)-free posets and obtain generating functions which count (3+1)-free posets with labelled or unlabelled vertices. Using this decomposition, w…
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…
We give a complete formula for the characteristic polynomial of hyperplane arrangements $\mathcal J_n$ consisting of the hyperplanes $x_i+x_j=1$, $x_k=0$, $x_l=1$, $ 1\leq i, j, k, l\leq n$. The formula is obtained by associating hyperplane arrangements with graphs, and then enumerating central graphs via generating functions for the number of bipartite graphs of given order, size and number of connected components.
We prove that the Cayley graphs $X(G,S)$ and $X^+(G,S)$ are equienergetic for any abelian group $G$ and any symmetric subset $S$. We then focus on the family of unitary Cayley graphs $G_R=X(R,R^*)$, where $R$ is a finite commutative ring with identity. We show that under mild conditions, $\{G_R, G_R^+\}$ are pairs of integral equienergetic non-isospectral graphs (generically connected and non-bipartite). Then, we obtain conditions such that $\{G_R, \bar G_R\}$ are equienergetic non-isospectral graphs. Finally, we characterize all integral equienergetic non-isospectral triples $\{G_R, G_R^+, \b…
In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the distinction between labeled and unlabeled structures. The cycle index series of a species encodes the labeled and unlabeled enumerative data of that species. Moreover, by using species operations, we are able to solve for the cycle index series of one species in terms of other, known cycle indices of other species. Section 3 is an exposition of species theory and Sect…
A Cayley graph of a group $H$ is a finite simple graph $Γ$ such that its automorphism group ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the partite sets of $Σ$. It is well-known that every Haar graph of finite abelian groups is a Cayley graph. In this paper, we prove that every finite non-abelian group admits a non-Cayley Haar graph except the dihedral groups $D_6$, $D_…
