Abelian Cayley Graphs Research Articles

The degree-diameter problem for circulant graphs of degrees 10 and 11

This paper considers the degree-diameter problem for undirected circulant graphs. For degrees 10 and 11, newly discovered families of circulant graphs of arbitrary diameter are presented which are largest known and are conjectured to be extremal. They are also the largest-known Abelian Cayley graphs of these degrees. For each such family, the order of every graph in the family is defined by a quintic polynomial function of the diameter which is specific to the family. The elements of the generating set for each graph are similarly defined by a set of polynomials in the diameter. The existence of the graphs in the degree 10 families has been proved for all diameters. These graphs are consistent with a conjecture on the order of extremal Abelian Cayley and circulant graphs of any degree and diameter.

Improved lower bounds on the degree–diameter problem

Let C(d, k) and AC(d, k) be the largest order of a Cayley graph and a Cayley graph based on an abelian group, respectively, of degree d and diameter k. It is well known that $$C(d,k)\le 1+d+d(d-1)+\cdots +d(d-1)^{k-1}$$ with equality satisfied if and only if the graph is a Moore graph. However, there is a much better upper bound for abelian Cayley graph. We have $$AC(d,2)\le \frac{d^{2}}{2}+d+1$$ and $$AC(d,k)\le \frac{d^{k}}{k!}+O(d^{k-1})$$ . On the other hand, the best currently lower bounds are $$C(d,2)\ge 0.684d^{2}$$ , $$AC(d,2)\ge \frac{25}{64}d^{2}-2.1d^{1.525}$$ and $$AC(d,k)\ge (\frac{d}{k})^{k}+O(d^{k-1})$$ for sufficiently large d. In this paper, we improve previous results on the degree–diameter problem. We show that $$C(d,2)\ge \frac{200}{289}d^{2}-5.4d^{1.525}$$ , $$AC(d,2)\ge \frac{27}{64}d^{2}-3.9d^{1.525}$$ and $$AC(d,k)\ge (\frac{3}{3k-1})^{k}d^{k}+O(d^{k-0.475})$$ for sufficiently large d.

A Conjecture of Norine and Thomas for Abelian Cayley Graphs

A graph $\Gamma_1$ is a matching minor of $\Gamma$ if some even subdivision of $\Gamma_1$ is isomorphic to a subgraph $\Gamma_2$ of $\Gamma$, and by deleting the vertices of $\Gamma_2$ from $\Gamma$ the left subgraph has a perfect matching. Motivated by the study of Pfaffian graphs (the numbers of perfect matchings of these graphs can be computed in polynomial time), we characterized Abelian Cayley graphs which do not contain a $K_{3,3}$ matching minor. Furthermore, the Pfaffian property of Cayley graphs on Abelian groups is completely characterized. This result confirms that the conjecture posed by Norine and Thomas in 2008 for Abelian Cayley graphs is true.

Improved Upper Bounds for the Order of Some Classes of Abelian Cayley and Circulant Graphs of Diameter Two

In the degree-diameter problem for Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d there is a wide gap between the best lower and upper bounds valid for all d, being quadratic functions with leading coefficient 1/4 and 1/2 respectively. Recent papers have presented constructions which increase the coefficient of the lower bound to be at or just below 3/8 for sparse sets of degree d related to primes of specific congruence classes. The constructions use the direct product of the multiplicative and additive subgroups of a Galois field and a specific cyclic group of coprime order. It was anticipated that this approach would be capable of yielding further improvement towards the upper bound value of 1/2. In this paper, however, it is proved that the quadratic coefficient of the order of families of Abelian Cayley graphs of this class of construction can never exceed the value of 3/8, establishing an asymptotic limit of 3/8 for the quadratic coefficient of families of extremal graphs of this class. By applying recent results from number theory these constructions can be extended to be valid for every degree above some threshold, establishing an improved asymptotic lower bound approaching 3/8 for general Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d.

The Degree/Diameter Problem for mixed abelian Cayley graphs

This paper investigates the upper bounds for the number of vertices in mixed abelian Cayley graphs with given degree and diameter. Additionally, in the case when the undirected degree is equal to one, we give a construction that provides a lower bound.

Locality of percolation for Abelian Cayley graphs

We prove that the value of the critical probability for percolation on an Abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function pcpc defined on the set of Cayley graphs of Abelian groups of rank at least 22 is continuous for the Benjamini–Schramm topology. The proof involves group-theoretic tools and a new block argument.

An inequality for the heat kernel on an Abelian Cayley graph

We demonstrate a relationship between the heat kernel on a finite weighted Abelian Cayley graph and Gaussian functions on lattices. This can be used to prove a new inequality for the heat kernel on such a graph: when $t \leq t'$, \[ \frac{H_t(u, v)} {H_t(u,u)} \leq \frac{H_{t'}(u, v)} {H_{t'}(u,u)}. \] This was an open problem posed by Regev and Shinkar.

A Note on Quadrangular Embedding of Abelian Cayley Graphs

The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound.

A Moore-like bound for mixed abelian Cayley graphs

We give an upper bound for the number of vertices in mixed abelian Cayley graphs with given degree and diameter.

Discrete Curvature and Abelian Groups

AbstractWe study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of “curvature” in discrete spaces. An appealing feature of this discrete version of the so-called Γ2-calculus (of Bakry-Émery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest, particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (à la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs, a result of independent interest.

Finite 2-Geodesic Transitive Abelian Cayley Graphs

In this paper, we first give a classification of the family of 2-geodesic transitive abelian Cayley graphs. Let $$\Gamma $$Γ be such a graph which is not 2-arc transitive. It is shown that one of the following holds: (1) $$\Gamma \cong \mathrm{K}_{m[b]}$$Γ?Km[b] for some $$m\ge 3$$m?3 and $$b\ge 2$$b?2; (2) $$\Gamma $$Γ is a normal Cayley graph of an elementary abelian group; (3) $$\Gamma $$Γ is a cover of Cayley graph $$\Gamma _K$$ΓK of an abelian group T / K, where either $$\Gamma _K$$ΓK is complete arc transitive or $$\Gamma _K$$ΓK is 2-geodesic transitive of girth 3, and A / K acts primitively on $$V(\Gamma _K)$$V(ΓK) of type Affine or Product Action. Second, we completely determine the family of 2-geodesic transitive circulants.

Restricted Connectivity for Some Interconnection Networks

A vertex-cut $$S$$S of a connected graph $$G$$G is called a restricted vertex-cut if no vertex $$u$$u of the graph $$G$$G satisfies $$N_G(u)\subseteq S$$NG(u)⊆S. The restricted connectivity $$\kappa '(G)$$??(G) of the graph $$G$$G is the cardinality of a minimum restricted vertex-cut of $$G$$G; this is a more refined index than the connectivity parameter $$\kappa (G)$$?(G). In this paper, we prove that $$\kappa '(K_m\times G_2)=2k_2+m-2$$??(Km×G2)=2k2+m-2, where $$G_2$$G2 is a $$k_2\ (\ge 2)$$k2(?2)-regular and maximally connected graph with girth $$g(G_2)\ge 4$$g(G2)?4; and $$\kappa '(G_1\times G_2)=2k_1+2k_2-2$$??(G1×G2)=2k1+2k2-2, where $$G_i$$Gi is a $$k_i\ (\ge 2)$$ki(?2)-regular and maximally connected graph with girth $$g(G_i)\ge 4$$g(Gi)?4 for $$1\le i\le 2$$1≤i≤2. Furthermore, we determine the restricted connectivity of a class of minimal Abelian Cayley graphs.

Abelian Cayley Graphs of Given Degree and Diameter 2 and 3

Let CC d,k be the largest possible number of vertices in a cyclic Cayley graph of degree d and diameter k, and let AC d,k be the largest order in an Abelian Cayley graph for given d and k. We show that $${CC_{d,2} \geq \frac{13}{36} (d + 2)(d -4)}$$ for any d= 6p−2 where p is a prime such that $${p \neq 13}$$ , $${p \not\equiv 1}$$ (mod 13), and $${AC_{d,3} \geq \frac{9}{128} (d + 3)^2(d - 5)}$$ for d = 8q−3 where q is a prime power.

Finite locally primitive abelian Cayley graphs

Let Γ be a finite connected locally primitive Cayley graph of an abelian group. It is shown that one of the following holds: (1) Γ = K n , K n,n , K n,n − nK2, K n × … × K n ; (2) Γ is the standard double cover of K n × … × K n ; (3) Γ is a normal or a bi-normal Cayley graph of an elementary abelian or a meta-abelian 2-group.

Bounding neighbor-connectivity of Abelian Cayley graphs

Bounding neighbor-connectivity of Abelian Cayley graphs

On perfect codes in Cartesian products of graphs

Assuming the existence of a partition in perfect codes of the vertex set of a finite or infinite bipartite graph G we give the construction of a perfect code in the Cartesian product G □ G □ P 2 . Such a partition is easily obtained in the case of perfect codes in Abelian Cayley graphs and we give some examples of applications of this result and its generalizations.

Nowhere-zero 3-flows in Cayley graphs and Sylow 2-subgroups

Tutte’s 3-Flow Conjecture suggests that every bridgeless graph with no 3-edge-cut can have its edges directed and labelled by the numbers 1 or 2 in such a way that at each vertex the sum of incoming values equals the sum of outgoing values. In this paper we show that Tutte’s 3-Flow Conjecture is true for Cayley graphs of groups whose Sylow 2-subgroup is a direct factor of the group; in particular, it is true for Cayley graphs of nilpotent groups. This improves a recent result of Potočnik et al. (Discrete Math. 297:119–127, 2005) concerning nowhere-zero 3-flows in abelian Cayley graphs.

Minimal cutwidth linear arrangements of abelian Cayley graphs

We find the minimal cutwidth and bisection width values for abelian Cayley graphs with up to 4 generators and present an algorithm for finding the corresponding optimal ordering. We also find minimal cuts of each order.

A new bound for neighbor-connectivity of abelian Cayley graphs

For the notion of neighbor-connectivity in graphs, whenever a vertex is “subverted” the entire closed neighborhood of the vertex is deleted from the graph. The minimum number of vertices whose subversion results in an empty, complete, or disconnected subgraph is called the neighbor-connectivity of the graph. Gunther, Hartnell, and Nowakowski have shown that for any graph, neighbor-connectivity is bounded above by κ . The main result of this paper is a sharpening of the bound for abelian Cayley graphs. In particular, we show by constructing an effective subversion strategy for such graphs, that neighbor-connectivity is bounded above by ⌈ δ / 2 ⌉ + 2 . Using a result of Watkins the new bound can be recast in terms of κ to get neighbor-connectivity bounded above by ⌈ 3 κ / 4 ⌉ + 2 for abelian Cayley graphs.

Closed walks and eigenvalues of Abelian Cayley graphs

We show that Abelian Cayley graphs contain many closed walks of even length. This implies that given k ⩾ 3 , for each ϵ > 0 , there exists C = C ( ϵ , k ) > 0 such that for each Abelian group G and each symmetric subset S of G with 1 ∉ S , the number of eigenvalues λ i of the Cayley graph X = X ( G , S ) such that λ i ⩾ k − ϵ is at least C ⋅ | G | . This can be regarded as an analogue for Abelian Cayley graphs of a theorem of Serre for regular graphs. To cite this article: S.M. Cioabă, C. R. Acad. Sci. Paris, Ser. I 342 (2006).