A class of Hamiltonian and edge symmetric Cayley graphs on symmetric groups

Abstract. Let Sn be the symmetric group     

Normal and non-normal Cayley graphs for symmetric groups

A Cayley graph is said to be an NNN-graph if its automorphism group contains two isomorphic regular subgroups where one is normal and the other is non-normal. In this paper, we show that there exist NNN-graphs among the Cayley graphs for symmetric groups $S_n$ if and only if $n?5$.     

Large bipartite Cayley graphs of given degree and diameter

Let BC_d,k be the largest possible number of vertices in a bipartite Cayley graph of degree d and diameter k. We show that BC_d,k≥2(k−1)((d−4)/3)^k−1 for any d≥6 and any even k≥4, and BC_d,k≥(k−1)((d−2)/3)^k−1 for d≥6 and k≥7 such that k≡3 (mod 4).     

The thresholds for diameter 2 in random Cayley graphs

Given a group G, the model denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any and any family of groups G_k of order n_k for which , a graph with high probability has diameter at most 2 if and with high probability has diameter greater than 2 if . We also provide examples of families of graphs which show that both of these results are best possible. Of particular interest is that for some families of groups, the corresponding random Cayley graphs achieve Diameter 2 significantly faster than the Erd?s‐Renyi random graphs. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 218–235, 2014     

Large Cayley graphs and vertex‐transitive non‐Cayley graphs of given degree and diameter

For any d?5 and k?3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d?3)/3)^k. By comparison with other available results in this area we show that our family gives the largest currently known Cayley graphs for a wide range of sufficiently large degrees and diameters. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 87–98, 2010     

On 7-valent symmetric Cayley graphs of finite simple groups

Journal of Algebraic Combinatorics - Quite a lot of attention has been paid recently to the classification of symmetric Cayley graphs of non-abelian simple groups. Besides the known complete...     

Isomorphisms of Cayley graphs of a free Abelian group

A group G is called a CI-group provided that the existence of some automorphism σ ∈ Aut(G) such that σ(A) = B follows from an isomorphism Cay(G, A) ? = Cay (G, B) between Cayley graphs, where A and B are two systems of generators for G. We prove that every finitely generated abelian group is a CI-group.     

Cayley graphs of the group ℤ4 and limits for minimal vertex-primitive graphs of HA-type

We point out a countable set of pairwise nonisomorphic Cayley graphs of the group ℤ⁴ that are limit for finite minimal vertex-primitive graphs admitting a vertex-primitive automorphism group containing a regular Abelian normal subgroup. Supported by RFBR grant No. 06-01-00378. __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 203–214, March–April, 2008.     

Nowhere-zero 3-flows in Cayley graphs on generalized dihedral group and generalized quaternion group

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111: 416–419]. We also generalizes an early result of M. Nánásiová and M. ?koviera [J. Algebraic Combin., 2009, 30: 103–110].     

Embedding graphs in Cayley graphs

IfY is a finite graph then it is known that every sufficiently large groupG has a Cayley graph containing an induced subgraph isomorphic toY. This raises the question as to what is sufficiently large. Babai and Sós have used a probabilistic argument to show that |G| > 9.5 |Y|³ suffices. Using a form of greedy algorithm we strengthen this to (2 + \sqrt 3 )|Y|^3 $$ " align="middle" border="0"> . Some related results on finite and infinite groups are included.     

Symmetry properties of Cayley graphs of small valencies on the alternating group A_5

The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal; that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.     

A note on Cayley graphs

An example is given of a finite group A of order 144, with a generating set X = {x, y} such that x³ = y² = 1 and such that the Cayley graph C(A, X) has genus 4 and characteristic −6 (both of which are small relative to the order of A), although there is no short relator of the form (xy)^r with r < 12 or of the form [x, y]^r with r < 6. Accordingly this and other possible examples do not fit into a pattern suggested by [5.], 244–268).     

Cayley graphs on abelian groups

Let A be an abelian group and let ι be the automorphism of A defined by: ι: a ? a^?1. A Cayley graph Γ = Cay(A,S) is said to have an automorphism group as small as possible if Aut(Γ)=A?<ι>. In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.     

Channel assignment on Cayley graphs

We address various channel assignment problems on the Cayley graphs of certain groups, computing the frequency spans by applying group theoretic techniques. In particular, we show that if G is the Cayley graph of an n‐generated group Γ with a certain kind of presentation, then λ(G;k, 1)≤2(k+n?1). For certain values of k this bound gives the obvious optimal value for any 2n‐regular graph. A large number of groups (for instance, even Artin groups and a number of Baumslag–Solitar groups) satisfy this condition. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 169‐177, 2011     

Bounds on the subdominant eigenvalue involving group inverse with applications to graphs

Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are ₁₂ ... _n. In this paper we derive several lower and upper bounds, in particular on ₂ and _n , but also, indirectly, on . The bounds are in terms of the diagonal entries of the group generalized inverse, Q ^#, of the singular and irreducible M-matrix Q = ₁ I – A. Our starting point is a spectral resolution for Q ^#. We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected graphs, where now Q becomes L, the Laplacian of the graph. In case the graph is a tree we find a graph-theoretic interpretation for the entries of L ^# and we also sharpen an upper bound on the algebraic connectivity of a tree, which is due to Fiedler and which involves only the diagonal entries of L, by exploiting the diagonal entries of L ^#.     

Spectra of Cayley graphs

By a result of L. Lovász, the determination of the spectrum of any graph with transitive automorphism group easily reduces to that of some Cayley graph.We derive an expression for the spectrum of the Cayley graph X(G,H) in terms of irreducible characters of the group G:

for any natural number t, where ξ_i is an irreducible character (over $\text{C}$), of degree n_i , and λ_i,1 ,…, λ_i,ni are eigenvalues of X(G, H), each one n_i times. (σn_i² = n = | G | is the total'number of eigenvalues.) Using this formula for t = 1,…, n_i one can obtain a polynomial of degree n_i whose roots are λ_i,1,…,λ_i,ni. The results are formulated for directed graphs with colored edges. We apply the results to dihedral groups and prove the existence of k nonisomorphic Cayley graphs of D_p with the same spectrum provided p > 64k, prime.     

Automorphisms of Cayley graphs

The aim of this paper is to characterize subgroups of non-regular automorphism groups of Cayley graphs. Relations between group automorphisms, graph automorphisms and permutations of the edge set of Cayley graphs are investigated.