Journal of Algebraic Combinatorics最新文献

A bipartite graph (Gamma ) is a bi-Cayley graph over a group H if (Hleqslant textrm{Aut}Gamma ) acts regularly on each part of (Gamma ). A bi-Cayley graph (Gamma ) is said to be a normal bi-Cayley graph over H if (Hunlhd textrm{Aut}Gamma ), and bi-primitive if the bipartition preserving subgroup of (textrm{Aut}Gamma ) acts primitively on each part of (Gamma ). In this paper, a classification is given for 2-arc-transitive bi-Cayley graphs which are bi-primitive and non-normal.

Let (Gamma = (V, E)) be a graph and a, b nonnegative integers. An (a, b)-regular set in (Gamma ) is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in (V{setminus }D) has exactly b neighbours in D. A (0, 1)-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset D of a group G is called an (a, b)-regular set of G if it is an (a, b)-regular set in some Cayley graph of G, and an (a, b)-regular set in a Cayley graph of G is called a subgroup (a, b)-regular set if it is also a subgroup of G. In this paper, we study (a, b)-regular sets in Cayley graphs with a focus on (0, k)-regular sets, where (k ge 1) is an integer. Among other things, we determine when a non-trivial proper normal subgroup of a group is a (0, k)-regular set of the group. We also determine all subgroup (0, k)-regular sets of dihedral groups and generalized quaternion groups. We obtain necessary and sufficient conditions for a hypercube or the Cartesian product of n copies of the cycle of length p to admit (0, k)-regular sets, where p is an odd prime. Our results generalize several known results from perfect codes to (0, k)-regular sets.

We first determine the structure of the power digraphs of completely 0-simple semigroups, and then some properties of their power graphs are given. As the main result in this paper, using Cameron and Ghosh’s theorem about power graphs of abelian groups, we obtain a characterization that two (G^{0})-normal completely 0-simple orthodox semigroups S and T with abelian group (mathcal {H})-classes are isomorphic based on their power graphs. We also present an algorithm to determine that S and T are isomorphic or not.

Given integers (k,lge 2), where either l is odd or k is even, let n(k, l) denote the largest integer n such that each element of (A_n) is a product of k many l-cycles. M. Herzog, G. Kaplan and A. Lev conjectured that (lfloor frac{2kl}{3} rfloor le n(k,l)le lfloor frac{2kl}{3}rfloor +1) [Herzog et al. in J Combin Theory Ser A, 115:1235-1245 2008]. It is known that the conjecture holds when (k=2,3,4). Moreover, it is also true when (3mid l). In this article, we determine the exact value of n(k, l) when (3not mid l) and (kge 5). As an immediate consequence, we get that (n(k,l)<lfloor frac{2kl}{3}rfloor ) when (kge 5) and (3not mid l), which shows that the above conjecture is not true in general. In fact in this case, the difference between the exact value of n(k, l) and the conjectured value grows linearly in terms of k. Our results complete the determination of n(k, l) for all values of k and l.

Abstract

We introduce the notion of an anti-dendriform algebra as a new approach of splitting the associativity. It is characterized as the algebra with two multiplications giving their left and right multiplication operators, respectively, such that the opposites of these operators define a bimodule structure on the sum of these two multiplications, which is associative. This justifies the terminology due to a closely analogous characterization of a dendriform algebra. The notions of anti- ({mathcal {O}}) -operators and anti-Rota–Baxter operators on associative algebras are introduced to interpret anti-dendriform algebras. In particular, there are compatible anti-dendriform algebra structures on associative algebras with nondegenerate commutative Connes cocycles. There is an important observation that there are correspondences between certain subclasses of dendriform and anti-dendriform algebras in terms of q-algebras. As a direct consequence, we give the notion of Novikov-type dendriform algebras as an analogue of Novikov algebras for dendriform algebras, whose relationship with Novikov algebras is consistent with the one between dendriform and pre-Lie algebras. Finally, we extend to provide a general framework of introducing the notions of analogues of anti-dendriform algebras, which interprets a new splitting of operations.

We consider networks on minimal Cayley graphs of irreducible complex reflection groups G(m, p, n). We show that resistance diameters of these graphs have asymptotic (Theta (1/n)) as (nrightarrow infty ) under fixed m, p. Non-trivial lower and upper asymptotic bounds for critical probabilities of percolation for there appearing a giant connected component have been obtained.

Let ({mathcal {A}}subseteq {[n]atopwithdelims ()a}) and ({mathcal {B}}subseteq {[n]atopwithdelims ()b}) be two families of subsets of [n], we say ({mathcal {A}}) and ({mathcal {B}}) are cross-intersecting if (Acap Bne emptyset ) for all (Ain {mathcal {A}}), (Bin {mathcal {B}}). In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.

Abstract

Given a finite abelian group G and cyclic subgroups A, B, C of G of the same order, we find necessary and sufficient conditions for A, B, C to admit a common transversal for the cosets they afford. For an arbitrary number of cyclic subgroups, we give a sufficient criterion when there exists a common complement. Moreover, in several cases where a common transversal exists, we provide concrete constructions.

A mixed dihedral group is a group H with two disjoint subgroups X and Y, each elementary abelian of order (2^n), such that H is generated by (Xcup Y), and (H/H'cong Xtimes Y). In this paper, we give a sufficient condition such that the automorphism group of the Cayley graph (textrm{Cay}(H,(Xcup Y){setminus }{1})) is equal to (Hrtimes A(H,X,Y)), where A(H, X, Y) is the setwise stabiliser in ({{,textrm{Aut},}}(H)) of (Xcup Y). We use this criterion to resolve a question of Li et al. (J Aust Math Soc 86:111-122, 2009), by constructing a 2-arc-transitive normal cover of order (2^{53}) of the complete bipartite graph ({{textbf {K}}}_{16,16}) and prove that it is not a Cayley graph.

Let (D_{n,k}) be the set of all permutations of the symmetric group (S_n) that have no cycles of length i for all (1 le i le k). In the paper mentioned above, Ku, Lau, and Wong prove that the set of all the largest independent sets of the Cayley graph (text {Cay}(S_n,D_{n,k})) is equal to the set of all the largest independent sets in the derangement graph (text {Cay}(S_n,D_{n,1})), provided n is sufficiently large in terms of k. We give a simpler proof that holds for all n, k and also applies to the alternating group.