Journal of Algebraic Combinatorics最新文献

Abstract

The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. In this paper, we determine the k-uniform hypertree with the minimal spectral radius among all k-uniform hypertrees with m edges and diameter d, where (3le dle m-1) .

This work is part of a research program to compute the Hochschild homology groups HH(_*({mathbb {C}}[x_1,ldots ,x_d]/(x_1,ldots ,x_d)^3;{mathbb {C}})) in the case (d = 2) through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism

provided by Larsen and Lindenstrauss. Here, (H_C^*) denotes Coxeter cohomology, (S_{i+j}) denotes the symmetric group on (i+j) letters, and V is the standard representation of (textrm{GL}_d({mathbb {C}})) on ({mathbb {C}}^d). We compute the Euler characteristic of the Coxeter cohomology (the alternating sum of the ranks of the Coxeter cohomology groups) of several representations of (S_n). In particular, the aforementioned tensor representation, and also several classes of irreducible representations of (S_n). Although the problem and its motivation are algebraic and topological in nature, the techniques used are largely combinatorial.

Abstract

A skew morphism of a finite group A is a permutation (varphi ) of A fixing the identity element and for which there is an integer-valued function (pi ) on A such that (varphi (ab)=varphi (a)varphi ^{pi (a)}(b)) for all (a, b in A) . A skew morphism (varphi ) of A is smooth if the associated power function (pi ) is constant on the orbits of (varphi ) , that is, (pi (varphi (a))equiv pi (a)pmod {|varphi |}) for all (ain A) . In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if (n=2^en_1) , where (0 le e le 4) and (n_1) is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.

Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In Wang and Suzuki (Discrete Math 264:225–236, 2003), the third author and Suzuki proposed a question when an orientation of a distance-regular graph defines a weakly distance-regular digraph. In this paper, we initiate this project and classify all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded n-cubes and Doob graphs, respectively.

Let G be a finite group with identity e and ( H ne {e}) be a subgroup of G. The generalized non-coprime graph (varGamma _{G,H}) of ( G ) with respect to (H) is the simple undirected graph with (G setminus {e }) as the vertex set and two distinct vertices ( x ) and ( y) are adjacent if and only if (gcd (|x|,|y|) ne 1) and either (x in H) or (y in H), where |x| is the order of (xin G). In this paper, we study certain graph theoretical properties of generalized non-coprime graphs of finite groups, concentrating on cyclic groups. More specifically, we obtain necessary and sufficient conditions for the generalized non-coprime graph of a cyclic group to be in the class of stars, paths, triangle-free, complete bipartite, complete, split, claw-free, chordal or perfect graphs. Then we show that widening the class of groups to all finite nilpotent groups gives us no new graphs, but we give as an example of contrasting behaviour the class of EPPO groups (those in which all elements have prime power order). We conclude with a connection to the Gruenberg–Kegel graph.

Each connected graded, graded-commutative algebra A of finite type over a field (Bbbk ) of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When (A=Bbbk langle Delta rangle ) is the exterior Stanley–Reisner algebra associated to a finite simplicial complex (Delta ), we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group ((G,+)) is a bijection (f:Arightarrow B) between two finite subsets A, B of G satisfying (a+f(a)notin A) for all (ain A). A group G has the matching property if for every two finite subsets (A,B subset G) of the same size with (0 notin B), there exists a matching from A to B. In Losonczy (Adv Appl Math 20(3):385–391, 1998) it was proved that an abelian group has the matching property if and only if it is torsion-free or cyclic of prime order. Here we consider a similar question in a matroid setting. We introduce an analogous notion of matching between matroids whose ground sets are subsets of an abelian group G, and we obtain criteria for the existence of such matchings. Our tools are classical theorems in matroid theory, group theory and additive number theory.

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.