Journal of Algebraic Combinatorics最新文献

The Terwilliger algebras of asymmetric association schemes of rank 3, whose nonidentity relations correspond to doubly regular tournaments, are shown to have thin irreducible modules, and to always be of dimension (4k+9) for some positive integer k. It is determined that asymmetric rank 3 association schemes of order up to 23 are determined up to combinatorial isomorphism by the list of their complex Terwilliger algebras at each vertex, but this is no longer true at order 27. To distinguish order 27 asymmetric rank 3 association schemes, it is shown using computer calculations that the list of rational Terwilliger algebras at each vertex will suffice.

The k-token graph (F_k(G)) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity of (F_k(G)) equals the algebraic connectivity of G with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of (F_k(G)) equals the one of G for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.

We introduce the concept of pseudocover, which is a counterpart of cover, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given by Praeger, Zhou and the first-named author a decade ago, which seems technical and hard to extend to obtain more examples. In this paper, we present a criterion for a symmetric extender of a symmetric graph to be a pseudocover, and then apply it to produce various examples of pseudocovers, including (1) with a single exception, each Praeger–Xu’s graph is a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph with vertex stabilizer of size divisible by 32 has connected pseudocovers.

The behavior of objects associated with general extended affine Lie algebras is typically distinct from their counterparts in affine Lie algebras. Our research focuses on studying characters and Cartan automorphisms, which appear in the study of Chevalley involutions and Chevalley bases for extended affine Lie algebras. We show that for almost all extended affine Lie algebras, any finite-order Cartan automorphism is diagonal, and its corresponding combinatorial map is a character.

Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n-fold coverings of a graph by Kwak and Lee. For regular graph coverings, Kwak, Chun and Lee enumerated the isomorphism classes of graph coverings when the covering transformation group is a finite abelian or a dihedral group in Kwak et al. (SIAM J Discrete Math 11:273–285, 1998). In 2018, the isomorphism classes of graph coverings are enumerated when the covering transformation groups are (mathbb {Z}_2)-extensions of a cyclic group. As a continuation of this work, we enumerate the isomorphism classes of coverings of a graph when the covering transformation groups are (mathbb {Z}_p)-extensions of a cyclic group for an odd prime integer p.

We study the effect of Feigin’s flat degeneration of the type (text {A}) flag variety on the defining ideals of its Schubert varieties. In particular, we describe two classes of Schubert varieties which stay irreducible under the degenerations and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some degenerate Schubert varieties (i.e. the vanishing sets of initial ideals of the ideals of Schubert varieties with respect to Feigin’s Gröbner degeneration) with Richardson varieties in higher rank partial flag varieties.

Let R be a finite commutative ring with identity. We study the structure of the zero-divisor graph of R and then determine its vertex connectivity when: (i) R is a local principal ideal ring, and (ii) R is a finite direct product of local principal ideal rings. For such rings R, we also characterize the vertices of minimum degree and the minimum cut-sets of the zero-divisor graph of R.

In this paper, we present results concerning the stabilizer (G_f) in ({{,mathrm{{GL}},}}(2,q^n)) of the subspace (U_f={(x,f(x)):xin mathbb {F}_{q^n}}), f(x) a scattered linearized polynomial in (mathbb {F}_{q^n}[x]). Each (G_f) contains the (q-1) maps ((x,y)mapsto (ax,ay)), (ain mathbb {F}_{q}^*). By virtue of the results of Beard (Duke Math J, 39:313–321, 1972) and Willett (Duke Math J 40(3):701–704, 1973), the matrices in (G_f) are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that (|G_f|>q-1) have a standard form of type (sum _{j=0}^{n/t-1}a_jx^{q^{s+jt}}) for some s and t such that ((s,t)=1), (t>1) a divisor of n; (ii) this standard form is essentially unique; (iii) for (n>2) and (q>3), the translation plane (mathcal {A}_f) associated with f(x) admits nontrivial affine homologies if and only if (|G_f|>q-1), and in that case those with axis through the origin form two groups of cardinality ((q^t-1)/(q-1)) that exchange axes and coaxes; (iv) no plane of type (mathcal {A}_f), f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.

Given an integer n, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in ({1,2,dots , n-1}). Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is ultimately periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic.

Abstract

Let ({mathcal {D}}) be a non-trivial G-block-transitive 3-(v, k, 1) design, where (Tle G le textrm{Aut}(T)) for some finite non-abelian simple group T. It is proved that if T is a simple exceptional group of Lie type, then T is either the Suzuki group ({}^2B_2(q)) or (G_2(q)) . Furthermore, if (T={}^2B_2(q)) then the design ({mathcal {D}}) has parameters (v=q^2+1) and (k=q+1) , and so ({mathcal {D}}) is an inverse plane of order q, and if (T=G_2(q)) then the point stabilizer in T is either (textrm{SL}_3(q).2) or (textrm{SU}_3(q).2) , and the parameter k satisfies very restricted conditions.