Journal of Algebraic Combinatorics最新文献

Denote by (rho (G)) and (kappa (G)) the spectral radius and the signless Laplacian spectral radius of a graph G, respectively. Let (kge 0) be a fixed integer and G be a graph of size m which is large enough. We show that if (rho (G)ge sqrt{m-k}), then (C_4subseteq G) or (K_{1,m-k}subseteq G). Moreover, we prove that if (kappa (G)ge m-k+1), then (K_{1,m-k}subseteq G). Both these results extend some known results.

Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. A graph is a semi-Cayley graph over a group G if it admits G as a semiregular subgroup of the full automorphism group with two orbits of equal size. A semi-Cayley graph SC(G, R, L, S) is called quasi-abelian if each of R, L and S is a union of some conjugacy classes of G. This paper establishes necessary and sufficient conditions for a quasi-abelian semi-Cayley graph to have perfect state transfer. As a corollary, it is shown that if a quasi-abelian semi-Cayley graph over a finite group G has perfect state transfer between distinct vertices g and h, and G has a faithful irreducible character, then (gh^{-1}) lies in the center of G and (gh=hg); in particular, G cannot be a non-abelian simple group. We also characterize quasi-abelian Cayley graphs over arbitrary groups having perfect state transfer, which is a generalization of previous works on Cayley graphs over abelian groups, dihedral groups, semi-dihedral groups and dicyclic groups.

Cactus groups (J_n) are currently attracting considerable interest from diverse mathematical communities. This work explores their relations to right-angled Coxeter groups and, in particular, twin groups (Tw_n) and Mostovoy’s Gauss diagram groups (D_n), which are better understood. Concretely, we construct an injective group 1-cocycle from (J_n) to (D_n) and show that (Tw_n) (and its k-leaf generalizations) inject into (J_n). As a corollary, we solve the word problem for cactus groups, determine their torsion (which is only even) and center (which is trivial), and answer the same questions for pure cactus groups, (PJ_n). In addition, we yield a 1-relator presentation of the first non-abelian pure cactus group (PJ_4). Our tools come mainly from combinatorial group theory.

Path algebras from quivers are a fundamental class of algebras with wide applications. Yet it is challenging to describe their universal properties since their underlying path semigroups are only partially defined. A new notion, called locality structures, was recently introduced to deal with partially defined operation, with motivation from locality in convex geometry and quantum field theory. We show that there is a natural correspondence between locality sets and quivers which leads to a concrete class of locality semigroups, called Brandt locality semigroups, which can be obtained by the paths of quivers. Further these path Brandt locality semigroups are precisely the free objects in the category of Brandt locality semigroups with a rigidity condition. This characterization gives a universal property of path algebras and at the same time a combinatorial realization of free rigid Brandt locality semigroups.

Let p be a prime, and let (Lambda _{2p}) be a connected cubic arc-transitive graph of order 2p. In the literature, a lot of works have been done on the classification of edge-transitive normal covers of (Lambda _{2p}) for specific (ple 7). An interesting problem is to generalize these results to an arbitrary prime p. In 2014, Zhou and Feng classified edge-transitive cyclic or dihedral normal covers of (Lambda _{2p}) for each prime p. In our previous work, we classified all edge-transitive N-normal covers of (Lambda _{2p}), where p is a prime and N is a metacyclic 2-group. In this paper, we give a classification of edge-transitive N-normal covers of (Lambda _{2p}), where (pge 5) is a prime and N is a metacyclic group of odd prime power order.

In this paper, we try to answer some questions raised by Cangelmi (Eur J Comb 33(7):1444–1448, 2012). We reinterpret the Riemann–Hurwitz theorem of orientable algebraic hypermaps by introducing tripartite graph morphisms and obtain Riemann–Roch theorems for orientable hypermaps by defining the divisor of a function f on darts. In addition, we extend Riemann–Roch theorem to non-orientable hypermaps by suitably replacing the orientable genus with the non-orientable genus. Finally, as an application of the Riemann–Hurwitz theorem, we establish the second main theorem from the viewpoint of Nevanlinna theory.

We study chains of nonzero edge ideals that are invariant under the action of the monoid ({{,textrm{Inc},}}) of increasing functions on the positive integers. We prove that the sequence of Castelnuovo–Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behavior sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of ({{,textrm{Inc},}})-invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of ({{,textrm{Inc},}})-invariant chains of edge ideals.

Let (mathcal{M}) be an orientably regular (resp. regular) map with the number n vertices. By (G^+) (resp. G) we denote the group of all orientation-preserving automorphisms (resp. all automorphisms) of (mathcal{M}). Let (pi ) be the set of prime divisors of n. A Hall (pi )-subgroup of (G^+)(resp. G) is meant a subgroup such that the prime divisors of its order all lie in (pi ) and the primes of its index all lie outside (pi ). It is mainly proved in this paper that (1) suppose that (mathcal{M}) is an orientably regular map where n is odd. Then (G^+) is solvable and contains a normal Hall (pi )-subgroup; (2) suppose that (mathcal{M}) is a regular map where n is odd. Then G is solvable if it has no composition factors isomorphic to (hbox {PSL}(2,q)) for any odd prime power (qne 3), and G contains a normal Hall (pi )-subgroup if and only if it has a normal Hall subgroup of odd order.

Let I be the edge ideal of a connected non-bipartite graph and R the base polynomial ring. Then, ({text {depth}}R/I ge 1) and ({text {depth}}R/I^t = 0) for (t gg 1). This paper studies the problem when ({text {depth}}R/I^t = 1) for some (t ge 1) and whether the depth function is non-increasing thereafter. Furthermore, we are able to give a simple combinatorial criterion for ({text {depth}}R/I^{(t)} = 1) for (t gg 1) and show that the condition ({text {depth}}R/I^{(t)} = 1) is persistent, where (I^{(t)}) denotes the t-th symbolic powers of I.