We give a rank augmentation technique for rank three string C-group representations of the symmetric group (S_n) and list the hypotheses under which it yields a valid string C-group representation of rank four thereof.
We give a rank augmentation technique for rank three string C-group representations of the symmetric group (S_n) and list the hypotheses under which it yields a valid string C-group representation of rank four thereof.
A generalized spline on an edge-labeled graph ((G,alpha )) is defined as a vertex labeling, such that the difference of labels on adjacent vertices lies in the ideal generated by the edge label. We study generalized splines over greatest common divisor domains and present a determinantal basis condition for generalized spline modules on arbitrary graphs. The main result of the paper answers a conjecture that appeared in several papers.
We construct a finite Young wall model for a certain irreducible module over (imath )quantum group ({textbf{U}}^{jmath }). Moreover, we show that this irreducible module is a highest weight module and is determined by a crystal structure on the set of finite Young walls.
We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the underlying lattice. When the lattice is planar, we show that the corresponding chain algebra is generated by a sortable set of monomials and is isomorphic to a Hibi ring of another finite distributive lattice. As a consequence, it has a defining toric ideal with a quadratic Gröbner basis, and its h-vector counts ascents in certain standard Young tableaux. If instead the lattice has dimension (n>2), we show that the defining ideal has minimal generators of degree at least n.
We construct cellular resolutions for monomial ideals via discrete Morse theory. In particular, we develop an algorithm to create homogeneous acyclic matchings and we call the cellular resolutions induced from these matchings Barile–Macchia resolutions. These resolutions are minimal for edge ideals of weighted oriented forests and (most) cycles. As a result, we provide recursive formulas for graded Betti numbers and projective dimension. Furthermore, we compare Barile–Macchia resolutions to those created by Batzies and Welker and some well-known simplicial resolutions. Under certain assumptions, whenever the above resolutions are minimal, so are Barile–Macchia resolutions.
((N,gamma )) -hyperelliptic semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated with totally ramified points of N-fold covers of curves of genus (gamma ) . Torres characterized ((2,gamma )) -hyperelliptic semigroups of maximal weight whenever their genus is large relative to (gamma ) . Here we do the same for ((3,gamma )) -hyperelliptic semigroups, and we formulate a conjecture about the general case whenever (N ge 3) is prime.
A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron—both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the Dodecahedron—answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graphs of a monoid with generating connection set of size two. This extends Nedela and Škoviera’s characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo.
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.
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