On the zero forcing number of the complement of graphs with forbidden subgraphs

Motivated in part by an observation that the zero forcing number for the complement of a tree on $n$ vertices is either $n-3$ or $n-1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under some conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs.Comment: 17 pages, 8 figure

Isomorphism of Modules for Type Dn

In this paper, for the current algebra associated with the Lie algebra \lie{so}_{2n}(\mathbb{C}), we connect particular CV-modules, indexed by pairs of integral weights $(\lambda,\mu)$ that satisfy particular conditions, with generalized Demazure modules which were introduced by [Naoi]. In particular, we show an isomorphism of a CV-module $V(\xi(\lambda,\mu))$ and the module generated by the tensor product of generating vectors $v_\lambda$, $v_\mu$ for the local Weyl modules W_{\loc}(\lambda) and W_{\loc}(\mu) respectively, i.e. V(\xi(\lambda,\mu))\cong \langle v_\lambda \otimes v_\mu\rangle \subset W_{\loc}(\lambda) \otimes W_{\loc}(\mu). To complete this proof, we construct short exact sequences of CV-modules. Moreover, through this construction, we obtain a Demazure character formula for the CV-module $V(\xi(\lambda,\mu))$