Abstract We consider a 2-homogeneous bipartite distance-regular graph $$\Gamma $$ Γ with diameter $$D \ge 3$$ D ≥ 3 . We assume that $$\Gamma $$ Γ is not a hypercube nor a cycle. We fix a Q -polynomial ordering of the primitive idempotents of $$\Gamma $$ Γ . This Q -polynomial ordering is described using a nonzero parameter $$q \in \mathbb {C}$$ q ∈ C that is not a root of unity. We investigate $$\Gamma $$ Γ using an $$S_3$$ S 3 -symmetric approach. In this approach one considers $$V^{\otimes 3} = V \otimes V \otimes V$$ V ⊗ 3 = V ⊗ V ⊗ V where V is the standard module of $$\Gamma $$ Γ . We construct a subspace $$\Lambda $$ Λ of $$V^{\otimes 3}$$ V ⊗ 3 that has dimension $$\left( {\begin{array}{c}D+3\\ 3\end{array}}\right) $$ D + 3 3 , together with six linear maps from $$\Lambda $$ Λ to $$\Lambda $$ Λ . Using these maps we turn $$\Lambda $$ Λ into an irreducible module for the nonstandard quantum group $$U^\prime _q(\mathfrak {so}_6)$$ U q ′ ( so 6 ) introduced by Gavrilik and Klimyk in 1991.

Abstract We associate with every graph a linear program for packings of vertex disjoint paths. We show that the optimal primal and dual values of the corresponding integer program are the binomial grade and height of the binomial edge ideal of the graph. We deduce from this a new combinatorial characterization of graphs of König type and use it to show that all trees are of König type. The log canonical threshold and the F-threshold are important invariants associated with the singularities of a variety in characteristic 0 and characteristic p. We show that the optimal value of the linear program (computed over the rationals) agrees with both the F-threshold and the log canonical threshold of the binomial edge ideal if the graph is a block graph or of König type. We conjecture that this linear program computes the log canonical threshold of the binomial edge ideal of any graph. Our results resemble theorems on monomial ideals arising from hypergraphs due to Howald and others.

Abstract We introduce the concept of a k-token signed graph and study some of its combinatorial and algebraic properties. We prove that two switching isomorphic signed graphs have switching isomorphic token graphs. Moreover, we show that the Laplacian spectrum of a balanced signed graph is contained in the Laplacian spectra of its k-token signed graph. Besides, we introduce and study the unbalance level of a signed graph, which is a new parameter that measures how far a signed graph is from being balanced. Moreover, we study the relation between the frustration index and the unbalance level of signed graphs and their token signed graphs.