Related papers: On rigidity of toric varieties arising from bipart…
We call a bipartite graph {\it homogeneous} if every finite partial automorphism which respects left and right can be extended to a total automorphism. A $(\kappa,{\lambda} )$ bipartite graph is a bipartite graph with left side of size…
We study semi-stable degenerations of toric varieties determined by certain partitions of their moment polytopes. Analyzing their defining equations we prove a property of uniqueness.
Recently, Ali et al. posed several open problems concerning extremal graphs with respect to the ABS index. These problems involve characterizing graphs that attain the maximum ABS index within specific graph classes, including: connected…
In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic…
The cohomological rigidity problem for toric orbifolds asks when an integral cohomology isomorphism implies a homotopy equivalence. In this paper we reformulate the cohomological rigidity problem in the context of $4$-dimensional toric…
In this paper, we explore the design and analysis of regular bipartite graphs motivated by their application in low-density parity-check (LDPC) codes specifically with constrained girth and in the high-rate regime. We focus on the relation…
Bipartite Graph is often a realistic model of complex networks where two different sets of entities are involved and relationship exist only two entities belonging to two different sets. Examples include the user-item relationship of a…
We prove a topological rigidity result for simple, thick, hyperbolic P-manifolds of dimension 2: isomorphism of the fundamental groups implies homeomorphism of the P-manifolds. An immediate application is a diagram rigidity theorem for…
It is known that the fundamental group homomorphism $\pi_1(T^2) \to \pi_1(S^3\setminus K)$ induced by the inclusion of the boundary torus into the complement of a knot $K$ in $S^3$ is a complete knot invariant. Many classical invariants of…
The classical multiplicative (Hirzebruch) genera of manifolds have the wonderful property which is called rigidity. Rigidity of a genus h means that if a compact connected Lie group G acts on a manifold X, then the equivariant genus h^G(X)…
For an arbitrary field K, let I be an ideal in the ring K[[x,y]] expressible as a polynomial in either the pair of ideals (x, y^4) and (x,y) or the pair (x,y^3) and (x^2, y). Let G be the group of automorphisms of K[[x,y]] sending the ideal…
Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which…
In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference…
Frei et al. [6] showed that the problem to decide whether a graph is stable with respect to some graph parameter under adding or removing either edges or vertices is $\Theta_2^{\text{P}}$-complete. They studied the common graph parameters…
In this paper we prove the scalar curvature extremality and rigidity for a class of warped product spaces that are possibly degenerate at the two ends. The leaves of these warped product spaces can be any closed Riemannian manifolds with…
For affine symmetric groups we construct finite $W$-graphs corresponding to two-row shapes, and prove their uniqueness. This gives the first non-trivial family of examples of finite $W$-graphs in an affine type. We compare our construction…
On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete…
We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by Mirror Symmetry, we give conditions for the limit toric variety to be a…
A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely-many realisations can be seen as a solution to a…
Let $G$ be an $r$-partite graph such that the edge density between any two parts is at least $\alpha$. How large does $\alpha$ need to be to guarantee that $G$ contains a connected transversal, that is, a tree on $r$ vertices meeting each…