Family Of Polytopes Research Articles

Stratified operations on maniplexes

There is an increasingly extensive literature on the problem of describing the connection (monodromy) groups and automorphism groups of families of polytopes and maniplexes that are not regular or reflexible. Many such polytopes and maniplexes arise as the result of constructions such as truncations and products. Here we show that for a wide variety of these constructions, the connection group of the output can be described in a nice way in terms of the connection group of the input. We call such operations stratified. Moreover, we show that, if F is a maniplex operation in one of two broad subclasses of stratified operations, and if ℛ is the smallest reflexible cover of some maniplex ℳ, then the connection group of F(ℛ) is equal to the connection group of F(ℳ). In particular, we show that this is true for truncations and medials of maps, for products of polytopes (including pyramids and prisms over polytopes), and for the mix of maniplexes. As an application, we determine the smallest reflexible covers of the pyramids over the equivelar toroidal maps.

Extension complexity of low-dimensional polytopes

Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P P is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which P P can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more specifically considering various families of low-dimensional polytopes. First, we prove that for a fixed dimension d d , the extension complexity of a random d d -dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic n n -vertex polygon (whose vertices lie on a circle) has extension complexity at most 24 n 24\sqrt n . This bound is tight up to the constant factor 24 24 . Finally, we show that there exists an n o ( 1 ) n^{o(1)} -dimensional polytope with at most n n vertices and extension complexity n 1 − o ( 1 ) n^{1-o(1)} . Our theorems are proved with a range of different techniques, which we hope will be of further interest.

On bounded partition dimension of different families of convex polytopes with pendant edges

<abstract><p>Let $ \psi = (V, E) $ be a simple connected graph. The distance between $ \rho_1, \rho_2\in V(\psi) $ is the length of a shortest path between $ \rho_1 $ and $ \rho_2. $ Let $ \Gamma = \{\Gamma_1, \Gamma_2, \dots, \Gamma_j\} $ be an ordered partition of the vertices of $ \psi $. Let $ \rho_1\in V(\psi) $, and $ r(\rho_1|\Gamma) = \{d(\rho_1, \Gamma_1), d(\rho_1, \Gamma_2), \dots, d(\rho_1, \Gamma_j)\} $ be a $ j $-tuple. If the representation $ r(\rho_1|\Gamma) $ of every $ \rho_1\in V(\psi) $ w.r.t. $ \Gamma $ is unique then $ \Gamma $ is the resolving partition set of vertices of $ \psi $. The minimum value of $ j $ in the resolving partition set is known as partition dimension and written as $ pd(\psi). $ The problem of computing exact and constant values of partition dimension is hard so one can compute bound for the partition dimension of a general family of graph. In this paper, we studied partition dimension of the some families of convex polytopes with pendant edge such as $ R_n^P $, $ D_n^p $ and $ Q_n^p $ and proved that these graphs have bounded partition dimension.</p></abstract>

Computation of Metric Dimension of Certain Subdivided Convex Polytopes

The distance d(z1, z2) from vertex z1 ∈ V(G) to z2 ∈ V(G) is minimum length of (z1, z2)‐path in a given connected graph G having E(G) and V(G) edges and vertices’/nodes’ sets, respectively. Suppose Z = {z1, z2, z3, …, zm}⊆V(G) is an order set and c ∈ V(G), and the code of c with reference to Z is the m‐tuple {d(c, z1), d(c, z2), d(c, z13), …, d(c, zk)}. Then, Z is named as the locating set or resolving set if each node of G has unique code. A locating set of least cardinality is described as a basis set for the graph G, and its cardinal number is referred to as metric dimension symbolized by dim(G). Metric dimension of certain subdivided convex polytopes STn has been computed, and it is concluded that just four vertices are sufficient for unique coding of all nodes belonging to this family of convex polytopes.

An Algebraic Approach to Projective Uniqueness with an Application to Order Polytopes

A combinatorial polytope P is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling property but is difficult to verify. In this paper, we merge two approaches to projective uniqueness in the literature. One is primarily geometric and is due to McMullen, who showed that certain natural operations on polytopes preserve projective uniqueness. The other is more algebraic and is due to Gouveia, Macchia, Thomas, and Wiebe. They use certain ideals associated to a polytope to verify a property called graphicality that implies projective uniqueness. In this paper, we show that McMullen’s operations preserve not only projective uniqueness but also graphicality. As an application, we show that large families of order polytopes are graphic and thus projectively unique.

On dual unit balls of Thurston norms

AbstractThurston norms are invariants of 3-manifolds defined on their second homology and understanding the shape of their dual unit balls is a widely open problem. In this article, we provide a large family of polytopes in $\mathbb {R}^{2g}$ that appear like dual unit balls of Thurston norms, generalizing Thurston’s construction for polygons in $\mathbb {R}^2$ .À mes enseignants, premiers

Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties

The main goal of this paper is to give explicit descriptions of two maximal cones in the Gröbner fan of the Plücker ideal. These cones correspond to the monomial ideals given by semistandard and PBW-semistandard Young tableaux. For the first cone, as an intermediate result we obtain the description of a maximal cone in the Gröbner fan of any Hibi ideal. For the second, we generalize the notion of Hibi ideals by associating an ideal with every interpolating polytope. This is a family of polytopes that generalizes the order and chain polytopes of a poset (à la Fang–Fourier–Litza–Pegel). We then describe a maximal cone in the Gröbner fan of each of these ideals. We also establish some useful facts concerning PBW-semistandardness, in particular, we prove that it provides a new Hodge algebra structure on the Plücker algebra.

Combinatorial generation via permutation languages. II. Lattice congruences

This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406–420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustively generating various classes of combinatorial objects by encoding them as permutations. We also characterize which of these graphs are vertex-transitive or regular via their arc diagrams, give corresponding precise and asymptotic counting results, and we determine their minimum and maximum degrees.

On Hamilton-Connectivity and Detour Index of Certain Families of Convex Polytopes

A convex polytope is the convex hull of a finite set of points in the Euclidean space ℝ n . By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. The detour index is defined to be the sum of the lengths of longest distances, i.e., detours between vertices in a graph. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering, whereas the detour index has important applications in chemistry. Checking whether a graph is Hamilton-connected and computing the detour index of an arbitrary graph are both NP-complete problems. In this paper, we study these problems simultaneously for certain families of convex polytopes. We construct two infinite families of Hamilton-connected convex polytopes. Hamilton-connectivity is shown by constructing Hamiltonian paths between any pair of vertices. We then use the Hamilton-connectivity to compute the detour index of these families. A family of non-Hamilton-connected convex polytopes has also been constructed to show that not all convex polytope families are Hamilton-connected.

On Computation of Recently Defined Degree-Based Topological Indices of Some Families of Convex Polytopes via M-Polynomial

Topological indices are of incredible significance in the field of graph theory. Convex polytopes play a significant role both in various branches of mathematics and also in applied areas, most notably in linear programming. We have calculated some topological indices such as atom-bond connectivity index, geometric arithmetic index, K-Banhatti indices, and K-hyper-Banhatti indices and modified K-Banhatti indices from some families of convex polytopes through M-polynomials. The M-polynomials of the graphs provide us with a great help to calculate the topological indices of different structures.

On symmetry breaking of dual polyhedra of non-crystallographic group H3.

The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes {\cal V}_{H_{3}}(\lambda) constructed for each type of dominant point λ. Here a polytope {\cal V}_{H_{3}}(\lambda) is considered as a dual of a {\cal D}_{H_{3}}(\lambda) polytope obtained from the action of the Coxeter group H3 on a single point \lambda\in{\bb R}^{3}. The H3 symmetry is reduced to the symmetry of its two-dimensional subgroups H2, A1 × A1 and A2 that are used to examine the geometric structure of {\cal V}_{H_{3}}(\lambda) polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the `pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the {\cal V}_{H_{3}}(\lambda) structure into various nanotubes. Moreover, since a {\cal V}_{H_{3}}(\lambda) polytope may contain the orbits obtained by the action of H3 on the seed points (a, 0, 0), (0, b, 0) and (0, 0, c) within its structure, the stellations of flat-faced {\cal V}_{H_{3}}(\lambda) polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C20 has the dodecahedral structure of {\cal V}_{H_{3}}(a,0,0), the construction of the smallest fullerenes C24, C26, C28, C30 together with the nanotubes C20+6N, C20+10N is presented.

Combinatorial mutations and block diagonal polytopes

Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes, and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of giving rise to a toric degeneration of the Grassmannians, is preserved by mutation. Moreover, the polytopes arising through mutations are Newton-Okounkov bodies for the Grassmannians with respect to certain full-rank valuations. We produce a large family of such polytopes, extending the family of so-called block diagonal matching fields.

Hamilton Connectivity of Convex Polytopes with Applications to Their Detour Index

A connected graph is called Hamilton-connected if there exists a Hamiltonian path between any pair of its vertices. Determining whether a graph is Hamilton-connected is an NP-complete problem. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering. The detour index of a graph is defined to be the sum of lengths of detours between all the unordered pairs of vertices. The detour index has diverse applications in chemistry. Computing the detour index for a graph is also an NP-complete problem. In this paper, we study the Hamilton-connectivity of convex polytopes. We construct three infinite families of convex polytopes and show that they are Hamilton-connected. An infinite family of non-Hamilton-connected convex polytopes is also constructed, which, in turn, shows that not all convex polytopes are Hamilton-connected. By using Hamilton connectivity of these families of graphs, we compute exact analytical formulas of their detour index.

Hamilton-connectedness and Hamilton-laceability of planar geometric graphs with applications

<abstract> In this paper, we have used two different proof techniques to show the Hamilton-connectedness of graphs. By using the vertex connectivity and Hamiltoniancity of graphs, we construct an infinite family of Hamilton-connected convex polytope line graphs whose underlying family of convex polytopes is not Hamilton-connected. By definition, we constructed two more infinite families of Hamilton-connected convex polytopes. As a by-product of our results, we compute exact values of the detour index of the families of Hamilton-connected convex polytopes. Finally, we classify the Platonic solids according to their Hamilton-connectedness and Hamilton-laceability properties. </abstract>

The diagonal of the associahedra

This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps.

The arithmetic of Coxeter permutahedra

Ehrhart theory mesures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, ..... We compute the Ehrhart theory of four families of polytopes of great importance in several areas of mathematics: the standard Coxeter permutahedra for the classical Coxeter groups An, Bn, Cn, Dn. A central tool, of independent interest, is a description of the Ehrhart theory of a rational translate of an integer projection of a cube.

Two infinite families of chiral polytopes of type {4,4,4} with solvable automorphism groups

We construct two infinite families of locally toroidal chiral polytopes of type {4,4,4}, with 1024m2 and 2048m2 automorphisms for every positive integer m, respectively. The automorphism groups of these polytopes are solvable groups, and when m is a power of 2, they provide examples with automorphism groups of order 2n where n can be any integer greater than 9. (On the other hand, no chiral polytopes of type [4,4,4] exist for n≤9.) In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss in [21].

Fault-Tolerant Metric Dimension of Generalized Wheels and Convex Polytopes

For a graph G , an ordered set S ⊆ V G is called the resolving set of G , if the vector of distances to the vertices in S is distinct for every v ∈ V G . The minimum cardinality of S is termed as the metric dimension of G . S is called a fault-tolerant resolving set (FTRS) for G , if S \ v is still the resolving set ∀ v ∈ V G . The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of G . Due to enormous application in science such as mathematics and computer, the notion of the resolving set is being widely studied. In the present article, we focus on determining the FTMD of a generalized wheel graph. Moreover, a formula is developed for FTMD of a wheel and generalized wheels. Recently, some bounds of the FTMD of some of the convex polytopes have been computed, but here we come up with the exact values of the FTMD of two families of convex polytopes denoted as D k for k ≥ 4 and Q k for k ≥ 6 . We prove that these families of convex polytopes have constant FTMD. This brings us to pose a natural open problem about the existence of a polytope having nonconstant FTMD.

Galois connections for phylogenetic networks and their polytopes

We describe Galois connections which arise between two kinds of combinatorial structures, both of which generalize trees with labeled leaves, and then apply those connections to a family of polytopes. The graphs we study can be imbued with metric properties or associated with vectors. Famous examples are the Billera–Holmes–Vogtmann metric space of phylogenetic trees, and the Balanced Minimal Evolution polytopes of phylogenetic trees described by Eickmeyer, Huggins, Pachter and Yoshida. Recently, the space of trees has been expanded to split networks by Devadoss and Petti, while the definition of phylogenetic polytopes has been generalized to encompass 1-nested phylogenetic networks, by Durell and Forcey. The first Galois connection we describe is a reflection between the (unweighted) circular split networks and the 1-nested phylogenetic networks. Another Galois connection exists between certain metric versions of these structures. Reflection between the purely combinatorial posets becomes a coreflection in the geometric case. Our chief contributions here, beyond the discovery of the Galois connections, are: a translation between approaches using PC-trees and networks, a new way to look at weightings on networks, and a fuller characterization of faces of the phylogenetic polytopes.

The Martin Gardner Polytopes

In the chapter “Magic with a Matrix” in Hexaflexagons and Other Mathematical Diversions (1988); Martin Gardner describes a delightful “party trick” to fill the squares of a d-by-d chessboard with nonnegative integers such that the sum of the numbers covered by any placement of d nonthreatening rooks is a given number N. We consider such chessboards from a geometric perspective which gives rise to a family of lattice polytopes. The polyhedral structure of these Gardner polytopes explains the underlying trick and enables us to count such chessboards for given N in three different ways. We also observe a curious duality that relates Gardner polytopes to Birkhoff polytopes