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Open AccessArticle

The Properties of Topological Manifolds of Simplicial Polynomials

by Susmit Bagchi

Symmetry 2024, 16(1), 102; https://doi.org/10.3390/sym16010102 - 14 Jan 2024

Cited by 3 | Viewed by 2385

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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34 pages, 1112 KB

Open AccessArticle

Topological Methods for Studying Contextuality: N-Cycle Scenarios and Beyond

by Aziz Kharoof, Selman Ipek and Cihan Okay

Entropy 2023, 25(8), 1127; https://doi.org/10.3390/e25081127 - 27 Jul 2023

Cited by 9 | Viewed by 2245

Abstract

Simplicial distributions are combinatorial models describing distributions on spaces of measurements and outcomes that generalize nonsignaling distributions on contextuality scenarios. This paper studies simplicial distributions on two-dimensional measurement spaces by introducing new topological methods. Two key ingredients are a geometric interpretation of Fourier–Motzkin elimination and a technique based on the collapsing of measurement spaces. Using the first one, we provide a new proof of Fine’s theorem characterizing noncontextual distributions in N-cycle scenarios. Our approach goes beyond these scenarios and can describe noncontextual distributions in scenarios obtained by gluing cycle scenarios of various sizes. The second technique is used for detecting contextual vertices and deriving new Bell inequalities. Combined with these methods, we explore a monoid structure on simplicial distributions. Full article

(This article belongs to the Special Issue Quantum Contextuality and Non-locality: Foundations, Applications and Extensions)

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13 pages, 3219 KB

Open AccessArticle

A Poset-Based Approach to Curvature of Hypergraphs

by Yasharth Yadav, Areejit Samal and Emil Saucan

Symmetry 2022, 14(2), 420; https://doi.org/10.3390/sym14020420 - 20 Feb 2022

Cited by 7 | Viewed by 4567

Abstract

In this contribution, we represent hypergraphs as partially ordered sets or posets, and provide a geometric framework based on posets to compute the Forman–Ricci curvature of vertices as well as hyperedges in hypergraphs. Specifically, we first provide a canonical method to construct a two-dimensional simplicial complex associated with a hypergraph, such that the vertices of the simplicial complex represent the vertices and hyperedges of the original hypergraph. We then define the Forman–Ricci curvature of the vertices and the hyperedges as the scalar curvature of the associated vertices in the simplicial complex. Remarkably, Forman–Ricci curvature has a simple combinatorial expression and it can effectively capture the variation in symmetry or asymmetry over a hypergraph. Finally, we perform an empirical study involving computation and analysis of the Forman–Ricci curvature of hyperedges in several real-world hypergraphs. We find that Forman–Ricci curvature shows a moderate to high absolute correlation with standard hypergraph measures such as eigenvector centrality and cardinality. Our results suggest that the notion of Forman–Ricci curvature extended to hypergraphs in this work can be used to gain novel insights on the organization of higher-order interactions in real-world hypernetworks. Full article

(This article belongs to the Special Issue Topological Graph Theory and Discrete Geometry)

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16 pages, 389 KB

Open AccessArticle

Induced Matchings and the v-Number of Graded Ideals

by Gonzalo Grisalde, Enrique Reyes and Rafael H. Villarreal

Mathematics 2021, 9(22), 2860; https://doi.org/10.3390/math9222860 - 11 Nov 2021

Cited by 23 | Viewed by 5427

Abstract

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal

I (G)

of a graph G, the induced matching number of G is [...] Read more.

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal

I (G)

of a graph G, the induced matching number of G is an upper bound for the v-number of

I (G)

when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of

I (G)

is a lower bound for the regularity of the edge ring of G. We classify when the induced matching number of G is an upper bound for the v-number of

I (G)

when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of

W_{2}

-graphs. Full article

(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)

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15 pages, 773 KB

Open AccessArticle

Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K_2,2,2,2

by Serge Lawrencenko and Abdulkarim M. Magomedov

Symmetry 2021, 13(8), 1418; https://doi.org/10.3390/sym13081418 - 3 Aug 2021

Cited by 5 | Viewed by 4685

Abstract

Using the orbit decomposition, a new enumerative polynomial

P (x)

is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the [...] Read more.

Using the orbit decomposition, a new enumerative polynomial

P (x)

P (1)

is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative

P^{'} (1)

is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of

P (x)

from 0 to 1 is equal to the total number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial

P (x)

is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graph

G = K_{2, 2, 2, 2}

, in which specific case

P (x) = x^{31}

. The graph G embeds in the torus as a triangulation,

T (G)

. The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (12 in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group

Q_{8}

with the three imaginary quaternions i, j, k as generators. Full article

(This article belongs to the Special Issue Topological Graph Theory and Discrete Geometry)

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15 pages, 600 KB

Open AccessArticle

Approximate Triangulations of Grassmann Manifolds

by Kevin P. Knudson

Algorithms 2020, 13(7), 172; https://doi.org/10.3390/a13070172 - 17 Jul 2020

Cited by 2 | Viewed by 4544

Abstract

We define the notion of an approximate triangulation for a manifold M embedded in Euclidean space. The basic idea is to build a nested family of simplicial complexes whose vertices lie in M and use persistent homology to find a complex in the [...] Read more.

G_{k} (R^{n})

. Full article

(This article belongs to the Special Issue Topological Data Analysis)

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25 pages, 547 KB

Open AccessArticle

Graph Extremities Defined by Search Algorithms

by Anne Berry, Jean R.S. Blair, Jean-Paul Bordat and Geneviève Simonet

Algorithms 2010, 3(2), 100-124; https://doi.org/10.3390/a3020100 - 24 Mar 2010

Cited by 21 | Viewed by 9710

Abstract

Graph search algorithms have exploited graph extremities, such as the leaves of a tree and the simplicial vertices of a chordal graph. Recently, several well-known graph search algorithms have been collectively expressed as two generic algorithms called MLS and MLSM. In this paper, we investigate the properties of the vertex that is numbered 1 by MLS on a chordal graph and by MLSM on an arbitrary graph. We explain how this vertex is an extremity of the graph. Moreover, we show the remarkable property that the minimal separators included in the neighborhood of this vertex are totally ordered by inclusion. Full article

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