Research

8 pages, 251 KiB

Open AccessArticle

New Obstructions to Warped Product Immersions in Complex Space Forms

by Mohd. Aquib, Adela Mihai, Ion Mihai and Siraj Uddin

Symmetry 2022, 14(8), 1747; https://doi.org/10.3390/sym14081747 - 22 Aug 2022

Cited by 1 | Viewed by 1043

In this paper, we obtain a geometric inequality for warped product pointwise semi-slant submanifolds of complex space forms endowed with a semi-symmetric metric connection and discuss the equality case of this inequality. We provide some applications concerning the minimality and compactness of such [...] Read more.

In this paper, we obtain a geometric inequality for warped product pointwise semi-slant submanifolds of complex space forms endowed with a semi-symmetric metric connection and discuss the equality case of this inequality. We provide some applications concerning the minimality and compactness of such submanifolds. Full article

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

30 pages, 14306 KiB

Open AccessArticle

A Path-Curvature Measure for Word-Based Strategy Searches in Semantic Networks

by Haim Cohen, Yinon Nachshon, Anat Maril, Paz M. Naim, Jürgen Jost and Emil Saucan

Symmetry 2022, 14(8), 1737; https://doi.org/10.3390/sym14081737 - 19 Aug 2022

Cited by 1 | Viewed by 1439

Abstract

Building on a modified version of the Haantjes path-based curvature, this article offers a novel measure that considers the direction of a stream of associations in a semantic network and estimates the extent to which any single association attracts the upcoming associations to [...] Read more.

Building on a modified version of the Haantjes path-based curvature, this article offers a novel measure that considers the direction of a stream of associations in a semantic network and estimates the extent to which any single association attracts the upcoming associations to its environment—in other words, to what degree one explores that environment. We demonstrate that our measure differs from Haantjes curvature and confirm that it expresses the extent to which a stream of associations remains close to its starting point. Finally, we examine the relationship between our measure and accessibility to knowledge stored in memory. We demonstrate that a high degree of attraction facilitates the retrieval of upcoming words in the stream. By applying methods from differential geometry to semantic networks, this study contributes to our understanding of strategic search in memory. Full article

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

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12 pages, 304 KiB

Open AccessArticle

A_γ Eigenvalues of Zero Divisor Graph of Integer Modulo and Von Neumann Regular Rings

by Bilal Ahmad Rather, Fawad Ali, Asad Ullah, Nahid Fatima and Rahim Dad

Symmetry 2022, 14(8), 1710; https://doi.org/10.3390/sym14081710 - 17 Aug 2022

Cited by 9 | Viewed by 1436

Abstract

The

A_{γ}

matrix of a graph G is determined by

A_{γ} (G) = (1 - γ) A (G) + γ D (G),

where

0 \leq γ \leq 1

, [...] Read more.

The

A_{γ}

matrix of a graph G is determined by

A_{γ} (G) = (1 - γ) A (G) + γ D (G),

where

0 \leq γ \leq 1

,

A (G)

and

D (G)

are the adjacency and the diagonal matrices of node degrees, respectively. In this case, the

A_{γ}

matrix brings together the spectral theories of the adjacency, the Laplacian, and the signless Laplacian matrices, and many more

γ

adjacency-type matrices. In this paper, we obtain the

A_{γ}

eigenvalues of zero divisor graphs of the integer modulo rings and the von Neumann rings. These results generalize the earlier published spectral theories of the adjacency, the Laplacian and the signless Laplacian matrices of zero divisor graphs. Full article

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

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10 pages, 2048 KiB

Open AccessArticle

NM-polynomials and Topological Indices of Some Cycle-Related Graphs

by Özge Çolakoğlu

Symmetry 2022, 14(8), 1706; https://doi.org/10.3390/sym14081706 - 16 Aug 2022

Cited by 4 | Viewed by 1533

Abstract

Topological indices (molecular descriptors) are numerical values of a chemical structure and represented by a graph. Molecular descriptors are used in QSPR/QSAR modeling to determine a chemical structure’s physical, biological, and chemical properties. The cycle graphs are symmetric graphs for any number vertices. [...] Read more.

Topological indices (molecular descriptors) are numerical values of a chemical structure and represented by a graph. Molecular descriptors are used in QSPR/QSAR modeling to determine a chemical structure’s physical, biological, and chemical properties. The cycle graphs are symmetric graphs for any number vertices. In this paper, recently defined neighborhood degree sum-based molecular descriptors and polynomials are studied. NM-polynomials and molecular descriptors of some cycle-related graphs, which consist of the wheel graph, gear graph, helm graph, flower graph, and friendship graph, are computed and compared. Full article

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

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13 pages, 3219 KiB

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 1 | Viewed by 2116

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 [...] Read more.

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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12 pages, 3102 KiB

Open AccessArticle

On a Surface Associated with Pascal’s Triangle

by Valeriu Beiu, Leonard Dăuş, Marilena Jianu, Adela Mihai and Ion Mihai

Symmetry 2022, 14(2), 411; https://doi.org/10.3390/sym14020411 - 19 Feb 2022

Cited by 2 | Viewed by 1779

Abstract

An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the [...] Read more.

An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal’s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries. Full article

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

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13 pages, 561 KiB

Open AccessArticle

Expected Values of Some Molecular Descriptors in Random Cyclooctane Chains

by Zahid Raza and Muhammad Imran

Symmetry 2021, 13(11), 2197; https://doi.org/10.3390/sym13112197 - 17 Nov 2021

Cited by 11 | Viewed by 1795

Abstract

The modified second Zagreb index, symmetric difference index, inverse symmetric index, and augmented Zagreb index are among the molecular descriptors which have good correlations with some physicochemical properties (such as formation heat, total surface area, etc.) of chemical compounds. By a random cyclooctane [...] Read more.

The modified second Zagreb index, symmetric difference index, inverse symmetric index, and augmented Zagreb index are among the molecular descriptors which have good correlations with some physicochemical properties (such as formation heat, total surface area, etc.) of chemical compounds. By a random cyclooctane chain, we mean a molecular graph of a saturated hydrocarbon containing at least two rings such that all rings are cyclooctane, every ring is joint with at most two other rings through a single bond, and exactly two rings are joint with one other ring. In this article, our main purpose is to determine the expected values of the aforementioned molecular descriptors of random cyclooctane chains explicitly. We also make comparisons in the form of explicit formulae and numerical tables consisting of the expected values of the considered descriptors of random cyclooctane chains. Moreover, we outline the graphical profiles of these comparisons among the mentioned descriptors. Full article

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

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14 pages, 315 KiB

Open AccessArticle

Study of θ^ϕ Networks via Zagreb Connection Indices

by Muhammad Asif, Bartłomiej Kizielewicz, Atiq ur Rehman, Muhammad Hussain and Wojciech Sałabun

Symmetry 2021, 13(11), 1991; https://doi.org/10.3390/sym13111991 - 21 Oct 2021

Cited by 4 | Viewed by 1577

Abstract

Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network

θ^{ϕ}

formed by

ϕ

time [...] Read more.

Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network

θ^{ϕ}

formed by

ϕ

time repetition of the process of joining

θ

copies of a selected graph

Ω

in such a way that corresponding vertices of

Ω

in all the copies are joined with each other by a new edge. The symmetry of

θ^{ϕ}

is ensured by the involvement of complete graph

K_{θ}

in the construction process. The free hand to choose an initial graph

Ω

and formation of chemical graphs using

θ^{ϕ} Ω

enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of

θ^{ϕ} Ω

. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of

θ^{ϕ} Ω

to enable its emerging use. Full article

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

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17 pages, 11842 KiB

Open AccessArticle

Modelling Heterogeneity and Super Spreaders of the COVID-19 Spread through Malaysian Networks

by Fatimah Abdul Razak and Zamira Hasanah Zamzuri

Symmetry 2021, 13(10), 1954; https://doi.org/10.3390/sym13101954 - 16 Oct 2021

Cited by 3 | Viewed by 1792

Abstract

Malaysia is multi-ethnic and diverse country. Heterogeneity, in terms of population interactions, is ingrained in the foundation of the country. Malaysian policies and social distancing measures are based on daily infections and R0 (average number of infections per infected person), estimated from the [...] Read more.

Malaysia is multi-ethnic and diverse country. Heterogeneity, in terms of population interactions, is ingrained in the foundation of the country. Malaysian policies and social distancing measures are based on daily infections and R0 (average number of infections per infected person), estimated from the data. Models of the Malaysian COVID-19 spread are mostly based on the established SIR compartmental model and its variants. These models usually assume homogeneity and symmetrical full mixing in the population; thus, they are unable to capture super-spreading events which naturally occur due to heterogeneity. Moreover, studies have shown that when heterogeneity is present, R0 may be very different and even possibly misleading. The underlying spreading network is a crucial element, as it introduces heterogeneity for a more representative and realistic model of the spread through specific populations. Heterogeneity introduces more complexities in the modelling due to its asymmetrical nature of infection compared to the relatively symmetrical SIR compartmental model. This leads to a different way of calculating R0 and defining super-spreaders. Quantifying a super-spreader individual is related to the idea of importance in a network. The definition of a super-spreading individual depends on how super-spreading is defined. Even when the spreading is defined, it may not be clear that a single centrality always correlates with super-spreading, since centralities are network dependent. We proposed using a measure of super-spreading directly related to R0 and that will give a measure of ‘spreading’ regardless of the underlying network. We captured the vulnerability for varying degrees of heterogeneity and initial conditions by defining a measure to quantify the chances of epidemic spread in the simulations. We simulated the SIR spread on a real Malaysian network to illustrate the effects of this measure and heterogeneity on the number of infections. We also simulated super-spreading events (based on our definition) within the bounds of heterogeneity to demonstrate the effectiveness of the newly defined measure. We found that heterogeneity serves as a natural curve-flattening mechanism; therefore, the number of infections and R0 may be lower than expected. This may lead to a false sense of security, especially since heterogeneity makes the population vulnerable to super-spreading events. Full article

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

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11 pages, 310 KiB

Open AccessArticle

On 3-Rainbow Domination Number of Generalized Petersen Graphs P(6k,k)

by Rija Erveš and Janez Žerovnik

Symmetry 2021, 13(10), 1860; https://doi.org/10.3390/sym13101860 - 03 Oct 2021

Cited by 5 | Viewed by 1384

Abstract

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs

P (6 k, k)

. In some cases, for some infinite families, exact values are established; in all other cases, the lower and upper bounds with small gaps [...] Read more.

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs

P (6 k, k)

. In some cases, for some infinite families, exact values are established; in all other cases, the lower and upper bounds with small gaps are given. We also define singleton rainbow domination, where the sets assigned have a cardinality of, at most, one, and provide analogous results for this special case of rainbow domination. Full article

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

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12 pages, 1306 KiB

Open AccessArticle

Eulerian and Even-Face Graph Partial Duals

by Metrose Metsidik

Symmetry 2021, 13(8), 1475; https://doi.org/10.3390/sym13081475 - 11 Aug 2021

Viewed by 1712

Abstract

Eulerian and bipartite graph is a dual symmetric concept in Graph theory. It is well-known that a plane graph is Eulerian if and only if its geometric dual is bipartite. In this paper, we generalize the well-known result to embedded graphs and partial [...] Read more.

Eulerian and bipartite graph is a dual symmetric concept in Graph theory. It is well-known that a plane graph is Eulerian if and only if its geometric dual is bipartite. In this paper, we generalize the well-known result to embedded graphs and partial duals of cellularly embedded graphs, and characterize Eulerian and even-face graph partial duals of a cellularly embedded graph by means of half-edge orientations of its medial graph. Full article

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

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15 pages, 773 KiB

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 - 03 Aug 2021

Cited by 4 | Viewed by 2563

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)

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 following three useful properties. (I) The value

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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16 pages, 335 KiB

Open AccessArticle

Logical Contradictions in the One-Way ANOVA and Tukey–Kramer Multiple Comparisons Tests with More Than Two Groups of Observations

by Vladimir Gurvich and Mariya Naumova

Symmetry 2021, 13(8), 1387; https://doi.org/10.3390/sym13081387 - 30 Jul 2021

Cited by 11 | Viewed by 3891

Abstract

We show that the one-way ANOVA and Tukey–Kramer (TK) tests agree on any sample with two groups. This result is based on a simple identity connecting the Fisher–Snedecor and studentized probabilistic distributions and is proven without any additional assumptions; in particular, [...] Read more.

We show that the one-way ANOVA and Tukey–Kramer (TK) tests agree on any sample with two groups. This result is based on a simple identity connecting the Fisher–Snedecor and studentized probabilistic distributions and is proven without any additional assumptions; in particular, the standard ANOVA assumptions (independence, normality, and homoscedasticity (INAH)) are not needed. In contrast, it is known that for a sample with

k > 2

groups of observations, even under the INAH assumptions, with the same significance level

α

, the above two tests may give opposite results: (i) ANOVA rejects its null hypothesis

H_{0}^{A} : μ_{1} = \dots = μ_{k}

, while the TK one,

H_{0}^{TK} (i, j) : μ_{i} = μ_{j}

, is not rejected for any pair

i, j \in {1, \dots, k}

; (ii) the TK test rejects

H_{0}^{TK} (i, j)

for a pair

(i, j)

(with

i \neq j

), while ANOVA does not reject

H_{0}^{A}

. We construct two large infinite pseudo-random families of samples of both types satisfying INAH: in case (i) for any

k \geq 3

and in case (ii) for some larger k. Furthermore, case (ii) ANOVA, being restricted to the pair of groups

(i, j)

, may reject equality

μ_{i} = μ_{j}

with the same

α

. This is an obvious contradiction, since

μ_{1} = \dots = μ_{k}

implies

μ_{i} = μ_{j}

for all

i, j \in {1, \dots, k} .

Such contradictions appear already in the symmetric case for

k = 3

, or in other words, for three groups of

d, d

, and c observations with sample means

+ 1, - 1

, and 0, respectively. We outline conditions necessary and sufficient for this phenomenon. Similar contradictory examples are constructed for the multivariable linear regression (MLR). However, for these constructions, it seems difficult to verify the Gauss–Markov assumptions, which are standardly required for MLR. Mathematics Subject Classification: 62 Statistics. Full article

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

9 pages, 862 KiB

Open AccessArticle

Recursively Divided Pancake Graphs with a Small Network Cost

by Jung-Hyun Seo and Hyeong-Ok Lee

Symmetry 2021, 13(5), 844; https://doi.org/10.3390/sym13050844 - 10 May 2021

Viewed by 1715

Abstract

Graphs are often used as models to solve problems in computer science, mathematics, and biology. A pancake sorting problem is modeled using a pancake graph whose classes include burnt pancake graphs, signed permutation graphs, and restricted pancake graphs. The network cost is degree [...] Read more.

Graphs are often used as models to solve problems in computer science, mathematics, and biology. A pancake sorting problem is modeled using a pancake graph whose classes include burnt pancake graphs, signed permutation graphs, and restricted pancake graphs. The network cost is degree × diameter. Finding a graph with a small network cost is like finding a good sorting algorithm. We propose a novel recursively divided pancake (RDP) graph that has a smaller network cost than other pancake-like graphs. In the pancake graph P_n, the number of nodes is n!, the degree is n − 1, and the network cost is O(n²). In an RDP_n, the number of nodes is n!, the degree is 2log₂n − 1, and the network cost is O(n(log₂n)³). Because O(n(log₂n)³) < O(n²), the RDP is superior to other pancake-like graphs. In this paper, we propose an RDP_n and analyze its basic topological properties. Second, we show that the RDP_n is recursive and symmetric. Third, a sorting algorithm is proposed, and the degree and diameter are derived. Finally, the network cost is compared between the RDP graph and other classes of pancake graphs. Full article

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

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