Research

18 pages, 1036 KiB

Open AccessArticle

Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

by Hassan Raza, Sakander Hayat and Xiang-Feng Pan

Symmetry 2018, 10(12), 727; https://doi.org/10.3390/sym10120727 - 06 Dec 2018

Cited by 14 | Viewed by 3688

A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space

R^{d}

. Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, [...] Read more.

A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space

R^{d}

. Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, we study the problem of binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally. We provide an integer linear programming (ILP) formulation for the binary locating-dominating problem of graphs. We have determined the exact values of the binary locating-dominating number for two infinite families of convex polytopes. The exact values of the binary locating-dominating number are obtained for two rotationally-symmetric convex polytopes families. Moreover, certain upper bounds are determined for other three infinite families of convex polytopes. By using the ILP formulation, we show tightness in the obtained upper bounds. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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18 pages, 2058 KiB

Open AccessArticle

On Degree-Based Topological Indices of Symmetric Chemical Structures

by Jia-Bao Liu, Haidar Ali, Muhammad Kashif Shafiq and Usman Munir

Symmetry 2018, 10(11), 619; https://doi.org/10.3390/sym10110619 - 09 Nov 2018

Cited by 7 | Viewed by 3075

Abstract

A Topological index also known as connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph [...] Read more.

A Topological index also known as connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study HDCN1(m,n) and HDCN2(m,n) of dimension

m, n

and derive analytical closed results of general Randić index

R_{α} (G)

for different values of

α

. We also compute the general first Zagreb, ABC, GA,

A B C_{4}

and

G A_{5}

indices for these Hex derived cage networks for the first time and give closed formulas of these degree-based indices. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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15 pages, 280 KiB

Open AccessFeature PaperArticle

Harmonic Index and Harmonic Polynomial on Graph Operations

by Juan C. Hernández-Gómez, J. A. Méndez-Bermúdez, José M. Rodríguez and José M. Sigarreta

Symmetry 2018, 10(10), 456; https://doi.org/10.3390/sym10100456 - 01 Oct 2018

Cited by 8 | Viewed by 2816

Abstract

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. [...] Read more.

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity

O (n^{2})

. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

10 pages, 4879 KiB

Open AccessFeature PaperArticle

Secure Resolving Sets in a Graph

by Hemalathaa Subramanian and Subramanian Arasappan

Symmetry 2018, 10(10), 439; https://doi.org/10.3390/sym10100439 - 27 Sep 2018

Cited by 1 | Viewed by 2442

Abstract

Let G = (V, E) be a simple, finite, and connected graph. A subset S = {u₁, u₂, _…, u_k} of V(G) is called a resolving set (locating set) if for any [...] Read more.

Let G = (V, E) be a simple, finite, and connected graph. A subset S = {u₁, u₂, _…, u_k} of V(G) is called a resolving set (locating set) if for any x ∈ V(G), the code of x with respect to S that is denoted by C_S (x), which is defined as C_S (x) = (d(u₁, x), d(u₂, x), .., d(u_k, x)), is different for different x. The minimum cardinality of a resolving set is called the dimension of G and is denoted by dim(G). A security concept was introduced in domination. A subset D of V(G) is called a dominating set of G if for any v in V – D, there exists u in D such that u and v are adjacent. A dominating set D is secure if for any u in V – D, there exists v in D such that (D – {v}) ∪ {u} is a dominating set. A resolving set R is secure if for any s ∈ V – R, there exists r ∈ R such that (R – {r}) ∪ {s} is a resolving set. The secure resolving domination number is defined, and its value is found for several classes of graphs. The characterization of graphs with specific secure resolving domination number is also done. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

7 pages, 426 KiB

Open AccessArticle

General (α,2)-Path Sum-Connectivirty Indices of One Important Class of Polycyclic Aromatic Hydrocarbons

by Haiying Wang

Symmetry 2018, 10(10), 426; https://doi.org/10.3390/sym10100426 - 21 Sep 2018

Cited by 2 | Viewed by 2184

Abstract

The general

(α, t)

-path sum-connectivity index of a molecular graph originates from many practical problems, such as the three-dimensional quantitative structure–activity relationships (3D QSAR) and molecular chirality. For arbitrary nonzero real number

α

and arbitrary positive integer t, [...] Read more.

The general

(α, t)

-path sum-connectivity index of a molecular graph originates from many practical problems, such as the three-dimensional quantitative structure–activity relationships (3D QSAR) and molecular chirality. For arbitrary nonzero real number

α

and arbitrary positive integer t, it is defined as

^{t} χ_{α} (G) = \sum_{P^{t} = v_{i_{1}} v_{i_{2}} \dots v_{i_{t + 1}} \subseteq G}^{} {[d_{G} (v_{i_{1}}) d_{G} (v_{i_{2}}) \dots d_{G} (v_{i_{t + 1}})]}^{α},

where we take the sum over all possible paths of length t of G and two paths

v_{i_{1}} v_{i_{2}} \dots v_{i_{t + 1}}

and

v_{i_{t + 1}} \dots v_{i_{2}} v_{i_{1}}

are considered to be one path. In this work, one important class of polycyclic aromatic hydrocarbons and their structures are firstly considered, which play a role in organic materials and medical sciences. We try to compute the exact general

(α, 2)

-path sum-connectivity indices of these hydrocarbon systems. Furthermore, we exactly derive the monotonicity and the extremal values of these polycyclic aromatic hydrocarbons for any real number

α

. These valuable results could produce strong guiding significance to these applied sciences. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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10 pages, 247 KiB

Open AccessArticle

Hyperbolicity on Graph Operators

by J. A. Méndez-Bermúdez, Rosalío Reyes, José M. Rodríguez and José M. Sigarreta

Symmetry 2018, 10(9), 360; https://doi.org/10.3390/sym10090360 - 24 Aug 2018

Cited by 6 | Viewed by 2339

Abstract

A graph operator is a mapping

F : Γ \to Γ^{'}

, where

Γ

and

Γ^{'}

are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations [...] Read more.

A graph operator is a mapping

F : Γ \to Γ^{'}

, where

Γ

and

Γ^{'}

are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph,

Λ (G)

; subdivision graph,

S (G)

; total graph,

T (G)

; and the operators

R (G)

and

Q (G)

. In particular, we get relationships such as

δ (G) \leq δ (R (G)) \leq δ (G) + 1 / 2

,

δ (Λ (G)) \leq δ (Q (G)) \leq δ (Λ (G)) + 1 / 2

,

δ (S (G)) \leq 2 δ (R (G)) \leq δ (S (G)) + 1

and

δ (R (G)) - 1 / 2 \leq δ (Λ (G)) \leq 5 δ (R (G)) + 5 / 2

for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

11 pages, 275 KiB

Open AccessArticle

On the Distinguishing Number of Functigraphs

by Muhammad Fazil, Muhammad Murtaza, Zafar Ullah, Usman Ali and Imran Javaid

Symmetry 2018, 10(8), 332; https://doi.org/10.3390/sym10080332 - 09 Aug 2018

Cited by 3 | Viewed by 2849

Abstract

Let

G_{1}

and

G_{2}

be disjoint copies of a graph G and

g : V (G_{1}) \to V (G_{2})

be a function. A functigraph

F_{G}

consists of the vertex set [...] Read more.

Let

G_{1}

and

G_{2}

be disjoint copies of a graph G and

g : V (G_{1}) \to V (G_{2})

be a function. A functigraph

F_{G}

consists of the vertex set

V (G_{1}) \cup V (G_{2})

and the edge set

E (G_{1}) \cup E (G_{2}) \cup {u v : g (u) = v}

. In this paper, we extend the study of distinguishing numbers of a graph to its functigraph. We discuss the behavior of distinguishing number in passing from G to

F_{G}

and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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13 pages, 1503 KiB

Open AccessArticle

Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes

by Zafar Hussain, Mobeen Munir, Maqbool Chaudhary and Shin Min Kang

Symmetry 2018, 10(8), 300; https://doi.org/10.3390/sym10080300 - 25 Jul 2018

Cited by 36 | Viewed by 3355

Abstract

Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices [...] Read more.

Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for graph G and this minimum cardinality is known as metric dimension of G. Boron nanotubes with different lattice structures, radii and chirality’s have attracted attention due to their transport properties, electronic structure and structural stability. In the present article, we compute the metric dimension and metric basis of 2D lattices of alpha-boron nanotubes. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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25 pages, 739 KiB

Open AccessArticle

Hyperbolicity of Direct Products of Graphs

by Walter Carballosa, Amauris De la Cruz, Alvaro Martínez-Pérez and José M. Rodríguez

Symmetry 2018, 10(7), 279; https://doi.org/10.3390/sym10070279 - 12 Jul 2018

Cited by 2 | Viewed by 2827

Abstract

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several [...] Read more.

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product

G_{1} \times G_{2}

is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs). Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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20 pages, 22731 KiB

Open AccessArticle

Topological Properties of Crystallographic Structure of Molecules

by Jia-Bao Liu, Muhammad Kamran Siddiqui, Manzoor Ahmad Zahid, Muhammad Naeem and Abdul Qudair Baig

Symmetry 2018, 10(7), 265; https://doi.org/10.3390/sym10070265 - 05 Jul 2018

Cited by 20 | Viewed by 3970

Abstract

Chemical graph theory plays an important role in modeling and designing any chemical structure. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. In this paper, we study the chemical graph of [...] Read more.

Chemical graph theory plays an important role in modeling and designing any chemical structure. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. In this paper, we study the chemical graph of the crystal structure of titanium difluoride

T i F_{2}

and the crystallographic structure of cuprite

C u_{2} O

. Furthermore, we compute degree-based topological indices, mainly

A B C

,

G A

,

A B C_{4}

,

G A_{5}

and general Randić indices. Furthermore, we also give exact results of these indices for the crystal structure of titanium difluoride

T i F_{2}

and the crystallographic structure of cuprite

C u_{2} O

. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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12 pages, 306 KiB

Open AccessArticle

Dynamics on Binary Relations over Topological Spaces

by Chung-Chuan Chen, J. Alberto Conejero, Marko Kostić and Marina Murillo-Arcila

Symmetry 2018, 10(6), 211; https://doi.org/10.3390/sym10060211 - 11 Jun 2018

Cited by 7 | Viewed by 3076

Abstract

The existence of chaos and the quest of dense orbits have been recently considered for dynamical systems given by multivalued linear operators. We consider the notions of topological transitivity, topologically mixing property, hypercyclicity, periodic points, and Devaney chaos in the general case of [...] Read more.

The existence of chaos and the quest of dense orbits have been recently considered for dynamical systems given by multivalued linear operators. We consider the notions of topological transitivity, topologically mixing property, hypercyclicity, periodic points, and Devaney chaos in the general case of binary relations on topological spaces, and we analyze how they can be particularized when they are represented with graphs and digraphs. The relations of these notions with different types of connectivity and with the existence of Hamiltonian paths are also exposed. Special attention is given to the study of dynamics over tournaments. Finally, we also show how disjointness can be introduced in this setting. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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11 pages, 290 KiB

Open AccessFeature PaperArticle

Computing the Metric Dimension of Gear Graphs

by Shahid Imran, Muhammad Kamran Siddiqui, Muhammad Imran, Muhammad Hussain, Hafiz Muhammad Bilal, Imran Zulfiqar Cheema, Ali Tabraiz and Zeeshan Saleem

Symmetry 2018, 10(6), 209; https://doi.org/10.3390/sym10060209 - 08 Jun 2018

Cited by 15 | Viewed by 4008

Abstract

Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v in G. A set of vertices W resolves a graph G if every vertex [...] Read more.

Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). Let J_2n,m be a m-level gear graph obtained by m-level wheel graph W_2n,m ≅ mC_2n + k₁ by alternatively deleting n spokes of each copy of C_2n and J_3n be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W_3n. In this paper, the metric dimension of certain gear graphs J_2n,m and J_3n generated by wheel has been computed. Also this study extends the previous result given by Tomescu et al. in 2007. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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