Special Issue "Symmetry in Graph Theory"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 October 2018).

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A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Jose M. Rodriguez
E-Mail Website
Guest Editor
Department of Mathematics, Carlos III University of Madrid-Leganés Campus, Avenida de la Universidad 30, CP-28911, Leganés, Madrid, Spain
Interests: discrete mathematics; fractional calculus; topological indices; polynomials in graphs; geometric function theory; geometry; approximation theory
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Special Issue Information

Dear Colleagues,

Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including, but not limited to, Gromov hyperbolic graphs, metric dimension of graphs, domination theory, topological indices, and polynomials in graphs. This Special Issue invites contributions addressing new results on these topics, both from a theoretical and an applied point of view.

Prof. Jose M. Rodriguez
Guest Editor

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Keywords

  • Gromov hyperbolic graphs
  • Metric dimension of graphs
  • Topological indices
  • Domination theory
  • Polynomials in graphs

Published Papers (12 papers)

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Research

Open AccessArticle
Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes
Symmetry 2018, 10(12), 727; https://doi.org/10.3390/sym10120727 - 06 Dec 2018
Cited by 2
Abstract
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) Printed Edition available
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Open AccessArticle
On Degree-Based Topological Indices of Symmetric Chemical Structures
Symmetry 2018, 10(11), 619; https://doi.org/10.3390/sym10110619 - 09 Nov 2018
Cited by 2
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) Printed Edition available
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Open AccessFeature PaperArticle
Harmonic Index and Harmonic Polynomial on Graph Operations
Symmetry 2018, 10(10), 456; https://doi.org/10.3390/sym10100456 - 01 Oct 2018
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) Printed Edition available
Open AccessFeature PaperArticle
Secure Resolving Sets in a Graph
Symmetry 2018, 10(10), 439; https://doi.org/10.3390/sym10100439 - 27 Sep 2018
Abstract
Let G = (V, E) be a simple, finite, and connected graph. A subset S = {u1, u2, , uk} 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, , uk} of V(G) is called a resolving set (locating set) if for any xV(G), the code of x with respect to S that is denoted by CS (x), which is defined as CS (x) = (d(u1, x), d(u2, x), .., d(uk, 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 sV – R, there exists rR 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) Printed Edition available
Open AccessArticle
General (α,2)-Path Sum-Connectivirty Indices of One Important Class of Polycyclic Aromatic Hydrocarbons
Symmetry 2018, 10(10), 426; https://doi.org/10.3390/sym10100426 - 21 Sep 2018
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 ) = P t = v i 1 v i 2 v i t + 1 G [ d G ( v i 1 ) d G ( v i 2 ) 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 v i t + 1 and v i t + 1 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) Printed Edition available
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Open AccessArticle
Hyperbolicity on Graph Operators
Symmetry 2018, 10(9), 360; https://doi.org/10.3390/sym10090360 - 24 Aug 2018
Abstract
A graph operator is a mapping F : Γ 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 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 ) δ ( R ( G ) ) δ ( G ) + 1 / 2 , δ ( Λ ( G ) ) δ ( Q ( G ) ) δ ( Λ ( G ) ) + 1 / 2 , δ ( S ( G ) ) 2 δ ( R ( G ) ) δ ( S ( G ) ) + 1 and δ ( R ( G ) ) 1 / 2 δ ( Λ ( G ) ) 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) Printed Edition available
Open AccessArticle
On the Distinguishing Number of Functigraphs
Symmetry 2018, 10(8), 332; https://doi.org/10.3390/sym10080332 - 09 Aug 2018
Cited by 2
Abstract
Let G 1 and G 2 be disjoint copies of a graph G and g : V ( G 1 ) V ( G 2 ) be a function. A functigraph F G consists of the vertex set V ( G 1 [...] Read more.
Let G 1 and G 2 be disjoint copies of a graph G and g : V ( G 1 ) V ( G 2 ) be a function. A functigraph F G consists of the vertex set V ( G 1 ) V ( G 2 ) and the edge set E ( G 1 ) E ( G 2 ) { 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) Printed Edition available
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Open AccessArticle
Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes
Symmetry 2018, 10(8), 300; https://doi.org/10.3390/sym10080300 - 25 Jul 2018
Cited by 2
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) Printed Edition available
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Open AccessArticle
Hyperbolicity of Direct Products of Graphs
Symmetry 2018, 10(7), 279; https://doi.org/10.3390/sym10070279 - 12 Jul 2018
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 G1×G2 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) Printed Edition available
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Open AccessArticle
Topological Properties of Crystallographic Structure of Molecules
Symmetry 2018, 10(7), 265; https://doi.org/10.3390/sym10070265 - 05 Jul 2018
Cited by 5
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 TiF2 and the crystallographic structure of cuprite Cu2O. Furthermore, we compute degree-based topological indices, mainly ABC, GA, ABC4, GA5 and general Randić indices. Furthermore, we also give exact results of these indices for the crystal structure of titanium difluoride TiF2 and the crystallographic structure of cuprite Cu2O. Full article
(This article belongs to the Special Issue Symmetry in Graph Theory) Printed Edition available
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Open AccessArticle
Dynamics on Binary Relations over Topological Spaces
Symmetry 2018, 10(6), 211; https://doi.org/10.3390/sym10060211 - 11 Jun 2018
Cited by 2
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) Printed Edition available
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Open AccessFeature PaperArticle
Computing the Metric Dimension of Gear Graphs
Symmetry 2018, 10(6), 209; https://doi.org/10.3390/sym10060209 - 08 Jun 2018
Cited by 2
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 J2n,m be a m-level gear graph obtained by m-level wheel graph W2n,mmC2n + k1 by alternatively deleting n spokes of each copy of C2n and J3n be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W3n. In this paper, the metric dimension of certain gear graphs J2n,m and J3n 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) Printed Edition available
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