Special Issue "Graph-Theoretic Problems and Their New Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 31 January 2020

Special Issue Editor

Guest Editor
Prof. Dr. Frank Werner

Fakultät für Mathematik, Otto-von-Guericke-Universität, D-39016 Magdeburg, Germany
Website | E-Mail
Interests: discrete optimization; operations research; scheduling; graph theory; manufacturing systems

Special Issue Information

Dear Colleagues,

Nowadays, graph theory plays a vital role in many disciplines. We invite you to submit your latest original research results in all aspects of graph theory to this Special Issue. We are looking both for new and innovative approaches for traditional graph-theoretic problems and well as for new applications of graph theory in emerging fields, such as in network security, computer science and data analysis, bioinformatics, operations research, engineering and manufacturing, physics and chemistry, linguistics, or social sciences. Both theoretical results, as well as new graph algorithms, with investigations of their computational complexity will be considered. Below, we give an exemplary, but not exhaustive, list of potential subjects for submissions to this Special Issue.

Prof. Dr. Frank Werner
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 850 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Deterministic, randomized, exact and approximative graph algorithms
  • Paths, cycles, and trees
  • Network theory
  • Graph labeling
  • Graph coloring
  • Symmetric graphs
  • Polyhedral graphs
  • Topological indices
  • Domination in graphs
  • Applications of graph theory

Published Papers (10 papers)

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Research

Open AccessArticle Reformulated Zagreb Indices of Some Derived Graphs
Mathematics 2019, 7(4), 366; https://doi.org/10.3390/math7040366
Received: 15 March 2019 / Revised: 13 April 2019 / Accepted: 15 April 2019 / Published: 22 April 2019
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Abstract
A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. [...] Read more.
A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Open AccessArticle The Bounds of Vertex Padmakar–Ivan Index on k-Trees
Mathematics 2019, 7(4), 324; https://doi.org/10.3390/math7040324
Received: 26 January 2019 / Revised: 11 March 2019 / Accepted: 11 March 2019 / Published: 1 April 2019
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Abstract
The Padmakar–Ivan (PI) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges uv of a graph such that these vertices are not equidistant from u [...] Read more.
The Padmakar–Ivan ( P I ) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges u v of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of P I -indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the P I -values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
Open AccessArticle On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks
Mathematics 2019, 7(4), 314; https://doi.org/10.3390/math7040314
Received: 22 February 2019 / Revised: 24 March 2019 / Accepted: 25 March 2019 / Published: 28 March 2019
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Abstract
The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let Hn be the linear heptagonal networks. It is interesting to [...] Read more.
The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let H n be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of H n due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of H n by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to H n . Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Open AccessArticle Distance Degree Index of Some Derived Graphs
Mathematics 2019, 7(3), 283; https://doi.org/10.3390/math7030283
Received: 21 January 2019 / Revised: 4 March 2019 / Accepted: 14 March 2019 / Published: 19 March 2019
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Abstract
Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree (DD) index was introduced by Dobrynin and Kochetova in 1994 for characterizing [...] Read more.
Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree ( D D ) index was introduced by Dobrynin and Kochetova in 1994 for characterizing alkanes by an integer. In this paper, we have determined expressions for a D D index of some derived graphs in terms of the parameters of the parent graph. Specifically, we establish expressions for the D D index of a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph, and paraline graph. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Open AccessArticle More Results on the Domination Number of Cartesian Product of Two Directed Cycles
Mathematics 2019, 7(2), 210; https://doi.org/10.3390/math7020210
Received: 12 December 2018 / Revised: 12 February 2019 / Accepted: 21 February 2019 / Published: 24 February 2019
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Abstract
Let γ(D) denote the domination number of a digraph D and let CmCn denote the Cartesian product of Cm and Cn, the directed cycles of length nm3. Liu [...] Read more.
Let γ ( D ) denote the domination number of a digraph D and let C m C n denote the Cartesian product of C m and C n , the directed cycles of length n m 3 . Liu et al. obtained the exact values of γ ( C m C n ) for m up to 6 [Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36–39]. Shao et al. determined the exact values of γ ( C m C n ) for m = 6 , 7 [On the domination number of Cartesian product of two directed cycles, Journal of Applied Mathematics, Volume 2013, Article ID 619695]. Mollard obtained the exact values of γ ( C m C n ) for m = 3 k + 2 [M. Mollard, On domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths, Discuss. Math. Graph Theory 33(2) (2013) 387–394.]. In this paper, we extend the current known results on C m C n with m up to 21. Moreover, the exact values of γ ( C n C n ) with n up to 31 are determined. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Open AccessArticle k-Rainbow Domination Number of P3Pn
Mathematics 2019, 7(2), 203; https://doi.org/10.3390/math7020203
Received: 12 January 2019 / Revised: 5 February 2019 / Accepted: 19 February 2019 / Published: 21 February 2019
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Abstract
Let k be a positive integer, and set [k]:={1,2,,k}. For a graph G, a k-rainbow dominating function (or kRDF) of G is a mapping f:V [...] Read more.
Let k be a positive integer, and set [ k ] : = { 1 , 2 , , k } . For a graph G, a k-rainbow dominating function (or kRDF) of G is a mapping f : V ( G ) 2 [ k ] in such a way that, for any vertex v V ( G ) with the empty set under f, the condition u N G ( v ) f ( u ) = [ k ] always holds, where N G ( v ) is the open neighborhood of v. The weight of kRDF f of G is the summation of values of all vertices under f. The k-rainbow domination number of G, denoted by γ r k ( G ) , is the minimum weight of a kRDF of G. In this paper, we obtain the k-rainbow domination number of grid P 3 P n for k { 2 , 3 , 4 } . Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Open AccessArticle Some Root Level Modifications in Interval Valued Fuzzy Graphs and Their Generalizations Including Neutrosophic Graphs
Mathematics 2019, 7(1), 72; https://doi.org/10.3390/math7010072
Received: 22 November 2018 / Revised: 26 December 2018 / Accepted: 3 January 2019 / Published: 10 January 2019
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Abstract
Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of [...] Read more.
Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of some counterexamples. To achieve this, first, we aim to improve the existing definition for interval-valued FG, interval-valued intuitionistic FG and their complements, as these existing definitions are not well-defined; i.e., one can obtain some senseless intervals using the existing definitions. The limitations of the existing definitions and the validity of the new definitions are supported with some examples. It is also observed that the notion of a single-valued neutrosophic graph (SVNG) is not well-defined either. The consequences of the existing definition of SVNG are discussed with the help of examples. A new definition of SVNG is developed, and its improvement is demonstrated with some examples. The definition of an interval-valued neutrosophic graph is also modified due to the shortcomings in the current definition, and the validity of the new definition is proved. An application of proposed work is illustrated through a decision-making problem under the framework of SVNG, and its performance is compared with existing work. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Open AccessArticle The Aα-Spectral Radii of Graphs with Given Connectivity
Mathematics 2019, 7(1), 44; https://doi.org/10.3390/math7010044
Received: 22 November 2018 / Revised: 23 December 2018 / Accepted: 24 December 2018 / Published: 4 January 2019
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Abstract
The Aα-matrix is Aα(G)=αD(G)+(1α)A(G) with α[0,1], given by Nikiforov in 2017, where A( [...] Read more.
The A α -matrix is A α ( G ) = α D ( G ) + ( 1 α ) A ( G ) with α [ 0 , 1 ] , given by Nikiforov in 2017, where A ( G ) is adjacent matrix, and D ( G ) is its diagonal matrix of the degrees of a graph G. The maximal eigenvalue of A α ( G ) is said to be the A α -spectral radius of G. In this work, we determine the graphs with largest A α ( G ) -spectral radius with fixed vertex or edge connectivity. In addition, related extremal graphs are characterized and equations satisfying A α ( G ) -spectral radius are proposed. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
Open AccessArticle The Bounds of the Edge Number in Generalized Hypertrees
Mathematics 2019, 7(1), 2; https://doi.org/10.3390/math7010002
Received: 11 November 2018 / Revised: 14 December 2018 / Accepted: 15 December 2018 / Published: 20 December 2018
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Abstract
A hypergraph H=(V,ε) is a pair consisting of a vertex set V, and a set ε of subsets (the hyperedges of H) of V. A hypergraph H is r-uniform if all the hyperedges [...] Read more.
A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T E is a fixed value k   ( 2 k r ) . We focus on the case that G H T E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n r + 1 edges if r 3   and   n 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T E has exactly k   ( 2 k r 1 ) components. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Open AccessArticle Kempe-Locking Configurations
Mathematics 2018, 6(12), 309; https://doi.org/10.3390/math6120309
Received: 27 October 2018 / Revised: 28 November 2018 / Accepted: 4 December 2018 / Published: 7 December 2018
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Abstract
The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored [...] Read more.
The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored in this article is that the connectivity and coloring properties are incompatible. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. All Kempe-locked triangulations that we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say x y , and (2) they have a Birkhoff diamond with endpoints x and y as a subgraph. On the strength of our investigations, we formulate a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample are indeed incompatible. It would also imply the appealing conclusion that the Birkhoff diamond configuration alone is responsible for the 4-colorability of planar triangulations. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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