Advances in Discrete Applied Mathematics and Graph Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 28806

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors


E-Mail Website
Guest Editor
1. Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia
2. Institute of Mathematics, Physics, and Mechanics, 1000 Ljubljana, Slovenia
Interests: graph theory; discrete optimization; algorithms; heuristics and metaheuristics
Special Issues, Collections and Topics in MDPI journals

E-Mail Website
Guest Editor
Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia
Interests: graph theory; topological indices; stochastic processes
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Since its origins in the 18th century, graph theory has been a branch o mathematics that is both motivated by and applied to real world problems. Research in discrete mathematics increased in the latter half of the twentieth century mainly due to development of digital computers. On the other side, the advances in technology of digital computers enables extensive application of new ideas from discrete mathematics to real-world problems.

This special issue intends to promote novel examples of application of graph theory and discrete mathematics, as well as purely  theoretical works with foreseen impact to applications. The editors do not intend to narrow the scope of applications, and also encourage studies in standard areas of application that may be of particular interest recently, such as discrete models in epidemiology or advanced graph theoretical models used in drug design, both in view of the current pandemy.

  • Discrete Optimization
  • Graph Algorithms
  • Graph Theory
  • Applied Mathematics and Modeling
  • Operations Research

Prof. Dr. Janez Žerovnik
Dr. Darja Rupnik Poklukar
Guest Editors

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 submissions that pass pre-check are 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 semimonthly 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 2600 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

  • Discrete Optimization
  • Graph Algorithms
  • Graph Theory
  • Applied Mathematics and Modeling
  • Operations Research

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue polices can be found here.

Published Papers (12 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

12 pages, 330 KiB  
Article
Domination Coloring of Graphs
by Yangyang Zhou, Dongyang Zhao, Mingyuan Ma and Jin Xu
Mathematics 2022, 10(6), 998; https://doi.org/10.3390/math10060998 - 21 Mar 2022
Cited by 3 | Viewed by 4123
Abstract
A domination coloring of a graph G is a proper vertex coloring of G, such that each vertex of G dominates at least one color class (possibly its own class), and each color class is dominated by at least one vertex. The [...] Read more.
A domination coloring of a graph G is a proper vertex coloring of G, such that each vertex of G dominates at least one color class (possibly its own class), and each color class is dominated by at least one vertex. The minimum number of colors among all domination colorings is called the domination chromatic number, denoted by χdd(G). In this paper, we study the complexity of the k-domination coloring problem by proving its NP-completeness for arbitrary graphs. We give basic results and properties of χdd(G), including the bounds and characterization results, and further research χdd(G) of some special classes of graphs, such as the split graphs, the generalized Petersen graphs, corona products, and edge corona products. Several results on graphs with χdd(G)=χ(G) are presented. Moreover, an application of domination colorings in social networks is proposed. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
Show Figures

Figure 1

10 pages, 474 KiB  
Article
Total Coloring of Dumbbell Maximal Planar Graphs
by Yangyang Zhou, Dongyang Zhao, Mingyuan Ma and Jin Xu
Mathematics 2022, 10(6), 912; https://doi.org/10.3390/math10060912 - 13 Mar 2022
Viewed by 1784
Abstract
The Total Coloring Conjecture (TCC) states that every simple graph G is totally (Δ+2)-colorable, where Δ denotes the maximum degree of G. In this paper, we prove that TCC holds for dumbbell maximal planar graphs. Especially, we [...] Read more.
The Total Coloring Conjecture (TCC) states that every simple graph G is totally (Δ+2)-colorable, where Δ denotes the maximum degree of G. In this paper, we prove that TCC holds for dumbbell maximal planar graphs. Especially, we divide the dumbbell maximal planar graphs into three categories according to the maximum degree: J9, I-dumbbell maximal planar graphs and II-dumbbell maximal planar graphs. We give the necessary and sufficient condition for I-dumbbell maximal planar graphs, and prove that any I-dumbbell maximal planar graph is totally 8-colorable. Moreover, a linear time algorithm is proposed to compute a total (Δ+2)-coloring for any I-dumbbell maximal planar graph. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
Show Figures

Figure 1

9 pages, 279 KiB  
Article
A Proof of a Conjecture on Bipartite Ramsey Numbers B(2,2,3)
by Yaser Rowshan, Mostafa Gholami and Stanford Shateyi
Mathematics 2022, 10(5), 701; https://doi.org/10.3390/math10050701 - 23 Feb 2022
Cited by 3 | Viewed by 1220
Abstract
The bipartite Ramsey number B(n1,n2,,nt) is the least positive integer b, such that any coloring of the edges of Kb,b with t colors will result in a [...] Read more.
The bipartite Ramsey number B(n1,n2,,nt) is the least positive integer b, such that any coloring of the edges of Kb,b with t colors will result in a monochromatic copy of Kni,ni in the i-th color, for some i, 1it. The values B(2,5)=17, B(2,2,2,2)=19 and B(2,2,2)=11 have been computed in several previously published papers. In this paper, we obtain the exact values of the bipartite Ramsey number B(2,2,3). In particular, we prove the conjecture on B(2,2,3) which was proposed in 2015—in fact, we prove that B(2,2,3)=17. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
12 pages, 300 KiB  
Article
More on Sombor Index of Graphs
by Wenjie Ning, Yuheng Song and Kun Wang
Mathematics 2022, 10(3), 301; https://doi.org/10.3390/math10030301 - 19 Jan 2022
Cited by 12 | Viewed by 2986
Abstract
Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let G=(V(G),E(G)) be a graph. Then, the Sombor index of G is defined as [...] Read more.
Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let G=(V(G),E(G)) be a graph. Then, the Sombor index of G is defined as SO(G)=uvE(G)dG2(u)+dG2(v). In this paper, we give some lemmas that can be used to compare the Sombor indices between two graphs. With these lemmas, we determine the graph with maximum SO among all cacti with n vertices and k cut edges. Furthermore, the unique graph with maximum SO among all cacti with n vertices and p pendant vertices is characterized. In addition, we find the extremal graphs with respect to SO among all quasi-unicyclic graphs. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
Show Figures

Figure 1

19 pages, 360 KiB  
Article
On the Double Roman Domination in Generalized Petersen Graphs P(5k,k)
by Darja Rupnik Poklukar and Janez Žerovnik
Mathematics 2022, 10(1), 119; https://doi.org/10.3390/math10010119 - 1 Jan 2022
Cited by 4 | Viewed by 2060
Abstract
A double Roman dominating function on a graph G=(V,E) is a function f:V{0,1,2,3} satisfying the condition that every vertex u for which [...] Read more.
A double Roman dominating function on a graph G=(V,E) is a function f:V{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u)=1 is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=vVf(v). The double Roman domination number γdR(G) of a graph G equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k,k). It is proven that γdR(P(5k,k))=8k for k2,3mod5 and 8kγdR(P(5k,k))8k+2 for k0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k). Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
Show Figures

Figure 1

11 pages, 280 KiB  
Article
On the Quasi-Total Roman Domination Number of Graphs
by Abel Cabrera Martínez, Juan C. Hernández-Gómez and José M. Sigarreta
Mathematics 2021, 9(21), 2823; https://doi.org/10.3390/math9212823 - 6 Nov 2021
Cited by 4 | Viewed by 1821
Abstract
Domination theory is a well-established topic in graph theory, as well as one of the most active research areas. Interest in this area is partly explained by its diversity of applications to real-world problems, such as facility location problems, computer and social networks, [...] Read more.
Domination theory is a well-established topic in graph theory, as well as one of the most active research areas. Interest in this area is partly explained by its diversity of applications to real-world problems, such as facility location problems, computer and social networks, monitoring communication, coding theory, and algorithm design, among others. In the last two decades, the functions defined on graphs have attracted the attention of several researchers. The Roman-dominating functions and their variants are one of the main attractions. This paper is a contribution to the Roman domination theory in graphs. In particular, we provide some interesting properties and relationships between one of its variants: the quasi-total Roman domination in graphs. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
Show Figures

Figure 1

12 pages, 272 KiB  
Article
Local Inclusive Distance Vertex Irregular Graphs
by Kiki Ariyanti Sugeng, Denny Riama Silaban, Martin Bača and Andrea Semaničová-Feňovčíková
Mathematics 2021, 9(14), 1673; https://doi.org/10.3390/math9141673 - 16 Jul 2021
Cited by 2 | Viewed by 2265
Abstract
Let G=(V,E) be a simple graph. A vertex labeling f:V(G){1,2,,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex [...] Read more.
Let G=(V,E) be a simple graph. A vertex labeling f:V(G){1,2,,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,yV(G) their weights are distinct, where the weight of a vertex xV(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d=1 and determine the precise values of the corresponding graph invariant for certain families of graphs. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
Show Figures

Figure 1

11 pages, 945 KiB  
Article
Free Cells in Hyperspaces of Graphs
by José Ángel Juárez Morales, Gerardo Reyna Hernández, Jesús Romero Valencia and Omar Rosario Cayetano
Mathematics 2021, 9(14), 1627; https://doi.org/10.3390/math9141627 - 10 Jul 2021
Viewed by 1623
Abstract
Often for understanding a structure, other closely related structures with the former are associated. An example of this is the study of hyperspaces. In this paper, we give necessary and sufficient conditions for the existence of finitely-dimensional maximal free cells in the hyperspace [...] Read more.
Often for understanding a structure, other closely related structures with the former are associated. An example of this is the study of hyperspaces. In this paper, we give necessary and sufficient conditions for the existence of finitely-dimensional maximal free cells in the hyperspace C(G) of a dendrite G; then, we give necessary and sufficient conditions so that the aforementioned result can be applied when G is a dendroid. Furthermore, we prove that the arc is the unique arcwise connected, compact, and metric space X for which the anchored hyperspace Cp(X) is an arc for some pX. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
Show Figures

Figure 1

12 pages, 247 KiB  
Article
Local Antimagic Chromatic Number for Copies of Graphs
by Martin Bača, Andrea Semaničová-Feňovčíková and Tao-Ming Wang
Mathematics 2021, 9(11), 1230; https://doi.org/10.3390/math9111230 - 27 May 2021
Cited by 12 | Viewed by 2565
Abstract
An edge labeling of a graph G=(V,E) using every label from the set {1,2,,|E(G)|} exactly once is a local antimagic labeling if the vertex-weights [...] Read more.
An edge labeling of a graph G=(V,E) using every label from the set {1,2,,|E(G)|} exactly once is a local antimagic labeling if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any local antimagic labeling induces a proper vertex coloring of G where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
17 pages, 275 KiB  
Article
Inequalities on the Generalized ABC Index
by Paul Bosch, Edil D. Molina, José M. Rodríguez and José M. Sigarreta
Mathematics 2021, 9(10), 1151; https://doi.org/10.3390/math9101151 - 20 May 2021
Cited by 5 | Viewed by 1787
Abstract
In this work, we obtained new results relating the generalized atom-bond connectivity index with the general Randić index. Some of these inequalities for ABCα improved, when α=1/2, known results on the ABC index. [...] Read more.
In this work, we obtained new results relating the generalized atom-bond connectivity index with the general Randić index. Some of these inequalities for ABCα improved, when α=1/2, known results on the ABC index. Moreover, in order to obtain our results, we proved a kind of converse Hölder inequality, which is interesting on its own. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
12 pages, 311 KiB  
Article
The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle
by Yaser Rowshan, Mostafa Gholami and Stanford Shateyi
Mathematics 2021, 9(7), 764; https://doi.org/10.3390/math9070764 - 1 Apr 2021
Cited by 8 | Viewed by 2133
Abstract
For given graphs G1,G2,,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,,Gn) is the smallest positive [...] Read more.
For given graphs G1,G2,,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1in, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n2 and mj(nK2,C7), for any j4 and n2. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
Show Figures

Figure 1

7 pages, 839 KiB  
Article
Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees
by Xinyue Liu, Huiqin Jiang, Pu Wu and Zehui Shao
Mathematics 2021, 9(3), 293; https://doi.org/10.3390/math9030293 - 2 Feb 2021
Cited by 1 | Viewed by 1946
Abstract
For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G){0,1,2,3} having the [...] Read more.
For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G){0,1,2,3} having the property that (i) wN(v)f(w)3 if f(v)=0; (ii) wN(v)f(w)2 if f(v)=1; and (iii) every vertex v with f(v)0 has a neighbor u with f(u)0 for every vertex vV(G). The weight of a TR3DF f is the sum f(V)=vV(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees. Full article
(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)
Show Figures

Figure 1

Back to TopTop