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Axioms
Special Issues
Recent Developments in Graph Theory
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Recent Developments in Graph Theory

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 27 May 2025 | Viewed by 3621

Special Issue Editor

Dr. Simon Brezovnik

E-Mail Website
Guest Editor
Department for Mathematics, University of Ljubljana, 1000 Ljubljana, Slovenia
Interests: chemical graph theory; resonance graphs; topological indices; Grundy domination; k-rainbow domaination

Special Issue Information

Dear Colleagues,

It brings me great pleasure to announce the commencement of our latest Special Issue, focusing on the forefront of advancements and research within the realm of graph theory and discrete applied mathematics, particularly spotlighting domination theory and chemical graph theory.

The realm of domination stands as a swiftly expanding domain within graph theory, bearing significant real-world applications spanning various sectors. These applications range from the analysis of electrical and communication networks to optimization and coding theory. Similarly, the realm of chemical graph theory holds immense application potential, employing graph theoretical constructs to model the physical and biological attributes of chemical compounds.

We extend a cordial invitation to researchers who are actively engaged in these fields to contribute their work to this Special Issue. We firmly believe that this compilation will serve as a pivotal repository of knowledge and inspiration for the researchers, professionals, and students vested in these dynamic and rapidly evolving domains.

Dr. Simon Brezovnik
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 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. Axioms 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 2400 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

  • chemical graph theory
  • topological indices
  • cut method
  • applied mathematics
  • domination
  • rainbow domination
  • domination games
  • independence domination
  • Grundy domination

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Published Papers (6 papers)

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Research

16 pages, 307 KiB  
Article
Horadam–Lucas Cubes
by Elif Tan, Luka Podrug and Vesna Iršič Chenoweth
Axioms 2024, 13(12), 837; https://doi.org/10.3390/axioms13120837 - 28 Nov 2024
Viewed by 330
Abstract
In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family [...] Read more.
In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family of the Horadam cubes, in a similar way the Lucas cubes relate to Fibonacci cubes or the Lucas-run graphs relate to Fibonacci-run graphs. As special cases, they also give rise to new graph families, such as Pell–Lucas cubes and Jacobsthal–Lucas cubes. We derive the several metric and enumerative properties of these cubes, including their diameter, periphery, radius, fundamental decomposition, number of edges, cube polynomials, and generating function of the cube polynomials. Full article
(This article belongs to the Special Issue Recent Developments in Graph Theory)
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14 pages, 311 KiB  
Article
Kekulé Structure of Angularly Connected Even Ring Systems
by Simon Brezovnik
Axioms 2024, 13(12), 827; https://doi.org/10.3390/axioms13120827 - 26 Nov 2024
Viewed by 374
Abstract
An even ring system G is a simple 2-connected plane graph with all interior vertices of degree 3, all exterior vertices of either degree 2 or 3, and all finite faces of an even length. G is angularly connected if all of the [...] Read more.
An even ring system G is a simple 2-connected plane graph with all interior vertices of degree 3, all exterior vertices of either degree 2 or 3, and all finite faces of an even length. G is angularly connected if all of the peripheral segments of G have odd lengths. In this paper, we show that every angularly connected even ring system G, which does not contain any triple of altogether-adjacent peripheral faces, has a perfect matching. This was achieved by finding an appropriate edge coloring of G, derived from the proof of the existence of a proper face 3-coloring of the graph. Additionally, an infinite family of graphs that are face 3-colorable has been identified. When interpreted in the context of the inner dual of G, this leads to the introduction of 3-colorable graphs containing cycles of lengths 4 and 6, which is a supplementation of some already known results. Finally, we have investigated the concept of the Clar structure and Clar set within the aforementioned family of graphs. We found that a Clar set of an angularly connected even ring system cannot in general be obtained by minimizing the cardinality of the set A. This result is in contrast to the previously known case for the subfamily of benzenoid systems, which admit a face 3-coloring. Our results open up avenues for further research into the properties of Clar and Fries sets of angularly connected even ring systems. Full article
(This article belongs to the Special Issue Recent Developments in Graph Theory)
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10 pages, 254 KiB  
Article
Extremal k-Connected Graphs with Maximum Closeness
by Fazal Hayat and Daniele Ettore Otera
Axioms 2024, 13(12), 810; https://doi.org/10.3390/axioms13120810 - 21 Nov 2024
Viewed by 383
Abstract
Closeness is a measure that quantifies how quickly information can spread from a given node to all other nodes in the network, reflecting the efficiency of communication within the network by indicating how close a node is to all other nodes. For a [...] Read more.
Closeness is a measure that quantifies how quickly information can spread from a given node to all other nodes in the network, reflecting the efficiency of communication within the network by indicating how close a node is to all other nodes. For a graph G, the subset S of vertices of V(G) is called vertex cut of G if the graph G−S becomes disconnected. The minimum cardinality of S for which G−S is either disconnected or contains precisely one vertex is called connectivity of G. A graph is called k-connected if it stays connected even when any set of fewer than k vertices is removed. In communication networks, a k-connected graph improves network reliability; even if up to k−1 nodes fail, the network remains operational, maintaining connectivity between devices. This paper aims to study the concept of closeness within n-vertex graphs with fixed connectivity. First, we identify the graphs that maximize the closeness among all graphs of order n with fixed connectivity k. Then, we determine the graphs that achieve the maximum closeness within all k-connected graphs of order n, given specific fixed parameters such as diameter, independence number, and minimum degree. Full article
(This article belongs to the Special Issue Recent Developments in Graph Theory)
14 pages, 283 KiB  
Article
Bounds for the Energy of Hypergraphs
by Liya Jess Kurian and Chithra Velu
Axioms 2024, 13(11), 804; https://doi.org/10.3390/axioms13110804 - 19 Nov 2024
Viewed by 379
Abstract
The concept of the energy of a graph has been widely explored in the field of mathematical chemistry and is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of a hypergraph is the trace [...] Read more.
The concept of the energy of a graph has been widely explored in the field of mathematical chemistry and is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of a hypergraph is the trace norm of its connectivity matrices, which generalize the concept of graph energy. In this paper, we establish bounds for the adjacency energy of hypergraphs in terms of the number of vertices, maximum degree, eigenvalues, and the norm of the adjacency matrix. Additionally, we compute the sum of squares of adjacency eigenvalues of linear k-hypergraphs and derive its bounds for k-hypergraph in terms of number of vertices and uniformity of the k-hypergraph. Moreover, we determine the Nordhaus–Gaddum type bounds for the adjacency energy of k-hypergraphs. Full article
(This article belongs to the Special Issue Recent Developments in Graph Theory)
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10 pages, 663 KiB  
Article
Total and Double Total Domination on Octagonal Grid
by Antoaneta Klobučar and Ana Klobučar Barišić
Axioms 2024, 13(11), 792; https://doi.org/10.3390/axioms13110792 - 16 Nov 2024
Viewed by 535
Abstract
A k-total dominating set is a set of vertices such that all vertices in the graph, including the vertices in the dominating set themselves, have at least k neighbors in the dominating set. The k-total domination number [...] Read more.
A k-total dominating set is a set of vertices such that all vertices in the graph, including the vertices in the dominating set themselves, have at least k neighbors in the dominating set. The k-total domination number γkt(G) is the cardinality of the smallest k-total dominating set. For k=1,2, the k-total dominating number is called the total and the double total dominating number, respectively. In this paper, we determine the exact values for the total domination number on a linear and on a double octagonal chain and an upper bound for the total domination number on a triple octagonal chain. Furthermore, we determine the exact values for the double total domination number on a linear and on a double octagonal chain and an upper bound for the double total domination number on a triple octagonal chain and on an octagonal grid Om,n,m≥3,n≥3. As each vertex in the octagonal system is either of degree two or of degree three, there is no k-total domination for k≥3. Full article
(This article belongs to the Special Issue Recent Developments in Graph Theory)
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10 pages, 363 KiB  
Article
Two-Matchings with Respect to the General Sum-Connectivity Index of Trees
by Roberto Cruz, Mateo Lopez and Juan Rada
Axioms 2024, 13(10), 658; https://doi.org/10.3390/axioms13100658 - 24 Sep 2024
Viewed by 813
Abstract
A vertex-degree-based topological index φ associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in [...] Read more.
A vertex-degree-based topological index φ associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in QSPR/QSAR investigations. A subset of k edges in G with no common vertices is called a k-matching of G, and the number of such subsets is denoted by mG,k. Recently, this number was naturally extended to weighted graphs, where the weight function is induced by the topological index φ. This number was denoted by mkG,φ and called the k-matchings of G with respect to the topological index φ. It turns out that m1G,φ=φG, and so for k≥2, the k-matching numbers mkG,φ can be viewed as kth order topological indices which involve both the topological index φ and the k-matching numbers. In this work, we solve the extremal value problem for the number of 2-matchings with respect to general sum-connectivity indices SCα, over the set Tn of trees with n vertices, when α is a real number in the interval −1,0. Full article
(This article belongs to the Special Issue Recent Developments in Graph Theory)
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