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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 9281

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Special Issue Editors

Dr. Hilal Ahmad

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Guest Editor

Department of Mathematics, University of Kashmir, Hazaratbal, Srinagar 190006, India
Interests: combinatorial matrix theory; eigenvalue analysis of graphs and digraphs; topological graph theory; linear algebra

Dr. Yilun Shang

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Guest Editor

Department of Computer and Information Sciences, Northumbria University, Newcastle-upon-Tyne, UK
Interests: mathematics; complex systems; networks; computer science; physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

By the middle of the 20th century, a new kind of mathematics, quite unlike the continuous mathematics of calculus, become prominent, called discrete mathematics. Discrete mathematics primarily became the study of the mathematical properties of sets, systems and structures having a countable collection of elements. The growing popularity of and interest in discrete mathematics was greatly influenced by its applications to areas such as computer science, engineering, communication and transportation, as well as its emergence as a fascinating area of mathematics. More unique to discrete mathematics, however, are topics involving procedures (algorithms) for solving problems and the number of objects possessing certain properties of interest—the area of discrete mathematics called enumerative combinatorics. Additionally, discrete mathematics is concerned with the study of relations and graph theory, a growing area of discrete mathematics with a much shorter history.

Graphs are discrete structures consisting of vertices and edges that connect these vertices. Problems in almost every conceivable discipline can be solved using graph models. One of the fascinating branches of graph theory that has attracted the attention of researchers all over the world is spectral graph theory. Spectral graph theory looks at the connection between the eigenvalues of a matrix associated with a graph/digraph and the corresponding structures of a graph/digraph. The study of spectral graph theory, in essence, is concerned with the relationships between the algebraic properties of the spectra of certain matrices associated with a graph/digraph and the topological properties of that graph/digraph. It lives on the crossroads of combinatorics and linear algebra. A number of matrices associated with the structure of the graph/digraph have been introduced and their spectral properties are discussed in detail. For example, the spectral properties of the adjacency matrix, the (signless) Laplacian matrix, the normalized Laplacian matrix, the randi\'{c} matrix, etc., their distance versions and their extension to digraphs are hot topics of the present research. Among the spectral properties of a graph/digraph matrix, the most studied topics include the spectral radius, the second largest eigenvalue, the smallest eigenvalue, the second smallest eigenvalue, the Ky-Fan k-norm (the sum of the k largest singular values, in the case of Hermitian positive semi-definite matrices it is the sum of the k largest eigenvalues), the trace norm (the sum of all the singular values, in the case of normal matrices it is the sum of the absolute values of the eigenvalues), the spread (the maximum of the absolute values of the difference between non-zero eigenvalues), etc. These topics have attracted much attention because of their applications to computer science, networking, and the fields of chemistry, biology, and graph coloring.

Investigating the eigenvalues of graph matrices has led to many interesting results in both combinatorics and matrix theory and in fact many open problems were solved in both the fields. One of the interesting problems in spectral graph theory is the extremal problems with respect to a given spectral invariant. Problems of this type include, for instance, finding the extremal value of some spectral parameter over a class of graphs, characterizing the elements of this class that achieve this extremal value, or ordering the elements in this class according to the value of this parameter. Another problem is the spectral determination of graph/digraph matrices.

Since different concepts are related to graphs and matrices, we can extend them to other concepts. One of these concepts is the Inverse Eigenvalue Problem (IEP). This problem concerns the reconstruction of a matrix from prescribed spectral data. The Nonnegative Inverse Eigenvalue Problem (NIEP) asks which collections of n complex numbers (counting multiplicities) occur as the eigenvalues of an n-by-n matrix, all of whose entries are non-negative real numbers. This is a long-standing problem that is very difficult and, perhaps, the most prominent problem in matrix analysis. Unlike many other inverse eigenvalue problems, tools are limited, and seemingly small results are very welcome. Nonetheless, the problem is very attractive and has been attacked by many excellent researchers.

Mathematical descriptors of molecular graphs, like the topological indices, have several uses in chemical studies. They play a very crucial role in theoretical chemistry, especially in quantitative structure–activity relationship (in short QSAR) and quantitative structure–property relationship (QSPR) related studies.

The theory of graphs is further extended to networks and systems, where the vertices can represent different entities in nature, society, and technology, and the links between them are modeled by edges.

Dr. Hilal Ahmad
Dr. Yilun Shang
Guest Editors

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Keywords

eigenvalues of a graph/digraph
sum of the k largest eigenvalues/singular values
energy of a graph/digraph
distance
distance energy
topological indices
extremal graphs/digraphs
spectral determination
complex networks
complex systems

Published Papers (11 papers)

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Research

21 pages, 3424 KiB

Open AccessArticle

Golden Laplacian Graphs

by Sadia Akhter, Mattia Frasca and Ernesto Estrada

Mathematics 2024, 12(4), 613; https://0-doi-org.brum.beds.ac.uk/10.3390/math12040613 - 19 Feb 2024

Viewed by 594

Abstract

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes

n \leq 10

. We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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11 pages, 1656 KiB

Open AccessArticle

On K-Banhatti, Revan Indices and Entropy Measures of MgO(111) Nanosheets via Linear Regression

by Norah Almalki and Hafsah Tabassum

Mathematics 2024, 12(4), 561; https://0-doi-org.brum.beds.ac.uk/10.3390/math12040561 - 13 Feb 2024

Viewed by 521

Abstract

The structure and topology of chemical compounds can be determined using chemical graph theory. Using topological indices, we may uncover much about connectivity, complexity, and other important aspects of molecules. Numerous research investigations have been conducted on the K-Banhatti indices and entropy measurements in various fields, including the study of natural polymers, nanotubes, and catalysts. At the same time, the Shannon entropy of a graph is widely used in network science. It is employed in evaluating several networks, including social networks, neural networks, and transportation systems. The Shannon entropy enables the analysis of a network’s topology and structure, facilitating the identification of significant nodes or structures that substantially impact network operation and stability. In the past decade, there has been a considerable focus on investigating a range of nanostructures, such as nanosheets and nanoparticles, in both experimental and theoretical domains. As a very effective catalyst and inert substrate, the

M g O

nanostructure has received a lot of interest. The primary objective of this research is to study different indices and employ them to look at entropy measures of magnesium oxide(111) nanosheets over a wide range of p values, including

p = 1, 2, 3, \dots, j

. Additionally, we conducted a linear regression analysis to establish the correlation between indices and entropies. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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21 pages, 337 KiB

Open AccessArticle

New Spectral Results for Laplacian Harary Matrix and the Harary Laplacian-Energy-like Applying a Matrix Order Reduction

by Luis Medina, Jonnathan Rodríguez and Macarena Trigo

Mathematics 2024, 12(1), 2; https://0-doi-org.brum.beds.ac.uk/10.3390/math12010002 - 19 Dec 2023

Viewed by 581

Abstract

In this paper, we introduce the concepts of Harary Laplacian-energy-like for a simple undirected and connected graph G with order n. We also establish novel matrix results in this regard. Furthermore, by employing matrix order reduction techniques, we derive upper and lower bounds utilizing existing graph invariants and vertex connectivity. Finally, we characterize the graphs that achieve the aforementioned bounds by considering the generalized join operation of graphs. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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14 pages, 423 KiB

Open AccessArticle

Spatio–Spectral Limiting on Replacements of Tori by Cubes

by Jeffrey A. Hogan and Joseph D. Lakey

Mathematics 2023, 11(23), 4714; https://0-doi-org.brum.beds.ac.uk/10.3390/math11234714 - 21 Nov 2023

Viewed by 554

Abstract

A class of graphs is defined in which each vertex of a discrete torus is replaced by a Boolean hypercube in such a way that vertices in a fixed subset of each replacement cube are adjacent to corresponding vertices of a neighboring replacement cube. Bases of eigenvectors of the Laplacians of the resulting graphs are described in a manner suitable for quantifying the concentration of a low-spectrum vertex function on a single vertex replacement. Functions that optimize this concentration on these graphs can be regarded as analogues of Slepian prolate functions that optimize concentration of a bandlimited signal on an interval in the classical setting of the real line. Comparison to the case of a simple discrete cycle shows that replacement allows for higher concentration. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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18 pages, 336 KiB

Open AccessArticle

Spectral Conditions, Degree Sequences, and Graphical Properties

by Xiao-Min Zhu, Weijun Liu and Xu Yang

Mathematics 2023, 11(20), 4264; https://0-doi-org.brum.beds.ac.uk/10.3390/math11204264 - 12 Oct 2023

Cited by 1 | Viewed by 543

Abstract

Integrity, tenacity, binding number, and toughness are significant parameters with which to evaluate network vulnerability and stability. However, we hardly use the definitions of these parameters to evaluate directly. According to the methods, concerning the spectral radius, we show sufficient conditions for a graph to be k-integral, k-tenacious, k-binding, and k-tough, respectively. In this way, the vulnerability and stability of networks can be easier to characterize in the future. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

23 pages, 4440 KiB

Open AccessArticle

Evaluation of Various Topological Indices of Flabellum Graphs

by Xiaolong Shi, Saeed Kosari, Uzma Ahmad, Saira Hameed and Sadia Akhter

Mathematics 2023, 11(19), 4167; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194167 - 05 Oct 2023

Viewed by 830

Abstract

Graph theory serves as an engaging arena for the investigation of proof methods within the field of discrete mathematics, and its findings find practical utility in numerous scientific domains. Chemical graph theory is a specialized branch of mathematics that uses graphs to represent and analyze the structure and properties of chemical compounds. Topological indices are mathematical properties of graphs that play a crucial role in chemistry. They provide a unique way to connect the structural characteristics of chemical compounds to their corresponding molecular graphs. The flabellum graph

F_{n}^{(k, j)}

is obtained with the help of

k \geq 2

duplicates of the cycle graph

C_{n}

with a common vertex (known as, central vertex). Then, in j of these duplicates, additional edges are added, joining the central vertex to all non-adjacent vertices. In this article, we compute different degree-based topological indices for flabellum graphs, including some well known indices, such as the Randić index, the atom bond connectivity index, the geometric–arithmetic index, and the Zagreb indices. This research provides an in-depth examination of these specific indices within the context of flabellum graphs. Moreover, the behavior of these indices is shown graphically, in terms of the parameters

j, k

, and n. Additionally, we have extended the concept of the first Zagreb index, to address the issue of cybercrime. This application enables us to identify criminals who exhibit higher levels of activity and engagement in multiple criminal activities when compared to their counterparts. Furthermore, we conducted a comprehensive comparative analysis of the first Zagreb index against the closeness centrality measure. This analysis sheds light on the effectiveness and relevance of the topological index in the context of cybercrime detection and network analysis. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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10 pages, 993 KiB

Open AccessArticle

Surface Family with Bertrand Curves as Joint Asymptotic Curves in 3D Galilean Space

by Awatif Al-Jedani and Rashad A. Abdel-Baky

Mathematics 2023, 11(19), 4100; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194100 - 28 Sep 2023

Viewed by 584

Abstract

The primary objective of this work is to discuss a surface family with the similarity of Bertrand curves in 3D Galilean space. Subsequently, by applying the Serret–Frenet frame, we estimate the sufficient and necessary statuses of a surface family with Bertrand curves as joint asymptotic curves. The dilation to ruled surfaces is also summarized. Meanwhile, the epitomes are illustrated to provide an explanation of the theoretical results. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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11 pages, 577 KiB

Open AccessArticle

On the Structure of the Mislin Genus of a Pullback

by Thandile Tonisi, Rugare Kwashira and Jules C. Mba

Mathematics 2023, 11(12), 2672; https://0-doi-org.brum.beds.ac.uk/10.3390/math11122672 - 12 Jun 2023

Viewed by 1001

Abstract

The notion of genus for finitely generated nilpotent groups was introduced by Mislin. Two finitely generated nilpotent groups Q and R belong to the same genus set

G (Q)

if and only if the two groups are nonisomorphic, but for each [...] Read more.

The notion of genus for finitely generated nilpotent groups was introduced by Mislin. Two finitely generated nilpotent groups Q and R belong to the same genus set

G (Q)

if and only if the two groups are nonisomorphic, but for each prime p, their p-localizations

Q_{p}

and

R_{p}

are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group Q has a finite commutator subgroup. In this study, we consider the class of finitely generated infinite nilpotent groups with a finite commutator subgroup. We construct a pullback

H_{t}

from the l-equivalences

H_{i} \to H

and

H_{j} \to H

t \equiv (i + j) m o d s

, where

s = | G (H) |

, and compare its genus to that of H. Furthermore, we consider a pullback L of a direct product

G \times K

of groups in this class. Here, we prove results on the group L and prove that its genus is nontrivial. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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16 pages, 2657 KiB

Open AccessArticle

Topological Properties and Entropy Calculations of Aluminophosphates

by Jeyaraj Sahaya Vijay, Santiago Roy, Bheeter Charles Beromeo, Mohamad Nazri Husin, Tony Augustine, R.U. Gobithaasan and Michael Easuraja

Mathematics 2023, 11(11), 2443; https://0-doi-org.brum.beds.ac.uk/10.3390/math11112443 - 25 May 2023

Cited by 8 | Viewed by 1068

Abstract

Topological indices are invariant numerical quantities of a graph that give facts about the structure of graphs and are found to be very helpful in predicting the physical properties of aluminophosphates. The characteristics of aluminophosphates are similar to the characteristics of zeolites. Two examples of current applications are natural gas dehydration and humidity sensors. Researchers in chemistry and materials science are synthesizing new frameworks. There are many layers and holes in these substances. The technique used to predict natural behaviors among the physicochemical characteristics of chemical molecules in their basic network is known as topological indices. This study explains the vertex version of distance-based topological indices, the entropy of topological indices and their numerical analysis. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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5 pages, 882 KiB

Open AccessArticle

Note on Discovering Doily in PG(2,5)

by Stefano Innamorati

Mathematics 2023, 11(9), 2210; https://0-doi-org.brum.beds.ac.uk/10.3390/math11092210 - 08 May 2023

Viewed by 740

Abstract

W. L. Edge proved that the internal points of a conic in PG(2,5), together with the collinear triples on the non-secant lines, form the Desargues configuration. M. Saniga showed an intimate connection between Desargues configurations and the generalized quadrangles of order 2, GQ(2,2), whose representation was dubbed “the doily” by Stan Payne in 1973. In this note, we prove that the external points of a conic in PG(2,5), together with the collinear and non-collinear triples on the non-tangent lines, form the generalized quadrangle of order 2. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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12 pages, 263 KiB

Open AccessArticle

Some New Bounds for α-Adjacency Energy of Graphs

by Haixia Zhang and Zhuolin Zhang

Mathematics 2023, 11(9), 2173; https://0-doi-org.brum.beds.ac.uk/10.3390/math11092173 - 05 May 2023

Viewed by 861

Abstract

Let G be a graph with the adjacency matrix

A (G)

, and let

D (G)

be the diagonal matrix of the degrees of G. Nikiforov first defined the matrix

A_{α} (G)

as [...] Read more.

Let G be a graph with the adjacency matrix

A (G)

, and let

D (G)

be the diagonal matrix of the degrees of G. Nikiforov first defined the matrix

A_{α} (G)

A_{α} (G) = α D (G) + (1 - α) A (G)

0 \leq α \leq 1

, which shed new light on

A (G)

and

Q (G) = D (G) + A (G)

, and yielded some surprises. The

α -

adjacency energy

E^{A_{α}} (G)

of G is a new invariant that is calculated from the eigenvalues of

A_{α} (G)

. In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for

E^{A_{α}} (G)

in terms of graph parameters (the order n, the edge size m, etc.) and characterize the corresponding extremal graphs. In addition, we obtain some relations between

E^{A_{α}} (G)

and other energies such as the energy

E (G)

. Some results can be applied to appropriately estimate the

α

-adjacency energy using some given graph parameters rather than by performing some tedious calculations. Full article

(This article belongs to the Special Issue Discrete Mathematics, Graph Theory and Applications)

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