Spectral Graph Theory, Molecular Graph Theory and Their Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 7189

Special Issue Editor

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Interests: spectral graph theory; extrema lGraph theory; molecular graph theory; graph labeling; graph algorithm; discrete mathematics; combinatorics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Graph Theory is the area of mathematics that studies networks or graphs. It arose from the need to analyze many diverse network-like structures like the internet, molecules, road networks, social networks, education networks, and electrical networks. Spectral graph theory (a branch of graph theory) deals with the connection between the eigenvalues of a matrix associated with a graph and the corresponding structure of the graph. The first practical need for studying graph eigenvalues was in quantum chemistry in the nineteen-thirties, forties, and fifties, specifically to describe the Hückel molecular orbital theory for unsaturated conjugated hydrocarbons. Several special kinds of graph matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix, distance matrix, etc.) are very popular in spectral graph theory. 

A graphical invariant is a function from the set of graphs to the set of reals which is invariant under graph automorphisms. In chemical graph theory, graphical invariants are most often referred to as topological indices. Among the oldest and most well-known topological indices, Wiener index, Randić index, and Zagreb indices, etc. were introduced in the literature. In the last twenty years, a large number of mathematical investigations were reported on graph invariants (topological indices) whose origin is in chemistry, and which are claimed to have chemical applications.

The aim of this Special Issue is to publish original research articles focusing on spectral graph theory and the topological indices of graphs, as well as applications in these areas. To find new connections between the eigenvalues and eigenvectors of graphs and graph theory, parameters such as average degree, maximum degree, cliques, chromatic number, and matching number are welcome. In addition, this Special Issue covers the characterization of extremal graphs with respect to the existing popular topological indices of graphs with fixed graph parameters. 

Prof. Dr. Kinkar Chandra Das
Guest Editor

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Keywords

  • spectral graph theory
  • topological indices of graphs
  • energy of graphs
  • combinatorics
  • extremal graph theory
  • graph polynomials

Published Papers (6 papers)

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Research

16 pages, 790 KiB  
Article
Eigenvalue −1 of the Vertex Quadrangulation of a 4-Regular Graph
by Vladimir R. Rosenfeld
Axioms 2024, 13(1), 72; https://doi.org/10.3390/axioms13010072 - 22 Jan 2024
Viewed by 881
Abstract
The vertex quadrangulation QG of a 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. In a previous work [JOMC 59, 1551–1569 (2021)], the [...] Read more.
The vertex quadrangulation QG of a 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. In a previous work [JOMC 59, 1551–1569 (2021)], the question was posed: does the spectrum of an arbitrary unweighted graph QG include the full spectrum {3,(1)3} of the tetrahedron graph (complete graph K4)? Previously, many bipartite and nonbipartite graphs QG with such a subspectrum have been found; for example, a nonbipartite variant of the graph QK5. Here, we present one of the variants of the nonbipartite vertex quadrangulation QO of the octahedron graph O, which has eigenvalue (1) of multiplicity 2 in the spectrum, while the spectrum of the bipartite variant QO contains eigenvalue (1) of multiplicity 3. Thus, in the case of nonbipartite graphs, the answer to the question posed depends on the particular graph QG. Here, we continue to explore the spectrum of graphs QG. Some possible connections of the mathematical theme to chemistry are also noted. Full article
(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)
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18 pages, 713 KiB  
Article
The Multivariable Zhang–Zhang Polynomial of Phenylenes
by Niko Tratnik
Axioms 2023, 12(11), 1053; https://doi.org/10.3390/axioms12111053 - 15 Nov 2023
Cited by 1 | Viewed by 788
Abstract
The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of [...] Read more.
The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of two variables was defined such that it also takes into account 10-cycles of a benzenoid system. The aim of this paper is to introduce and study a new variation of the Zhang–Zhang polynomial for phenylenes, which are important molecular graphs composed of 6-membered and 4-membered rings. In our case, Clar covers can contain 4-cycles, 6-cycles, 8-cycles, and edges. Since this new polynomial has three variables, we call it the multivariable Zhang–Zhang (MZZ) polynomial. In the main part of the paper, some recursive formulas for calculating the MZZ polynomial from subgraphs of a given phenylene are developed and an algorithm for phenylene chains is deduced. Interestingly, computing the MZZ polynomial of a phenylene chain requires some techniques that are different to those used to calculate the (generalized) Zhang–Zhang polynomial of benzenoid chains. Finally, we prove a result that enables us to find the MZZ polynomial of a phenylene with branched hexagons. Full article
(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)
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20 pages, 436 KiB  
Article
Conversion of Unweighted Graphs to Weighted Graphs Satisfying Properties R and SR
by Xiaolong Shi, Saira Hameed, Sadia Akhter, Aysha Khan and Maryam Akhoundi
Axioms 2023, 12(11), 1043; https://doi.org/10.3390/axioms12111043 - 09 Nov 2023
Viewed by 845
Abstract
Spectral graph theory is like a special tool for understanding graphs. It helps us find patterns and connections in complex networks, using the magic of eigenvalues. Let G be the graph and A(G) be its adjacency matrix, then G is [...] Read more.
Spectral graph theory is like a special tool for understanding graphs. It helps us find patterns and connections in complex networks, using the magic of eigenvalues. Let G be the graph and A(G) be its adjacency matrix, then G is singular if the determinant of the adjacency matrix A(G) is 0, otherwise it is nonsingular. Within the realm of nonsingular graphs, there is the concept of property R, where each eigenvalue’s reciprocal is also an eigenvalue of G. By introducing multiplicity constraints on both eigenvalues and their reciprocals, it becomes property SR. Similarly, the world of nonsingular graphs reveals property R, where the negative reciprocal of each eigenvalue also finds a place within the spectrum of G. Moreover, when the multiplicity of each eigenvalue and its negative reciprocal is equal, this results in a graph with a property of SR. Some classes of unweighted nonbipartite graphs are already constructed in the literature with the help of the complete graph Kn and a copy of the path graph P4 satisfying property R but not SR. This article takes this a step further. The main aim is to construct several weighted classes of graphs which satisfy property R but not SR. For this purpose, the weight functions are determined that enable these nonbipartite graph classes to satisfy the SR and R properties, even if the unweighted graph does not satisfy these properties. Some examples are presented to support the investigated results. These examples explain how certain weight functions make these special types of graphs meet the properties R or SR, even when the original graphs without weights do not meet these properties. Full article
(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)
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14 pages, 497 KiB  
Article
A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices
by Rashad Ismail, Muhammad Azeem, Yilun Shang, Muhammad Imran and Ali Ahmad
Axioms 2023, 12(7), 675; https://doi.org/10.3390/axioms12070675 - 09 Jul 2023
Cited by 4 | Viewed by 924
Abstract
The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are [...] Read more.
The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for αR+\{1}. Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for αR+\{1}, we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices. Full article
(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)
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12 pages, 999 KiB  
Article
On Graphs with c2-c3 Successive Minimal Laplacian Coefficients
by Yue Xu and Shi-Cai Gong
Axioms 2023, 12(5), 464; https://doi.org/10.3390/axioms12050464 - 11 May 2023
Viewed by 895
Abstract
Let G be a graph of order n and L(G) be its Laplacian matrix. The Laplacian polynomial of G is defined as [...] Read more.
Let G be a graph of order n and L(G) be its Laplacian matrix. The Laplacian polynomial of G is defined as P(G;λ)=det(λIL(G))=i=0n(1)ici(G)λni, where ci(G) is called the i-th Laplacian coefficient of G. Denoted by Gn,m the set of all (n,m)-graphs, in which each of them contains n vertices and m edges. The graph G is called uniformly minimal if, for each i(i=0,1,,n), H is ci(G)-minimal in Gn,m. The Laplacian matrix and eigenvalues of graphs have numerous applications in various interdisciplinary fields, such as chemistry and physics. Specifically, these matrices and eigenvalues are widely utilized to calculate the energy of molecular energy and analyze the physical properties of materials. The Laplacian-like energy shares a number of properties with the usual graph energy. In this paper, we investigate the existence of uniformly minimal graphs in Gn,m because such graphs have minimal Laplacian-like energy. We determine that the c2(G)-c3(G) successive minimal graph is exactly one of the four classes of threshold graphs. Full article
(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)
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13 pages, 305 KiB  
Article
Extremal Graphs to Vertex Degree Function Index for Convex Functions
by Dong He, Zhen Ji, Chenxu Yang and Kinkar Chandra Das
Axioms 2023, 12(1), 31; https://doi.org/10.3390/axioms12010031 - 27 Dec 2022
Cited by 1 | Viewed by 1356
Abstract
The vertex-degree function index Hf(Γ) is defined as Hf(Γ)=vV(Γ)f(d(v)) for a function f(x) defined on non-negative [...] Read more.
The vertex-degree function index Hf(Γ) is defined as Hf(Γ)=vV(Γ)f(d(v)) for a function f(x) defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degree function index in the set of all n-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus–Gaddum-type results for Hf(Γ)+Hf(Γ¯) and Hf(Γ)·Hf(Γ¯). Full article
(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)
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