Symmetric Matrices of Graphs: Topics and Advances

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 3899

Special Issue Editors

Associate Professor, Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, United Arab Emirates
Interests: distance in graphs; graph labeling; chemical graph theory; spectral graph theory
Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, United Arab Emirates
Interests: spectral properties of graphs matrices; algebraic graphs of groups and rings

Special Issue Information

Dear Colleagues,

For a graph, G, a general vertex-degree-based topological index, Φ, is defined as \({{\phi}(G) = \sum_{u v \in E(G)}}{\phi_{d_{u} d_{v}}}\), where \({\phi_{d_{u} d_{v}}}\) is a symmetric function (\({\phi_{d_{u} d_{v}}=\phi_{d_{v} d_{u}}}\)). For particular values of \({\phi_{d_{u} d_{v}}}\), we obtain well-known topological indices, such as the arithmetic–geometric index \({\phi_{d_{u} d_{v}}={\frac{d_{u}+d_{v}}{2 \sqrt{d_{u} d_{v}}}}}\), the general Randic index \({\phi_{d_{u} d_{v}}=({d_{u} d_{v}})^α}\) (for \({α={\frac{1}{2}}}\), we obtain the ordinary Randic index \({R=\sum_{u v \in E(G)}\frac{1}{\sqrt{d_{u} d_{v}}}}\)), the general Sombor index \({\phi_{d_{u} d_{v}}=(d_{u}^2+d_{v}^2)^α}\), and several other indices.

The general adjacency matrix associated with the Φ of G is a real symmetric matrix, defined as \begin{eqnarray} A_{\phi}(G) & = & \left(a_{\phi}\right)_{i j} & = & \left\{\begin{array}{c} \phi_{d_{u} d_{v}}~if~u v \in E(G) \\ 0 ~otherwise~\end{array}\right. \end{eqnarray}.

The set of all the eigenvalues of \({A_{\phi}(G)}\) is known as the general adjacency spectrum of G and is denoted by λ1(AΦ(G)) ≥ λ2(AΦ(G)) ≥... ≥ λn(AΦ(G)), where λ1(AΦ(G)) is the general adjacency spectral radius of G. For bipartite graphs its spectrum is symmetric towards the origin. The energy of \({A_{\phi}(G)}\) associated with the topological index, Φ, is defined as \({\varepsilon_{\phi}(G)=\sum_{i=1}^{n}\left|\lambda_{i}\left(\mathrm{~A}_{\phi}(\mathrm{G})\right)\right|}\).

If \({\phi_{d_{u} d_{v}}=1}\), when u is adjacent to v, then \({A_{\phi}(G)}\) is the much-studied adjacency A(G) matrix and εΦ(G) is the usual graph energy \({\varepsilon(G)=\sum_{i=1}^{n}\left|\lambda_{i}\left(\mathrm{~A}(\mathrm{G})\right)\right|}\), where λ1(A(G)) ≥ λ2(A(G)) ≥... ≥ λn(A(G)) are the eigenvalues of A(G) and is known as the spectrum of G. Similarly, putting particular values of \({\phi_{d_{u} d_{v}}}\), we obtain various well-known matrices such as the arithmetic–geometric matrix, Somber matrix, ABC matrix, and others. Mathematical descriptors of molecular graphs, such as topological indices, have several uses in chemical studies. They play a very crucial role in theoretical chemistry, especially in quantitative structure–activity relationship (QSAR)- and quantitative structure–property relationship (QSPR)-related studies. The distance between two vertices in a graph is the length of the shortest path connecting them, and this distance satisfied the famous symmetric property of a metric space in addition to giving rise to various types of symmetric matrices, including the distance matrix, eccentricity matrix, and their variations.

Dr. Muhammad Imran
Dr. Bilal Ahmad Rather
Guest Editors

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Keywords

  • topological indices
  • symmetric graph matrices
  • graph energy
  • distance
  • adjacency eigenvalues
  • laplacian energy
  • QSPR/QSAR
  • distance energy
  • extremal graphs
  • graph spectrum

Published Papers (3 papers)

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18 pages, 1914 KiB  
Article
Spectral Characterization of Graphs with Respect to the Anti-Reciprocal Eigenvalue Property
by Hao Guan, Aysha Khan, Sadia Akhter and Saira Hameed
Symmetry 2023, 15(6), 1240; https://doi.org/10.3390/sym15061240 - 10 Jun 2023
Cited by 2 | Viewed by 960
Abstract
Let G=(V,E) be a simple connected graph with vertex set V and edge set E, respectively. The term “anti-reciprocal eigenvalue property“ refers to a non-singular graph G for which, [...] Read more.
Let G=(V,E) be a simple connected graph with vertex set V and edge set E, respectively. The term “anti-reciprocal eigenvalue property“ refers to a non-singular graph G for which, 1λσ(G), whenever λσ(G), λσ(G). Here, σ(G) is the multiset of all eigenvalues of A(G). Moreover, if multiplicities of eigenvalues and their negative reciprocals are equal, then that graph is said to have strong anti-reciprocal eigenvalue properties, and the graph is referred to as a strong anti-reciprocal graph (or (SR) graph). In this article, a new family of graphs Fn(k,j) is introduced and the energy of F5(k,k2)k2 is calculated. Furthermore, with the help of F5(k,k2), some families of (SR) graphs are constructed. Full article
(This article belongs to the Special Issue Symmetric Matrices of Graphs: Topics and Advances)
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24 pages, 5133 KiB  
Article
Ordering Unicyclic Connected Graphs with Girth g ≥ 3 Having Greatest SK Indices
by Wang Hui, Adnan Aslam, Salma Kanwal, Mahnoor Akram, Tahira Sumbal Shaikh and Xuewu Zuo
Symmetry 2023, 15(4), 871; https://doi.org/10.3390/sym15040871 - 06 Apr 2023
Viewed by 1056
Abstract
For a graph, the SK index is equal to the half of the sum of the degrees of the vertices, the SK1 index is equal to the half of the product of the degrees of the vertices, and the SK2 index [...] Read more.
For a graph, the SK index is equal to the half of the sum of the degrees of the vertices, the SK1 index is equal to the half of the product of the degrees of the vertices, and the SK2 index is equal to the half of the square of the sum of the degrees of the vertices. This paper shows a simple and unified approach to the greatest SK indices for unicyclic graphs by using some transformations and characterizes these graphs with the first, second, and third SK indices having order r ≥ 5 and girth g ≥ 3, where girth is the length of the shortest cycle in a graph. Full article
(This article belongs to the Special Issue Symmetric Matrices of Graphs: Topics and Advances)
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10 pages, 310 KiB  
Article
Kirchhoff Index and Additive Kirchhoff Index Based on Multiplicative Degree for a Random Polyomino Chain
by Meilian Li, Muhammad Asif, Haidar Ali, Fizza Mahmood and Parvez Ali
Symmetry 2023, 15(3), 718; https://doi.org/10.3390/sym15030718 - 14 Mar 2023
Cited by 1 | Viewed by 1006
Abstract
Several topological indices are known to have widespread implications in a variety of research areas. Over the years, the Kirchhoff index has turned out to be an extremely significant and efficient index. In this paper, we propose the exact formulas for the expected [...] Read more.
Several topological indices are known to have widespread implications in a variety of research areas. Over the years, the Kirchhoff index has turned out to be an extremely significant and efficient index. In this paper, we propose the exact formulas for the expected values of the random polyomino chain to construct the multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index. We also carefully examine the highest degree of the expected values for a random polyomino chain through the multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index. Full article
(This article belongs to the Special Issue Symmetric Matrices of Graphs: Topics and Advances)
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