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 6242
Special Issue Editors
Interests: distance in graphs; graph labeling; chemical graph theory; spectral graph theory
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
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