On the spectrum, energy and Laplacian energy of graphs with self-loops
The total energy of a conjugated molecule's π-electrons is a quantum-theoretical feature that has been known since the 1930s. It is determined using the Huckel tight-binding molecular orbital (HMO) method. In 1978, a modified definition of the total π-electron energy was introduced, which is no...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Elsevier
2023-07-01
|
| Series: | Heliyon |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2405844023042081 |
| _version_ | 1827892382678384640 |
|---|---|
| author | Ugasini Preetha P M. Suresh Ebenezer Bonyah |
| author_facet | Ugasini Preetha P M. Suresh Ebenezer Bonyah |
| author_sort | Ugasini Preetha P |
| collection | DOAJ |
| description | The total energy of a conjugated molecule's π-electrons is a quantum-theoretical feature that has been known since the 1930s. It is determined using the Huckel tight-binding molecular orbital (HMO) method. In 1978, a modified definition of the total π-electron energy was introduced, which is now known as the graph energy. It is calculated by summing the absolute values of the eigenvalues of the adjacency matrix. Quiet Recently in the year 2022, Gutman extended the concept of conjugated systems to hetero-conjugated systems which is the extension of ordinary graph energy to energy of graph with self loops. Let Gσ be an order (vertices) ‘p’ graph with ‘q’ edges and σ− self loops. The adjacency matrix of Gσ is defined by A(Gσ)=(aij) if vi∼adjvj then aij=1; if vi=vj where vi∈Vσ then aii=1 and zero otherwise, where Vσ represents the set of all vertices with loops. Then the energy of graph with self loops is defined as E(Gσ)=∑|λi−σ/p|. In this paper, we aim to analyze the adjacency and Laplacian spectra of certain non-simple standard graphs that contain self-loops. We also calculate the energy and Laplacian energy of these graphs with loops. Furthermore, we derive lower bounds for the energy of any graph containing loops and develop a MATLAB algorithm to calculate these quantities for selected non-simple standard graphs with self-loops. Our study evaluates the strength of a graph by considering the presence of loops, which are edges that connect a vertex to itself. This approach accounts for the impact of each vertex on the entire structure of the graph. By analyzing the energy of a graph with loops, we can gain a better understanding of its distinctive characteristics and behavior. |
| first_indexed | 2024-03-12T21:39:58Z |
| format | Article |
| id | doaj.art-f67b2406769249a3a4c5b779508f49c7 |
| institution | Directory Open Access Journal |
| issn | 2405-8440 |
| language | English |
| last_indexed | 2024-03-12T21:39:58Z |
| publishDate | 2023-07-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Heliyon |
| spelling | doaj.art-f67b2406769249a3a4c5b779508f49c72023-07-27T05:56:16ZengElsevierHeliyon2405-84402023-07-0197e17001On the spectrum, energy and Laplacian energy of graphs with self-loopsUgasini Preetha P0M. Suresh1Ebenezer Bonyah2Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur 603203, Tamilnadu, IndiaDepartment of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur 603203, Tamilnadu, IndiaDepartment of Mathematics Education, Akenten Appiah Menka University of Skills Training and Entrepreneurial Development, Kumasi, Ghana; Corresponding author.The total energy of a conjugated molecule's π-electrons is a quantum-theoretical feature that has been known since the 1930s. It is determined using the Huckel tight-binding molecular orbital (HMO) method. In 1978, a modified definition of the total π-electron energy was introduced, which is now known as the graph energy. It is calculated by summing the absolute values of the eigenvalues of the adjacency matrix. Quiet Recently in the year 2022, Gutman extended the concept of conjugated systems to hetero-conjugated systems which is the extension of ordinary graph energy to energy of graph with self loops. Let Gσ be an order (vertices) ‘p’ graph with ‘q’ edges and σ− self loops. The adjacency matrix of Gσ is defined by A(Gσ)=(aij) if vi∼adjvj then aij=1; if vi=vj where vi∈Vσ then aii=1 and zero otherwise, where Vσ represents the set of all vertices with loops. Then the energy of graph with self loops is defined as E(Gσ)=∑|λi−σ/p|. In this paper, we aim to analyze the adjacency and Laplacian spectra of certain non-simple standard graphs that contain self-loops. We also calculate the energy and Laplacian energy of these graphs with loops. Furthermore, we derive lower bounds for the energy of any graph containing loops and develop a MATLAB algorithm to calculate these quantities for selected non-simple standard graphs with self-loops. Our study evaluates the strength of a graph by considering the presence of loops, which are edges that connect a vertex to itself. This approach accounts for the impact of each vertex on the entire structure of the graph. By analyzing the energy of a graph with loops, we can gain a better understanding of its distinctive characteristics and behavior.http://www.sciencedirect.com/science/article/pii/S2405844023042081Complete graphStar graphSpectrumEnergyLaplacian energySelf loops |
| spellingShingle | Ugasini Preetha P M. Suresh Ebenezer Bonyah On the spectrum, energy and Laplacian energy of graphs with self-loops Heliyon Complete graph Star graph Spectrum Energy Laplacian energy Self loops |
| title | On the spectrum, energy and Laplacian energy of graphs with self-loops |
| title_full | On the spectrum, energy and Laplacian energy of graphs with self-loops |
| title_fullStr | On the spectrum, energy and Laplacian energy of graphs with self-loops |
| title_full_unstemmed | On the spectrum, energy and Laplacian energy of graphs with self-loops |
| title_short | On the spectrum, energy and Laplacian energy of graphs with self-loops |
| title_sort | on the spectrum energy and laplacian energy of graphs with self loops |
| topic | Complete graph Star graph Spectrum Energy Laplacian energy Self loops |
| url | http://www.sciencedirect.com/science/article/pii/S2405844023042081 |
| work_keys_str_mv | AT ugasinipreethap onthespectrumenergyandlaplacianenergyofgraphswithselfloops AT msuresh onthespectrumenergyandlaplacianenergyofgraphswithselfloops AT ebenezerbonyah onthespectrumenergyandlaplacianenergyofgraphswithselfloops |