Abstract
Based on the extended adjacency matrix, the spectral radius and the energy of the extended adjacency matrix are found that they possess high discriminating power and correlate well with a number of physicochemical properties and biological activities of organic compounds. In this research, by establishing a relationship between directed graphs and undirected graphs, we mainly present the Sachs theorem and the Coulson’s integral formula of the energy of the extended adjacency matrix of G. By using the formula, we can compare the energies of the extended adjacency matrices of two graphs and we find that the graph energy is less than the extended energy in the case of a tree.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Enquiries about data availability should be directed to the authors.
Change history
08 April 2024
This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1007/s10878-024-01156-4
References
Balakrishnan R (2004) The energy of a graph. Linear Algebra Appl 387:287–295
Bhat MA (2017) Energy of weighted digraphs. Discret Appl Math 223:1–14
Bondy JA, Murty USR (2008) Graph theory. Springer, New York
Chen L, Shi Y (2015) Maximal matching energy of tricyclic graphs. MATCH Commun Math Comput Chem 73:105–119
Cvetković DM, Doob M, Sachs H (1980) Spectra of graphs: theory and application. Academic Press, New York
Das KC, Sorgun S (2014) On Randić energy of graphs. MATCH Commun Math Comput Chem 72:227–238
Das KC, Sorgun S, Gutman I (2015) On Randić energy. MATCH Commun Math Comput Chem 73:81–92
Das KC, Gutman I, Furtula B (2017) On spectral radius and energy of extended adjacency matrix of graphs. Appl Math Comput 296:116–123
Deng B, Li X (2021) Energies for the complements of borderenergetic graphs. MATCH Commun Math Comput Chem 85:181–194
Deng B, Li X, Zhao H (2019) (Laplacian) borderenergetic graphs and bipartite graphs. MATCH Commun Math Comput Chem 82:481–489
Gutman I (1978) The energy of a graph. Ber Math Statist Statist Sekt Forschungsz Graz 103:1–22
Gutman I, Li X (2016) Graph energies-theory and applications. University of Kragujevac, Kragujevac
Gutman I, Zhou B (2006) Laplacian energy of a graph. Linear Algebra Appl 414:29–37
Gutman I, Furtula B, Das KC (2017) Extended energy and its dependence on molecular structure. Can J Chem 95(5):1–4
Li X, Shi YT, Gutman I (2012) Graph energy. Springer, New York
Lv X, Deng B, Li X (2021) Laplacian borderenergetic graphs and their complements. MATCH Commun Math Comput Chem 86:587–596
Lv X, Deng B, Li X (2022) On the borderenergeticity of line graphs. MATCH Commun Math Comput Chem 87:693–702
Modanli M, Faraj BM, Ahmed FW (2020) Using matrix stability for variable telegraph partial differential equation. Int J Optim Control Theor Appl. https://doi.org/10.11121/ijocta.01.2020.00870
Wang Z, Mao Y, Furtula B, Wang X (2021) Bounds for the spectral radius and energy of extended adjacency matrix of graphs. Linear Multilinear Algebra 69(10)
Xu K, Zheng Z, Das KC (2015) Extremal \(t\)-apex trees with respect to matching energy. Complexity 21(5):238–247
Yang YQ, Xu L, Hu CY (1994) Extended adjacency matrix indices and their applications. J Chem Inf Comput Sci 34(5):1140–1145
Zhou B (2004) Energy of a graph. MATCH Commun Math Comput Chem 51:111–118
Acknowledgements
This research was funded by Qinghai Office of Science and Technology (Grant No. 2022-ZJ-T02), The Key Laboratory of Tibetan Information Processing Ministry of Education, Tibetan Information Processing Engineering Technology and Research Center of Qinghai Province, The 111 Project (D20035), NSFC No.12261073; K. C. Das is supported by National Research Foundation funded by the Korean government (Grant No. 2021R1F1A1050646).
Funding
This work was funded by Qinghai Office of Science and Technology (Grant No. 2022-ZJ-T02).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article has been retracted. Please see the retraction notice for more detail:https://doi.org/10.1007/s10878-024-01156-4
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Deng, B., Chang, C. & Das, K.C. RETRACTED ARTICLE: The Sachs theorem and its application on extended adjacency matrix of graphs. J Comb Optim 45, 23 (2023). https://doi.org/10.1007/s10878-022-00938-y
Accepted:
Published:
DOI: https://doi.org/10.1007/s10878-022-00938-y