Some relations between energy and Seidel energy of a graph

Harishchandra S. Ramane; B. Parvathalu; K. Ashoka; Daneshwari Patil

doi:10.2478/ausi-2023-0005

Abstract

The energy E(G) of a graph G is the sum of the absolute values of eigenvalues of G and the Seidel energy E_S(G) is the sum of the absolute values of eigenvalues of the Seidel matrix S of G. In this paper, some relations between the energy and Seidel energy of a graph in terms of different graph parameters are presented. Also, the inertia relations between the graph eigenvalue and Seidel eigenvalue of a graph are given. The results in this paper generalize some of the existing results.

References

S. Akbari, J. Askari, K. C. Das, Some properties of eigenvalues of the Seidel matrix, Linear Multilinear Algebra (2020) 1–12. DOI:10.1080/03081087.2020.1790481. ⇒48, 51
Open DOI Search in Google Scholar Back to article
S. Akbari, M. Einollahzadeh, M. M. Karkhaneei, M. A. Nematollahi, Proof of a conjecture on the Seidel energy of graphs, European J. Combin. 86 (2020) 103078. ⇒48
Search in Google Scholar Back to article
A. E. Brouwer, W. H. Haemers, Spectra of Graphs, Springer, Berlin, 2012. ⇒48
Search in Google Scholar Back to article
K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci USA. 37 (1951) 760–766. ⇒49
Search in Google Scholar Back to article
M. Ghorbani, M. Hakimi-Nezhaad, B. Zhou, Graphs with at most four Seidel eigenvalues, Kragujevac J. Math. 47 (2023) 173–186. ⇒53
Search in Google Scholar Back to article
I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, 2012. ⇒47
Search in Google Scholar Back to article
I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz. 103 (1978) 1–22. ⇒47
Search in Google Scholar Back to article
W. H. Haemers, Seidel switching and graph energy, MATCH Commun. Math. Comput. Chem. 68, 3 (2012) 653-659. ⇒47
Search in Google Scholar Back to article
F. Harary, Graph Theory, Narosa Publishing House, New Delhi, 1998. ⇒48
Search in Google Scholar Back to article
R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2012. ⇒48
Search in Google Scholar Back to article
G. Indulal, Distance equienergetic graphs, International Conference on Graph Connections (ICGC)-2020, Bishop Chulaparambil Memorial College and Mahatma Gandhi University. https://icgc2020.wordpress.com/invitedlectures (2020). Accessed 8 August 2020. ⇒47
Search in Google Scholar Back to article
X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. ⇒47
Search in Google Scholar Back to article
H. S. Ramane, K. Ashoka, D. Patil, B. Parvathalu, On the Seidel Laplacian and Seidel signless Laplacian polynomials of graphs, Kyungpook Math. J. 61 (2021) 155–168. ⇒48
Search in Google Scholar Back to article
H. S. Ramane, K. Ashoka, B. Parvathalu, D. Patil, On A-energy and S-energy of certain class of graphs, Acta Univ. Sapientiae Informatica 13, 2 (2021) 195–219. DOI:10.2478/ausi-2021-0009 ⇒48
Open DOI Search in Google Scholar Back to article
H. S. Ramane, B. Parvathalu, K. Ashoka, Seidel energy of k-fold and strong k-fold graphs, Commun. Combin., Cryptogr. & Computer Sci. 1(2021) 152–159. DOI:10.5666/KMJ.0000.00.0.000. ⇒48
Open DOI Search in Google Scholar Back to article
H. S. Ramane, B. Parvathalu, K. Ashoka, S. Pirzada, On families of graphs which are both adjacency equienergetic and distance equienergetic, Indian J Pure Appl Math. DOI:10.1007/s13226-022-00355-1. ⇒47
Open DOI Search in Google Scholar Back to article
H. S. Ramane, B. Parvathalu, D. Patil, K. Ashoka, Iterated line graphs with only negative eigenvalues −2, their complements and energy, Preprint (2022). DOI:10.48550/arXiv.2205.02276. ⇒52
Open DOI Search in Google Scholar Back to article
H. S. Ramane, M. M. Gundloor, S. M. Hosamani, Seidel equienergetic graphs, Bull. Math. Sci. Appl. 16 (2016) 62–69. ⇒48
Search in Google Scholar Back to article
H. S. Ramane, I. Gutman, M. M. Gundloor, Seidel energy of iterated line graphs of regular graphs, Kragujevac J. Math., 39, 1 (2015), 7–12. ⇒52
Search in Google Scholar Back to article
H. S. Ramane, H. B. Walikar, S. B. Rao, B. D. Acharya, P. R. Hampiholi, S. R. Jog, I. Gutman, Spectra and energies of iterated line graphs of regular graphs, Appl. Math. Lett. 18 (2005) 679–682. ⇒52
Search in Google Scholar Back to article
J. J. Seidel, Strongly regular graphs with (−1, 1, 0) adjacency matrix having eigenvalue 3, Linear Algebra Appl. 1 (1968) 281–298. ⇒47
Search in Google Scholar Back to article
G. X. Tian, Y. Li, S. Y. Cui, The change of Seidel energy of tripartite Turán graph due to edge deletion, Linear Multilinear Algebra (2021) 1–18. DOI:10.1080/03081087.2021.1888858. ⇒48
Open DOI Search in Google Scholar Back to article
S. K. Vaidya, K. M. Popat, Some new results on Seidel equienergetic graphs, Kyungpook Math. J. 59, 2 (2019) 335—340. ⇒48
Search in Google Scholar Back to article
J. H. van Lint, J. J. Seidel, Equilateral point sets in elliptic geometry, Indag Math. 28 (1966) 335–348. ⇒47
Search in Google Scholar Back to article
B. Zhou, I. Gutman, Nordhaus-Gaddum type relations for the energy and Laplacian energy of graphs, Bull. Cl. Sci. Math. Nat. Sci. Math. (2007) 1–11. DOI: 10.1071/ASEG2007ab174. ⇒49
Open DOI Search in Google Scholar Back to article

Some relations between energy and Seidel energy of a graph

Abstract

Paradigm

My account