On the energy and spectral properties of the he matrix of hexagonal systems

Faqir M. Bhatti; Kinkar Ch. Das; Syed A. Ahmed

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 1, page 47-63
  • ISSN: 0011-4642

Abstract

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The He matrix, put forward by He and He in 1989, is designed as a means for uniquely representing the structure of a hexagonal system (= benzenoid graph). Observing that the He matrix is just the adjacency matrix of a pertinently weighted inner dual of the respective hexagonal system, we establish a number of its spectral properties. Afterwards, we discuss the number of eigenvalues equal to zero of the He matrix of a hexagonal system. Moreover, we obtain a relation between the number of triangles and the eigenvalues of the He matrix of a hexagonal system. Finally, we present an upper bound on the He energy of hexagonal systems.

How to cite

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Bhatti, Faqir M., Das, Kinkar Ch., and Ahmed, Syed A.. "On the energy and spectral properties of the he matrix of hexagonal systems." Czechoslovak Mathematical Journal 63.1 (2013): 47-63. <http://eudml.org/doc/252533>.

@article{Bhatti2013,
abstract = {The He matrix, put forward by He and He in 1989, is designed as a means for uniquely representing the structure of a hexagonal system (= benzenoid graph). Observing that the He matrix is just the adjacency matrix of a pertinently weighted inner dual of the respective hexagonal system, we establish a number of its spectral properties. Afterwards, we discuss the number of eigenvalues equal to zero of the He matrix of a hexagonal system. Moreover, we obtain a relation between the number of triangles and the eigenvalues of the He matrix of a hexagonal system. Finally, we present an upper bound on the He energy of hexagonal systems.},
author = {Bhatti, Faqir M., Das, Kinkar Ch., Ahmed, Syed A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {molecular graph; hexagonal system; inner dual; He matrix; spectral radius; eigenvalue; energy of graph; molecular graph; hexagonal graph; He matrix},
language = {eng},
number = {1},
pages = {47-63},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the energy and spectral properties of the he matrix of hexagonal systems},
url = {http://eudml.org/doc/252533},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Bhatti, Faqir M.
AU - Das, Kinkar Ch.
AU - Ahmed, Syed A.
TI - On the energy and spectral properties of the he matrix of hexagonal systems
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 1
SP - 47
EP - 63
AB - The He matrix, put forward by He and He in 1989, is designed as a means for uniquely representing the structure of a hexagonal system (= benzenoid graph). Observing that the He matrix is just the adjacency matrix of a pertinently weighted inner dual of the respective hexagonal system, we establish a number of its spectral properties. Afterwards, we discuss the number of eigenvalues equal to zero of the He matrix of a hexagonal system. Moreover, we obtain a relation between the number of triangles and the eigenvalues of the He matrix of a hexagonal system. Finally, we present an upper bound on the He energy of hexagonal systems.
LA - eng
KW - molecular graph; hexagonal system; inner dual; He matrix; spectral radius; eigenvalue; energy of graph; molecular graph; hexagonal graph; He matrix
UR - http://eudml.org/doc/252533
ER -

References

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  7. Gantmacher, F. R., The Theory of Matrices, Vol. 2, Reprint of the 1959 translation AMS Chelsea Publishing (1974). (1974) MR0107649
  8. Gutman, I., Zhou, B., Laplacian energy of a graph, Linear Algebra Appl. 414 (2006), 29-37. (2006) Zbl1092.05045MR2209232
  9. Horn, R. A., Johnson, C. R., Matrix Analysis, Cambridge University Press (1985). (1985) Zbl0576.15001MR0832183
  10. Sachs, H., 10.1007/BF02579161, Combinatorica 4 (1984), 89-99. (1984) Zbl0542.05048MR0739417DOI10.1007/BF02579161
  11. So, W., Robbiano, M., Abreu, N. M. M. de, Gutman, I., Applications of a theorem by Ky Fan in the theory of graph energy, Linear Algebra Appl. 432 (2010), 2163-2169. (2010) Zbl1218.05100MR2599850

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