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Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 47-63, 2013
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On the energy and spectral properties of the He matrix of hexagonal systems

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

* Faqir M. Bhatti*, Lahore University of Management Sciences, Sector U, DHA, Lahore 54792, Pakistan, e-mail: ` fmbhatti@lums.edu.pk`; * Kinkar Ch. Das* (corresponding author), Sungkyunkwan University, Suwon 440-746, Republic of Korea, e-mail: ` kinkar@lycos.com`; * Syed A. Ahmed*, Lahore University of Management Sciences, Sector U, DHA, Lahore 54792, Pakistan, e-mail: ` aliahmed@lums.edu.pk`

**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.

**Keywords:** molecular graph, hexagonal system, inner dual, He matrix, spectral radius, eigenvalue, energy of graph

**Classification (MSC 2010):** 05C30, 68R10, 81Q30, 05C10

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