Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 47-63, 2013

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

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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