Faqir M. Bhatti, Lahore University of Management Sciences, Sector U, DHA, Lahore 54792, Pakistan, e-mail: firstname.lastname@example.org; Kinkar Ch. Das (corresponding author), Sungkyunkwan University, Suwon 440-746, Republic of Korea, e-mail: email@example.com; Syed A. Ahmed, Lahore University of Management Sciences, Sector U, DHA, Lahore 54792, Pakistan, e-mail: firstname.lastname@example.org
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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