Strong B-T inverse.

New-typed matrix inverses based on the Hartwig-Spindelböck decomposition were investigated, which is called the strong B-T inverse, as a generalization of the B-T inverse. The relationships between the above inverse and other matrix inverses were established. Several sufficient and necessary conditions of the strong B-T inverse are obtained via the column and null spaces.

Computing the Laplacian spectrum and Wiener index of pentagonal-derivation cylinder/Möbius network.

The Laplacian spectrum significantly contributes the study of the structural features of non-regular networks. Actually, it emphasizes the interaction among the network eigenvalues and their structural properties. Let Pn(Pn') represent the pentagonal-derivation cylinder (Möbius) network. In this article, based on the decomposition techniques of the Laplacian characteristic polynomial, we initially determine that the Laplacian spectra of Pn contain the eigenvalues of matrices LR and LS. Furthermore, using the relationship among the coefficients and roots of these two matrices, explicit calculations of the Kirchhoff index and spanning trees of Pn are determined. The relationship between the Wiener and Kirchhoff indices of Pn is also established.

Weak dual generalized inverse of a dual matrix and its applications.

Recently, the dual Moore-Penrose generalized inverse has been applied to study the linear dual equation when the dual Moore-Penrose generalized inverse of the coefficient matrix of the linear dual equation exists. Nevertheless, the dual Moore-Penrose generalized inverse exists only in partially dual matrices. In this paper, to study more general linear dual equation, we introduce the weak dual generalized inverse described by four dual equations, and is a dual Moore-Penrose generalized inverses for it when the latter exists. Any dual matrix has the weak dual generalized inverse and is unique. We obtain some basic properties and characterizations of the weak dual generalized inverse. Also, we investigate relationships among the weak dual generalized inverse, the Moore-Penrose dual generalized inverses and the dual Moore-Penrose generalized inverses, give the equivalent characterization and use some numerical examples to show that the three are different dual generalized inverse. Afterwards, by applying the weak dual generalized inverse we solve two special linear dual equations, one of which is consistent and the other is inconsistent. Neither of the coefficient matrices of the above two linear dual equations has dual Moore-Penrose generalized inverses.