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Updated: 16 hours 34 min ago

On the construction of Q-controllable graphs

Sun, 2017-10-15 08:53

A connected graph is called Q-controllable if its signless Laplacian eigenvalues are mutually distinct and main. Two graphs G and H are said to be Q-cospectral if they share the same signless Laplacian spectrum. In this paper, infinite families of Q-controllable graphs are constructed, by using the operator of rooted product introduced by Godsil and McKay. In the process, innitely many non-isomorphic Q-cospectral graphs are also constructed, especially, including those graphs whose signless Laplacian eigenvalues are mutually distinct.

On the Location of Eigenvalues of Real Matrices

Mon, 2017-10-02 10:47

The research in this paper is motivated by a recent work of I. Barany and J. Solymosi [I. Barany and J. Solymosi. Gershgorin disks for multiple eigenvalues of non-negative matrices. Preprint arXiv no. 1609.07439, 2016.] about the location of eigenvalues of nonnegative matrices with geometric multiplicity higher than one. In particular, an answer to a question posed by Barany and Solymosi, about how the location of the eigenvalues can be improved in terms of their geometric multiplicities is obtained. New inclusion sets for the eigenvalues of a real square matrix, called Ger\v{s}gorin discs of the second type, are introduced. It is proved that under some conditions, an eigenvalue of a real matrix is in a Ger\v{s}gorin disc of the second type. Some relationships between the geometric multiplicities of eigenvalues and these new inclusion sets are established. Some other related results, consequences, and examples are presented. The results presented here apply not only to nonnegative matrices, but extend to all real matrices, and some of them do not depend on the geometric multiplicity.