746 research outputs found
Randi\'c energy and Randi\'c eigenvalues
Let be a graph of order , and the degree of a vertex of
. The Randi\'c matrix of is defined by if the vertices and are adjacent in and
otherwise. The normalized signless Laplacian matrix is
defined as , where is the identity matrix. The
Randi\'c energy is the sum of absolute values of the eigenvalues of .
In this paper, we find a relation between the normalized signless Laplacian
eigenvalues of and the Randi\'c energy of its subdivided graph . We
also give a necessary and sufficient condition for a graph to have exactly
and distinct Randi\'c eigenvalues.Comment: 7 page
On a Problem of Harary and Schwenk on Graphs with Distinct Eigenvalues
Harary and Schwenk posed the problem forty years ago: Which graphs have
distinct adjacency eigenvalues? In this paper, we obtain a necessary and
sufficient condition for an Hermitian matrix with simple spectral radius and
distinct eigenvalues. As its application, we give an algebraic characterization
to the Harary-Schwenk's problem. As an extension of their problem, we also
obtain a necessary and sufficient condition for a positive semidefinite matrix
with simple least eigenvalue and distinct eigenvalues, which can provide an
algebraic characterization to their problem with respect to the (normalized)
Laplacian matrix.Comment: 11 page
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