318 research outputs found
Laplacian Distribution and Domination
Let denote the number of Laplacian eigenvalues of a graph in an
interval , and let denote its domination number. We extend the
recent result , and show that isolate-free graphs also
satisfy . In pursuit of better understanding Laplacian
eigenvalue distribution, we find applications for these inequalities. We relate
these spectral parameters with the approximability of , showing that
. However, for -cyclic graphs, . For trees ,
Uniform density in matroids, matrices and graphs
We give new characterizations for the class of uniformly dense matroids, and
we describe applications to graphic and real representable matroids. We show
that a matroid is uniformly dense if and only if its base polytope contains a
point with constant coordinates, and if and only if there exists a measure on
the bases such that every element of the ground set has equal probability to be
in a random basis with respect to this measure. As one application, we derive
new spectral, structural and classification results for uniformly dense graphic
matroids. In particular, we show that connected regular uniformly dense graphs
are -tough and thus contain a (near-)perfect matching. As a second
application, we show that strictly uniformly dense real representable matroids
can be represented by projection matrices with constant diagonal and that they
are parametrized by a subvariety of the real Grassmannian.Comment: 23 page
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