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The domination number and the least -eigenvalue
A vertex set of a graph is said to be a dominating set if every
vertex of is adjacent to at least a vertex in , and the
domination number (, for short) is the minimum cardinality
of all dominating sets of . For a graph, the least -eigenvalue is the
least eigenvalue of its signless Laplacian matrix. In this paper, for a
nonbipartite graph with both order and domination number , we show
that , and show that it contains a unicyclic spanning subgraph
with the same domination number . By investigating the relation between
the domination number and the least -eigenvalue of a graph, we minimize the
least -eigenvalue among all the nonbipartite graphs with given domination
number.Comment: 13 pages, 3 figure
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 ,
Universality for general Wigner-type matrices
We consider the local eigenvalue distribution of large self-adjoint random matrices with centered independent entries.
In contrast to previous works the matrix of variances is not assumed to be stochastic. Hence the density of states is
not the Wigner semicircle law. Its possible shapes are described in the
companion paper [1]. We show that as grows, the resolvent,
, converges to a diagonal matrix, , where
solves the vector equation that has
been analyzed in [1]. We prove a local law down to the smallest spectral
resolution scale, and bulk universality for both real symmetric and complex
hermitian symmetry classes.Comment: Changes in version 3: The format of pictures was changed to resolve a
conflict with certain pdf viewer
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