Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Partially ordering the class of invertible trees   A tree T is invertible if and only if T has a perfect matching. Godsil
considers an invertible tree T and finds that the inverse of the adjacency
matrix of T has entries in {0, 1, -1} and is the signed adjacency matrix of a
graph which contains T. In this paper, we give a new proof of this theorem,
which gives rise to a partial ordering relation on the class of all invertible
trees on 2n vertices. In particular, we show that given an invertible tree T
whose inverse graph has strictly more edges, we can remove an edge from T and
add another edge to obtain an invertible tree T' whose median eigenvalue is
strictly greater. This extends naturally to a partial ordering. We characterize
the maximal and minimal elements of this poset and explore the implications
about the median eigenvalues of invertible trees.

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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th