Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Descriptive complexity of graph spectra   Two graphs are co-spectral if their respective adjacency matrices have the
same multi-set of eigenvalues. A graph is said to be determined by its spectrum
if all graphs that are co-spectral with it are isomorphic to it. We consider
these properties in relation to logical definability. We show that any pair of
graphs that are elementarily equivalent with respect to the three-variable
counting first-order logic $C^3$ are co-spectral, and this is not the case with
$C^2$, nor with any number of variables if we exclude counting quantifiers. We
also show that the class of graphs that are determined by their spectra is
definable in partial fixed-point logic with counting. We relate these
properties to other algebraic and combinatorial problems.

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The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
2
The categories of this paper are: math
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
19
The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int