Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebraic properties of perfect structures   A perfect structure is a triple $(M,P,S)$ of matrices $M, P$ and $S$ of
consistent sizes such that $MP = PS$. Perfect structures comprise similar
matrices, eigenvectors, perfect colorings (equitable partitions) and graph
coverings. In this paper we study general algebraic properties of perfect
structures and characterize all perfect structures with identity or unity
matrix $M$. Next, we consider a graph product generalizing most standard
products (e.g. Cartesian, tensor, normal, lexicographic graph products). For
this product we propose a construction of perfect structures and prove that it
can be reversed for eigenvectors. Finally, we apply obtained results to
calculate the spectra of several classes of graphs and to prove some properties
of perfect colorings.

11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga