Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exact solutions for the two- and all-terminal reliabilities of the
  Brecht-Colbourn ladder and the generalized fan   The two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the
generalized fan have been calculated exactly for arbitrary size as well as
arbitrary individual edge and node reliabilities, using transfer matrices of
dimension four at most. While the all-terminal reliabilities of these graphs
are identical, the special case of identical edge ($p$) and node ($\rho$)
reliabilities shows that their two-terminal reliabilities are quite distinct,
as demonstrated by their generating functions and the locations of the zeros of
the reliability polynomials, which undergo structural transitions at $\rho =
\displaystyle {1/2}$.

2
The categories of this paper are: math
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org