Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The graph bottleneck identity   A matrix $S=(s_{ij})\in{\mathbb R}^{n\times n}$ is said to determine a
\emph{transitional measure} for a digraph $G$ on $n$ vertices if for all
$i,j,k\in\{1,\...,n\},$ the \emph{transition inequality} $s_{ij} s_{jk}\le
s_{ik} s_{jj}$ holds and reduces to the equality (called the \emph{graph
bottleneck identity}) if and only if every path in $G$ from $i$ to $k$ contains
$j$. We show that every positive transitional measure produces a distance by
means of a logarithmic transformation. Moreover, the resulting distance
$d(\cdot,\cdot)$ is \emph{graph-geodetic}, that is, $d(i,j)+d(j,k)=d(i,k)$
holds if and only if every path in $G$ connecting $i$ and $k$ contains $j$.
Five types of matrices that determine transitional measures for a digraph are
considered, namely, the matrices of path weights, connection reliabilities,
route weights, and the weights of in-forests and out-forests. The results
obtained have undirected counterparts. In [P. Chebotarev, A class of
graph-geodetic distances generalizing the shortest-path and the resistance
distances, Discrete Appl. Math., URL
http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to
fill the gap between the shortest path distance and the resistance distance.

21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics