Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Shortest-weight paths in random regular graphs   Consider a random regular graph with degree $d$ and of size $n$. Assign to
each edge an i.i.d. exponential random variable with mean one. In this paper we
establish a precise asymptotic expression for the maximum number of edges on
the shortest-weight paths between a fixed vertex and all the other vertices, as
well as between any pair of vertices. Namely, for any fixed $d \geq 3$, we show
that the longest of these shortest-weight paths has about $\hat{\alpha}\log n$
edges where $\hat{\alpha}$ is the unique solution of the equation $\alpha
\log(\frac{d-2}{d-1}\alpha) - \alpha = \frac{d-3}{d-2}$, for $\alpha >
\frac{d-1}{d-2}$.

1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con