Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the roots of Wiener polynomials of graphs   The Wiener polynomial of a connected graph $G$ is defined as $W(G;x)=\sum
x^{d(u,v)}$, where $d(u,v)$ denotes the distance between $u$ and $v$, and the
sum is taken over all unordered pairs of distinct vertices of $G$. We examine
the nature and location of the roots of Wiener polynomials of graphs, and in
particular trees. We show that while the maximum modulus among all roots of
Wiener polynomials of graphs of order $n$ is $\binom{n}{2}-1$, the maximum
modulus among all roots of Wiener polynomials of trees of order $n$ grows
linearly in $n$. We prove that the closure of the collection of real roots of
Wiener polynomials of all graphs is precisely $(-\infty, 0]$, while in the case
of trees, it contains $(-\infty, -1]$. Finally, we demonstrate that the
imaginary parts and (positive) real parts of roots of Wiener polynomials can be
arbitrarily large.

18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph