Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Steiner $k$-Wiener index of graphs with given minimum degree   Let $G$ be a connected graph. The Steiner distance $d(S)$ of a set $S$ of
vertices is the minimum size of a connected subgraph of $G$ containing all
vertices of $S$. For $k\in \mathbb{N}$, the Steiner $k$-Wiener index $SW_k(G)$
is defined as $\sum_S d(S)$, where the sum is over all $k$-element subsets of
the vertex set of $G$. The average Steiner $k$-distance $\mu_k(G)$ of $G$ is
defined as $\binom{n}{k}^{-1} SW_k(G)$.
  In this paper we prove upper bounds on the Steiner Wiener index and the
average Steiner distance of graphs with given order $n$ and minimum degree
$\delta$. Specifically we show that $SW_k(G) \leq
\frac{k-1}{k+1}\frac{3n}{\delta+1} \binom{n}{k} + O(n^{k})$, and that $\mu_k(G)
\leq \frac{k-1}{k+1}\frac{3n}{\delta+1} + O(1)$. We improve this bound for
triangle-free graphs to $SW_k(G) \leq \frac{k-1}{k+1}\frac{2n}{\delta}
\binom{n}{k} + O(n^{k})$, and $\mu_k(G) \leq \frac{k-1}{k+1}\frac{2n}{\delta} +
O(1)$. All bounds are best possible.

8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc