# Steiner Distance in Product Networks

## TL;DR

This paper explores the Steiner distance and Steiner k-diameter in product graphs, providing new bounds and properties relevant to network analysis and design.

## Contribution

It introduces new results on Steiner distances and k-diameters specifically for Cartesian and lexicographical product graphs, expanding understanding of these metrics in complex networks.

## Key findings

- Derived bounds for Steiner k-diameter in product graphs
- Analyzed Steiner distances in specific network classes
- Extended results to practical network models

## Abstract

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.