Steiner distance matrix of caterpillar graphs

TL;DR

This paper investigates the properties of the 2-Steiner distance matrix in caterpillar graphs, establishing a precise formula for its rank based on the number of vertices and pendant vertices.

Contribution

It provides a novel explicit formula for the rank of the 2-Steiner distance matrix specifically for caterpillar graphs.

Findings

Rank of 2-Steiner distance matrix is 2N - p - 1 for caterpillar graphs.

The formula relates graph structure to matrix rank.

Enhances understanding of Steiner distance matrices in specific graph classes.

Abstract

For a connected graph $G:=(V,E)$, the Steiner distance $d_G(X)$ among a set of vertices $X$ is the minimum size among all the connected subgraphs of $G$ whose vertex set contains $X$. The $k-$Steiner distance matrix $D_k(G)$ of $G$ is a matrix whose rows and columns are indexed by $k-$subsets of $V$. For $k$-subsets $X_1$ and $X_2$, the $(X_1,X_2)-$entry of $D_k(G)$ is $d_G(X_1 \cup X_2)$. In this paper, we show that the rank of $2-$Steiner distance matrix of a caterpillar graph on $N$ vertices and with $p$ pendant veritices is $2N-p-1$.