Most, And Least, Compact Spanning Trees of a Graph

TL;DR

This paper introduces methods to find the most and least compact spanning trees in a graph based on average shortest path distances, using a greedy iterative approach with empirical validation on various graph types.

Contribution

It presents a novel greedy algorithm for identifying extremal spanning trees based on average shortest path distances, utilizing forest accessibility matrices.

Findings

The method effectively finds extremal spanning trees in different graph families.

Empirical results support the algorithm's applicability to standard graph models.

The approach has polynomial time complexity.

Abstract

We introduce the concept of Most, and Least, Compact Spanning Trees - denoted respectively by $T^*(G)$ and $T^\#(G)$ - of a simple, connected, undirected and unweighted graph $G(V, E, W)$. For a spanning tree $T(G) \in \mathcal{T}(G)$ to be considered $T^*(G)$, where $\mathcal{T}(G)$ represents the set of all the spanning trees of the graph $G$, it must have the least average inter-vertex pair (shortest path) distances from amongst the members of the set $\mathcal{T}(G)$. Similarly, for it to be considered $T^\#(G)$, it must have the highest average inter-vertex pair (shortest path) distances. In this work, we present an iteratively greedy rank-and-regress method that produces at least one $T^*(G)$ or $T^\#(G)$ by eliminating one extremal edge per iteration. The rank function for performing the elimination is based on the elements of the matrix of relative forest accessibilities of a graph and the related forest distance. We provide empirical evidence in support of our methodology using some standard graph families: complete graphs, the Erd\H{o}s-Renyi random graphs and the Barab\'{a}si-Albert scale-free graphs; and discuss computational complexity of the underlying methods which incur polynomial time costs.