Some Applications of Spanning Trees in Complete and Complete Bipartite Graphs

M. M.

doi:10.3844/ajassp.2012.584.592

Abstract

Problem statement: The number of spanning trees τ(G) in graphs (networks) is an important invariant, it is also an important measure of reliability of a network. Approach: Using linear algebra and matrix analysis techniques to evaluate the associated determinants. Results: In this study we derive simple formulas for the number of spanning trees of complete graph Kn and complete bipartite graph Kn,m and some of their applications. A large number of theorems of number of the spanning trees of known operations on complete graph Kn and complete bipartite graph Kn,m are obtained. Conclusion: The evaluation of number of spanning trees is not only interesting from a mathematical (computational) perspective, but also, it is an important measure of reliability of a network and designing electrical circuits. Some computationally hard problems such as the travelling salesman problem can be solved approximately by using spanning trees. Due to the high dependence of the network design and reliability on the graph theory we introduced the following important theorems and lemmas and their proofs.

Highlights

We consider finite undirected graph with no loops or multiple edges
A spanning tree for a graph G is a subgraph of G that is a tree and contains all vertices of G
Whenever one wants to find a simple, cheap, yet efficient way to connect a set of terminals, be they computers, telephones, factories, or cities, a solution is normally one kind of spanning trees

Summary

Introduction

Let G be such a graph on n vertices. A spanning tree for a graph G is a subgraph of G that is a tree and contains all vertices of G. A classic result of Kirchhoff, (Cayley, 1889) can be used to determine the number of spanning trees for for a few special families of graphs there exists simple formulas that make it much easier to calculate and determine the number of corresponding spanning trees especially when these numbers are very large.

Results

Conclusion