Complexity of Cocktail Party Graph and Crown Graph

Khalid Khalid

doi:10.3844/ajassp.2012.202.207

Abstract

Problem statement: The number of spanning trees τ(G) in graphs (networks) was an important invariant. Approach: Using the properties of the Chebyshev polynomials of the second kind and the linear algebra techniques to evaluate the associated determinants. Results: The complexity, number of spanning trees, of the cocktail party graph on 2n vertices, given in detail in the text was proved. Also the complexity of the crown graph on 2n vertices was shown to had the value nn-2 (n-1) (n-2)n-1. Conclusion: The number of spanning trees τ(G) in graphs (networks) is an important invariant. The evaluation of this number and analyzing its behavior 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 above important theorems and lemmas and their proofs.

Highlights

There are other methods for calculating τ(G)
As in e.g., (Clark, 2003; Qiao and Chen, 2009). Another result is due to Guy (1970) who derived a formula for the wheel on n+1 vertices, Wn+1, which is formed from a cycle Cn on n vertices by adding a vertex adjacent to every vertex of Cn
Another class of graphs for which an explicit formula has been derived is based on a prism (Boesch and Bogdanowicz, 1987; Boesch and Prodinger, 1986)

Summary

Introduction

There are other methods for calculating τ(G). Let μ1≥μ1≥...≥μp denote the eignvalues of H matrix of a p point graph. Spanning trees in a d-regular graph G can be expressed The formula for the number of due to Cayley (1889) who showed that complete graph on n vertices, Kn has nn-2 spanning trees that he showed τ(Kn)= nn-2, n≥2. Τ(Kp,q ) = pq−1qp−1, p, q ≥ 1 , where Kp,q is the complete bipartite graph with bipartite sets containing P and q vertices, respectively. (1970) later derived a formula for the number of spanning trees in a Mobius ladder.

Results

Conclusion