Number of Spanning Trees of Circulant Graphs C&lt;sub&gt;6n&lt;/sub&gt; and their Applications

Helmi Helmi

doi:10.3844/jmssp.2012.24.31

Abstract

Problem statement: The number of spanning trees of a graph G is usually denoted by τ(G). A circulant graph with n vertices and k jumps is Cn (a1,…..,ak). Approach: In this study the number τ(G) of spanning trees of the circulant graphs C6n with some non-fixed jumps such as C6n (1, n), C6n (1, n, 2n), C6n (1, n, 3n), C6n (1, 2n 3n), C6n (1, n, 2n, 3n), are evaluated using Chebyshev polynomials. A large number of theorems of number of the spanning trees of circulate graphs C12n are obtained. Results: The number t(G) of spanning trees of the circulant graphs C6n(1, n), C6n(1, n, 2n), C6n(1, n, 3n), C6n(1, 2n, 3n), C6n(1, n, 2n, 3n), C12n(1, 2n, 3n), C12n(1, 3n, 6n), C12n(1, 3n, 4n), C12n(1, 2n, 3n, 4n), C12n(1, 2n, 3n, 6n), C12n(1, 3n, 4n, 6n) and C12n(1, 2n, 3n, 4n, 6n) are evaluated. 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 following important theorems and their proofs.

Highlights

The number of spanning trees τ(G) in graphs is an important invariant
1 p p −1 k =1 (d where μ0 = d, μ1, μ2,....,μp-1 are the an be used to determine the number of spanning trees eigenvalues of the corresponding adjacency matrix of for G = (V, E)
Chebyshev polynomial: we introduce some some relations concerning Chebyshev polynomials of the first and second kind which we use it in our computations

Summary

Introduction

The number of spanning trees τ(G) in graphs (networks) is an important invariant. G and A is the adjacency matrix of G, H = [aij] is defined as follows: (i) aij = -1, when viand vj are adjacent and i≠j, (ii) aij equal the degree of vertex vi if i≠j and (iii) aij = 0 otherwise. The formula for the number of spanning trees in a d-regular graph G can be expressed as

Results

Conclusion