The Number of Spanning Trees in the Composition Graphs

Feng Li

doi:10.1155/2014/613685

Abstract

Using the composition of some existing smaller graphs to construct some large graphs, the number of spanning trees and the Laplacian eigenvalues of such large graphs are also closely related to those of the corresponding smaller ones. By using tools from linear algebra and matrix theory, we establish closed formulae for the number of spanning trees of the composition of two graphs with one of them being an arbitrary complete 3-partite graph and the other being an arbitrary graph. Our results extend some of the previous work, which depend on the structural parameters such as the number of vertices and eigenvalues of the small graphs only.

Highlights

Given the class Ω(n, m) of all connected graphs with n vertices and m edges, it is very important to seek graphs with the most number of spanning trees, so the number of spanning trees is closely connected to reliable network design [1, 2]
It is well known that the number of spanning trees of some specific family of graphs can be given explicitly, which include the complete graph Kn, the path Pn, the cycle Cn, the wheel Wn, and the Mobius ladders; “almost-complete” graphs, the threshold graphs, and some multicomplete/star related graphs can be obtained [6,7,8,9]
Mathematic structures can be properly understood if one has a grasp of their symmetries; it helps to know whether they can be constructed from smaller constituents, since many large graphs are usually composed from some existing smaller graphs through graph operations

Summary

Introduction

Given the class Ω(n, m) of all connected graphs with n vertices and m edges, it is very important to seek graphs (known as the t-optimal graphs) with the most number of spanning trees, so the number of spanning trees is closely connected to reliable network design [1, 2]. Let G1 and G2 be two simple graphs; the composition graphs of G1 and G2, denoted by G1 ⊙ G2, are a graph with vertex set V(G1) × V(G2), and there is an edge between (u1, u2) and (V1, V2) if there is an edge connecting u1 and V1 in G1, or if u1 = V1 and there is an edge connecting u2 and V2 in G2 Fiedler [12] gave the eigenvalues of nonnegative symmetric matrices where he obtains the description of the eigenspace of some matrix operations. This has been applied for some graph product in [13]. For more details on the composition graphs and matrix operations the reader is referred to [14,15,16]

Preliminaries

Some Corollaries