On the number of spanning trees of a multi-complete/star related graph

Chung, Kuo–Liang; Yan, Wen-Ming

doi:10.1016/s0020-0190(00)00135-6

Cited by 11 publications

(13 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The results extend some of the previous results [8,13,17,21,22]. However, it is still a problem remained unsolved to derive a formula for t(G 1 G 2 ), in which G 1 and G 2 are arbitrary graphs.…”

Section: Resultssupporting

confidence: 86%

“…For example, the closed formula for counting the number of spanning trees of graphs, including complete graphs, the triangle graphs, the Möbius laders, the complete multipartite graphs, and the "almost-complete" graphs, can be referred in [10,11]. Recently, the number of spanning trees of some graphs, for example, the circulant graphs, the square of a cycle, the threshold graphs, some multicomplete/star-related graphs, and spanning trees with few leaves in weighted graphs can also be obtained [6,[12][13][14]. In the design and analysis of network reliability with failure of lines in the network, it is very important and necessary to calculate the number of spanning trees in the certain graph.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The number of spanning trees in a new lexicographic product of graphs

Liang

Zhao

2014

Sci. China Inf. Sci.

View full text Add to dashboard Cite

The problem of counting the number of spanning trees is an old topic in graph theory with important applications to reliable network design. Usually, it is desirable to put forward a formula of the number of spanning trees for various graphs, which is not only interesting in its own right but also in practice. Since some large graphs can be composed of some existing smaller graphs by using the product of graphs, the number of spanning trees of such large graph is also closely related to that of the corresponding smaller ones. In this article, we establish a formula for the number of spanning trees in the lexicographic product of two graphs, in which one graph is an arbitrary graph G and the other is a complete multipartite graph. The results extend some of the previous work, which is closely related to the number of vertices and Lapalacian eigenvalues of smaller graphs only.

show abstract

Section: Resultssupporting

confidence: 86%

Section: Introductionmentioning

confidence: 99%

The number of spanning trees in a new lexicographic product of graphs

Liang

Zhao

2014

Sci. China Inf. Sci.

View full text Add to dashboard Cite

show abstract

“…Thus, for both theoretical and practical purposes, we are interested in deriving formulas for the number of spanning trees of classes of graphs of the form K n − H. Many cases have already been examined. For example there exist formulas for the cases when H is a pairwise disjoint set of edges [20], when it is a star [17], when it is a complete graph [1], when it is a path [5], when it is a cycle [5], when it is a multi-star [3,16,22], and so on (see Berge [1] for an exposition of the main results).…”

Section: Introductionmentioning

confidence: 99%

The Number of Spanning Trees in Kn-Complements of Quasi-Threshold Graphs

Nikolopoulos

Papadopoulos

2004

Graphs and Combinatorics

View full text Add to dashboard Cite

In this paper we examine the classes of graphs whose K n -complements are trees and quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of K n , the K n -complement of H is the graph K n − H which is obtained from K n by removing the edges of H. Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of the graph. Our results generalize previous results and extend the family of graphs of the form K n − H admitting formulas for the number of their spanning trees.

show abstract

“…Denote by .G/ the number of spanning trees in G. Enumeration of spanning trees in graphs with certain symmetry and fractals has been widely studied via ad hoc techniques capitalizing on the particular structures [6][7][8][9][10][11]. In general, we often have to resort to Kirchhoff's celebrated matrixtree theorem [12], which asserts that n .G/ equals the product of all nonzero eigenvalues of Laplacian matrix of G, i.e., .G/ D 1 n Q n 1 i D1 i .G/.…”

Section: Introductionmentioning

confidence: 99%

On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs

Shang

2016

Open Mathematics

View full text Add to dashboard Cite

As a generalization of the Sierpiński-like graphs, the subdivided-line graph .G/ of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in .G/, employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of .G/ in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition, we establish upper and lower bounds for the Laplacian Estrada index of .G/ based on the vertex degrees of G. These bounds are also connected with the number of spanning trees in .G/.

show abstract

On the number of spanning trees of a multi-complete/star related graph

Cited by 11 publications

References 5 publications

The number of spanning trees in a new lexicographic product of graphs

The number of spanning trees in a new lexicographic product of graphs

The Number of Spanning Trees in Kn-Complements of Quasi-Threshold Graphs

On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs

Contact Info

Product

Resources

About