The number of spanning trees of a graph
Loading...
Date
2013-08
Authors
Das, Kinkar Chandra
Çevik, Ahmet Sinan
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
Abstract
Let G be a simple connected graph of order n, m edges, maximum degree Delta(1) and minimum degree delta. Li et al. (Appl. Math. Lett. 23: 286-290, 2010) gave an upper bound on number of spanning trees of a graph in terms of n, m, Delta(1) and delta:
t(G) <= delta (2m-Delta(1)-delta-1/n-3)(n-3).
The equality holds if and only if G congruent to K-1,K-n-1, G congruent to K-n, G congruent to K-1 boolean OR (K-1 boolean OR Kn-2) or G congruent to K-n - e, where e is any edge of K-n. Unfortunately, this upper bound is erroneous. In particular, we show that this upper bound is not true for complete graph K-n. In this paper we obtain some upper bounds on the number of spanning trees of graph G in terms of its structural parameters such as the number of vertices (n), the number of edges (m), maximum degree (Delta(1)), second maximum degree (Delta(2)), minimum degree (delta), independence number (alpha), clique number (omega). Moreover, we give the Nordhaus-Gaddum-type result for number of spanning trees.
Description
Keywords
Mathematics, Graph, Spanning trees, Independence number, Clique number, First Zagreb index, Molecular-orbitals, Zagreb indexes
Citation
Das, K. C. vd. (2013). “The number of spanning trees of a graph”. Journal of Inequalities and Applications, 2013.