The number of spanning trees of a graph

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.Faculty research Fund, Sungkyunkwan UniversityKorean Government (2013R1A1A2009341)Selçuk ÜniversitesiGlaucoma Research FoundationHong Kong Baptist Universit

Taylor Expansion Proof of the Matrix Tree Theorem - Part I

The Matrix-Tree Theorem states that the number of spanning trees of a graph is given by the absolute value of any cofactor of the Laplacian matrix of the graph. We propose a very short proof of this result which amounts to comparing Taylor expansions

Can One Hear the Spanning Trees of a Quantum Graph?

Kirchhoff showed that the number of spanning trees of a graph is the spectral determinant of the combinatorial Laplacian divided by the number of vertices; we reframe this result in the quantum graph setting. We prove that the spectral determinant of the Laplace operator on a finite connected metric graph with standard (Neummann-Kirchhoff) vertex conditions determines the number of spanning trees when the lengths of the edges of the metric graph are sufficiently close together. To obtain this result, we analyze an equilateral quantum graph whose spectrum is closely related to spectra of discrete graph operators and then use the continuity of the spectral determinant under perturbations of the edge lengths.Comment: 15 pages, 1 figur