The number of spanning trees of a graph

Abstract

Let G be a simple connected graph of order n, m edges, maximum degreeΔ1 and minimum degree δ. 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,Δ1 and δ: t(G) ≤ δ (2m-Δ1 - δ - 1/n - 3 )n-3 . The equality holds if and only if G≅K1, n-1, G≅Kn, G≅ K1 ∨ (K1 ∪ Kn-2) or G≅Kn - 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 Kn. 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 (Δ1), second maximum degree (Δ2), minimum degree (δ), independence number (α), clique number (ω). Moreover, we give the Nordhaus-Gaddum-type result for number of spanning trees. © 2013 Das et al.; licensee Springer.