Upper bounds for the number of spanning trees of graphs

In this paper, we present some upper bounds for the number of spanning trees of graphs in terms of the number of vertices, the number of edges and the vertex degrees. MSC:05C05, 05C50.


Introduction
Let G be a simple graph with n vertices and e edges. Let V (G) = {v  , v  , . . . , v n } be the vertex set of G. If two vertices v i and v j are adjacent, then we use the notation v i ∼ v j .
For v i ∈ V (G), the degree of the vertex v i , denoted by d i , is the number of vertices adjacent to v i . Throughout this paper, we assume that the vertex degrees are ordered by The complete graph, the complete bipartite graph and the star of order n are denoted by K n , K p,q (p + q = n) and S n , respectively. Let Gm be the graph obtained by deleting any edge m from the graph G and let G be the complement of G. Let G ∪ H be the vertex-  is the matrix which is obtained by taking (-  )-power of each entry of D(G). The Laplacian eigenvalues and the normalized Laplacian eigenvalues of G are the eigenvalues of L(G) and L, respectively. Let μ  ≥ μ  ≥ · · · ≥ μ n be the Laplacian eigenvalues and λ  ≥ λ  ≥ · · · ≥ λ n be the normalized Laplacian eigenvalues of G. It is well known that It is known that the number of spanning trees of G is also expressed by the normalized Laplacian eigenvalues as follows (see [], p.): Now we list some known upper bounds for t(G).
-Grone and Merris []: where e is the number of edges of G. and (   ) http://www.journalofinequalitiesandapplications.com/content/2012/1/269 In [] Grimmett observed that () is the generalization of (). Grone and Merris [] stated that by the application of arithmetic-geometric mean inequality, () leads to (). In [] Das indicated that () is sharp for S n or K n , but (), (), () and () are sharp only for K n . Li et al. [] pointed out that () is sharp for S n , K n , G ∼ = K  ∨ (K  ∪ K n- ) or K nm, but () is sharp only for K n , () and () are sharp for S n or K n . In [, ] the authors showed that () is always better than (), and () is always better than () and ().
This paper is organized as follows. In Section , we give some useful lemmas. In Section , we obtain some upper bounds for the number of spanning trees of graphs in terms of the number of vertices, the number of edges and the vertex degrees of graphs. We also show that one of these upper bounds is always better than the upper bound ().

Preliminary lemmas
In this section, we give some lemmas which will be used later. Firstly, we introduce an auxiliary quantity of a graph G on the vertex set V (G) = {v  , v  , . . . , v n } as where d i is the degree of the vertex v i of G.

Lemma  [] Let G be a graph with n vertices and normalized Laplacian eigenvalues
Moreover, the equality holds in () if and only if G is a complete graph K n .

Lemma  []
The lower bound () is always better than the lower bound ().

Lemma  []
Let G be a connected graph with n >  vertices. Then λ  = λ  = · · · = λ n- if and only if G ∼ = K n or G ∼ = K p,q .

Lemma  [] Let G be a graph with n vertices and without isolated vertices. Suppose G has the maximum vertex degree equal to d  .
Then

Main results
Now we present the main results of this paper following the ideas in [] and []. Note that P was defined earlier in the previous section.

Theorem  Let G be a graph with n vertices and without isolated vertices. Then
Proof If G is disconnected, then t(G) =  and () follows. Now we assume that G is connected. From (), we have since λ n- > . Let q = n n- and x i = λ i q - for  ≤ i ≤ n -. Then x i > -. Moreover, by Lemma  and Lemma , we get Then by Lemma , we obtain Therefore, we arrive at Hence, the result holds. Proof From () and Lemma , we get By Lemma  and Lemma , we have that Hence, f (x) takes its maximum value at x = P and () follows. If the equality holds in (), then all inequalities in the above argument must be equalities. Hence, we have λ  = P and λ  = · · · = λ n- .
Then by Lemma  and Lemma , we conclude that G is the complete graph K n . Conversely, we can easily see that the equality holds in () for the complete graph K n . Now we consider the bipartite graph case of the above theorem.