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Upper bounds for the number of spanning trees of graphs

Abstract

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.

1 Introduction

Let G be a simple graph with n vertices and e edges. Let V(G)={ v 1 , v 2 ,, 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 d 1 d 2 d n .

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 GH be the vertex-disjoint union of the graphs G and H and let GH be the graph obtained from GH by adding all possible edges from vertices of G to vertices of H, i.e., GH= G ¯ H ¯ ¯ [1].

Let L(G)=D(G)A(G) be the Laplacian matrix of the graph G, where A(G) and D(G) are the adjacency matrix and the diagonal matrix of the vertex degrees of G, respectively. The normalized Laplacian matrix of G is defined as L=D ( G ) 1 2 L(G)D ( G ) 1 2 , where D ( G ) 1 2 is the matrix which is obtained by taking ( 1 2 )-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 1 μ 2 μ n be the Laplacian eigenvalues and λ 1 λ 2 λ n be the normalized Laplacian eigenvalues of G. It is well known that μ n =0, λ n =0 and the multiplicities of these zero eigenvalues are equal to the number of connected components of G; see [2, 3].

The number of spanning trees (also known as complexity), t(G), of G is given by the following formula in terms of the Laplacian eigenvalues (see [1], p.39):

t(G)= 1 n i = 1 n 1 μ i .
(1)

It is known that the number of spanning trees of G is also expressed by the normalized Laplacian eigenvalues as follows (see [1], p.49):

t(G)= ( i = 1 n d i 2 e ) i = 1 n 1 λ i .
(2)

Now we list some known upper bounds for t(G).

  • Grimmett [4]:

    t(G) 1 n ( 2 e n 1 ) n 1 .
    (3)
  • Grone and Merris [5]:

    t(G) ( n n 1 ) n 1 ( i = 1 n d i 2 e ) .
    (4)
  • Nosal [6]: For r-regular graphs,

    t(G) n n 2 ( r n 1 ) n 1 .
    (5)
  • Kelmanns ([1], p.222):

    t(G) n n 2 ( 1 2 n ) e ¯ ,
    (6)

where e ¯ is the number of edges of G ¯ .

  • Das [7]:

    t(G) ( 2 e d 1 1 n 2 ) n 2 .
    (7)
  • Zhang [8]:

    t(G) ( 1 + ( n 2 ) a ) ( 1 a ) n 2 1 n ( 2 e n 1 ) n 1 ,
    (8)

where a= ( n ( n 1 ) 2 e 2 e n ( n 2 ) ) 1 / 2 .

  • Feng et al. [9]:

    t(G) ( d 1 + 1 n ) ( 2 e d 1 1 n 2 ) n 2
    (9)

and

t(G) ( i = 1 n d i 2 + 2 e ( d 1 + 1 ) 2 n 2 ) n 2 2 .
(10)
  • Li et al. [10]:

    t(G) d n ( 2 e d 1 1 d n n 3 ) n 3 .
    (11)

In [4] Grimmett observed that (3) is the generalization of (5). Grone and Merris [5] stated that by the application of arithmetic-geometric mean inequality, (4) leads to (3). In [7] Das indicated that (7) is sharp for S n or K n , but (3), (4), (5) and (6) are sharp only for  K n . Li et al. [10] pointed out that (11) is sharp for S n , K n , G K 1 ( K 1 K n 2 ) or K n m, but (3) is sharp only for K n , (7) and (9) are sharp for S n or K n . In [8, 9] the authors showed that (8) is always better than (3), and (9) is always better than (7) and (10).

This paper is organized as follows. In Section 2, we give some useful lemmas. In Section 3, 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 (4).

2 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 1 , v 2 ,, v n } as

P=1+ 2 n ( n 1 ) v i v j 1 d i d j ,

where d i is the degree of the vertex v i of G.

Lemma 1 [11]

Let G be a graph with n vertices and normalized Laplacian matrix L without isolated vertices. Then

i = 1 n λ i =tr(L)=n

and

i = 1 n λ i 2 =tr ( L 2 ) =n+2 v i v j 1 d i d j .

Lemma 2 [3]

Let G be a graph with n vertices and normalized Laplacian eigenvalues λ 1 λ 2 λ n =0. Then

0 λ i 2.

Moreover, λ 1 =2 if and only if a connected component of G is bipartite and nontrivial.

Lemma 3 [3]

Let G be a graph with n vertices and normalized Laplacian eigenvalues λ 1 λ 2 λ n =0. Then

λ 1 n n 1 .
(12)

Moreover, the equality holds in (12) if and only if G is a complete graph K n .

Lemma 4 [12]

Let G be a graph with n vertices and normalized Laplacian eigenvalues λ 1 λ 2 λ n =0. Then

λ 1 P.
(13)

Moreover, the equality holds in (13) if and only if G is a complete graph K n .

Lemma 5 [12]

The lower bound (13) is always better than the lower bound (12).

Lemma 6 [12]

Let G be a connected graph with n>2 vertices. Then λ 2 = λ 3 == λ n 1 if and only if G K n or G K p , q .

Lemma 7 [13]

Let G be a graph with n vertices and without isolated vertices. Suppose G has the maximum vertex degree equal to d 1 . Then

v i v j 1 d i d j n 2 d 1 .
(14)

Moreover, the equality holds in (14) if and only if G is a regular graph.

Lemma 8 [14]

Let x i >1 for 1in. If i = 1 n x i =0 and i = 1 n x i 2 c 2 (1 n 1 ), then

i = 1 n ln(1+ x i )ln ( 1 + c c n 1 ) +(n1)ln ( 1 c n 1 ) .

3 Main results

Now we present the main results of this paper following the ideas in [8] and [9]. Note that P was defined earlier in the previous section.

Theorem 1 Let G be a graph with n vertices and without isolated vertices. Then

t(G) ( 1 + ( n 2 ) b ) ( 1 b ) n 2 ( n n 1 ) n 1 ( i = 1 n d i 2 e ) ,
(15)

where b= ( n 1 d 1 n ( n 2 ) d 1 ) 1 / 2 .

Proof If G is disconnected, then t(G)=0 and (15) follows. Now we assume that G is connected. From (2), we have

0<t(G)= ( i = 1 n d i 2 e ) λ 1 λ n 1

since λ n 1 >0. Let q= n n 1 and x i = λ i q 1 for 1in1. Then x i >1. Moreover, by Lemma 1 and Lemma 7, we get

i = 1 n 1 x i = i = 1 n 1 ( λ i q 1 ) =0

and

i = 1 n 1 x i 2 = i = 1 n 1 ( λ i q 1 ) 2 = ( n 1 ) 2 i = 1 n 1 λ i q + i = 1 n 1 λ i 2 q 2 ( n 1 ) 2 ( n 1 ) + ( n 1 n ) 2 ( n + n d 1 ) = ( n 1 ) 2 n d 1 ( n 1 n ) = ( n 1 ) 2 ( n 1 d 1 ) n ( n 2 ) d 1 ( 1 1 n 1 ) = ( ( n 1 ) b ) 2 ( 1 1 n 1 ) .

Then by Lemma 8, we obtain

i = 1 n 1 (1+ x i ) ( 1 + ( n 1 ) b ( n 1 ) b n 1 ) ( 1 b ) n 2 .

Therefore, we arrive at

i = 1 n 1 λ i ( 1 + ( n 2 ) b ) ( 1 b ) n 2 ( n n 1 ) n 1

and

t(G) ( 1 + ( n 2 ) b ) ( 1 b ) n 2 ( n n 1 ) n 1 ( i = 1 n d i 2 e ) .

Hence, the result holds. □

Remark 1 Let f(b)=(1+(n2)b) ( 1 b ) n 2 . Then

f (b)=(n2)(n1)b ( 1 b ) n 3 0

for 0b1. Therefore, f(b)f(0)=1; see [8]. Hence, we conclude that the upper bound (15) is always better than the upper bound (4). Moreover, if G is the complete graph K n , then the equality holds in (15).

Theorem 2 Let G be a connected graph with n>2 vertices. Then

t(G)P ( n P n 2 ) n 2 ( i = 1 n d i 2 e ) .
(16)

Moreover, the equality holds in (16) if and only if G is the complete graph K n .

Proof From (2) and Lemma 1, we get

t ( G ) = ( i = 1 n d i 2 e ) i = 1 n 1 λ i = ( i = 1 n d i 2 e ) λ 1 i = 2 n 1 λ i ( i = 1 n d i 2 e ) λ 1 ( i = 2 n 1 λ i n 2 ) n 2 = ( i = 1 n d i 2 e ) λ 1 ( i = 1 n 1 λ i λ 1 n 2 ) n 2 = ( i = 1 n d i 2 e ) λ 1 ( n λ 1 n 2 ) n 2 .

For Px2, let

f(x)=x ( n x ) n 2 .

By Lemma 4 and Lemma 5, we have that

λ 1 P n n 1

and

f (x)=f(x) n ( n 1 ) x x ( n x ) 0

for Px2. Hence, f(x) takes its maximum value at x=P and (16) follows.

If the equality holds in (16), then all inequalities in the above argument must be equalities. Hence, we have

λ 1 =Pand λ 2 == λ n 1 .

Then by Lemma 4 and Lemma 6, we conclude that G is the complete graph K n .

Conversely, we can easily see that the equality holds in (16) for the complete graph K n . □

Now we consider the bipartite graph case of the above theorem.

Theorem 3 Let G be a connected bipartite graph with n>2 vertices. Then

t(G) i = 1 n d i e .
(17)

Moreover, the equality holds in (17) if and only if G K p , q .

Proof Since G is a connected bipartite graph, by Lemma 2, we have λ 1 =2. Considering this, (2) and Lemma 1, we obtain

t ( G ) = ( i = 1 n d i 2 e ) i = 1 n 1 λ i = ( i = 1 n d i 2 e ) λ 1 i = 2 n 1 λ i ( i = 1 n d i e ) ( i = 2 n 1 λ i n 2 ) n 2 = ( i = 1 n d i e ) ( n λ 1 n 2 ) n 2 = i = 1 n d i e .

Moreover, the equality holds in (17) if and only if λ 2 == λ n 1 , by Lemma 6, i.e., if and only if G K p , q . □

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Acknowledgements

The author thanks the referees for their helpful comments and suggestions concerning the presentation of this paper. The author is also thankful to TUBITAK and the Office of Selcuk University Scientific Research Project (BAP). This study is based on a part of the author’s PhD thesis.

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Correspondence to Ş Burcu Bozkurt.

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Bozkurt, Ş.B. Upper bounds for the number of spanning trees of graphs. J Inequal Appl 2012, 269 (2012). https://doi.org/10.1186/1029-242X-2012-269

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Keywords

  • graph
  • spanning trees
  • normalized Laplacian eigenvalues