 Research
 Open Access
Upper bounds for the number of spanning trees of graphs
 Ş Burcu Bozkurt^{1}Email author
https://doi.org/10.1186/1029242X2012269
© Bozkurt; licensee Springer 2012
 Received: 15 August 2012
 Accepted: 6 November 2012
 Published: 22 November 2012
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.
Keywords
 graph
 spanning trees
 normalized Laplacian eigenvalues
1 Introduction
Let G be a simple graph with n vertices and e edges. Let $V(G)=\{{v}_{1},{v}_{2},\dots ,{v}_{n}\}$ be the vertex set of G. If two vertices ${v}_{i}$ and ${v}_{j}$ are adjacent, then we use the notation ${v}_{i}\sim {v}_{j}$. For ${v}_{i}\in V(G)$, 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 ${d}_{1}\ge {d}_{2}\ge \cdots \ge {d}_{n}$.
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 $\overline{G}$ be the complement of G. Let $G\cup H$ be the vertexdisjoint union of the graphs G and H and let $G\vee H$ be the graph obtained from $G\cup H$ by adding all possible edges from vertices of G to vertices of H, i.e., $G\vee H=\overline{\overline{G}\cup \overline{H}}$ [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 $L=D{(G)}^{\frac{1}{2}}L(G)D{(G)}^{\frac{1}{2}}$, where $D{(G)}^{\frac{1}{2}}$ is the matrix which is obtained by taking $(\frac{1}{2})$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 ${\mu}_{1}\ge {\mu}_{2}\ge \cdots \ge {\mu}_{n}$ be the Laplacian eigenvalues and ${\lambda}_{1}\ge {\lambda}_{2}\ge \cdots \ge {\lambda}_{n}$ be the normalized Laplacian eigenvalues of G. It is well known that ${\mu}_{n}=0$, ${\lambda}_{n}=0$ and the multiplicities of these zero eigenvalues are equal to the number of connected components of G; see [2, 3].

Grimmett [4]:$t(G)\le \frac{1}{n}{\left(\frac{2e}{n1}\right)}^{n1}.$(3)

Grone and Merris [5]:$t(G)\le {\left(\frac{n}{n1}\right)}^{n1}\left(\frac{{\prod}_{i=1}^{n}{d}_{i}}{2e}\right).$(4)

Nosal [6]: For rregular graphs,$t(G)\le {n}^{n2}{\left(\frac{r}{n1}\right)}^{n1}.$(5)

Kelmanns ([1], p.222):$t(G)\le {n}^{n2}{(1\frac{2}{n})}^{\overline{e}},$(6)

Feng et al. [9]:$t(G)\le \left(\frac{{d}_{1}+1}{n}\right){\left(\frac{2e{d}_{1}1}{n2}\right)}^{n2}$(9)

Li et al. [10]:$t(G)\le {d}_{n}{\left(\frac{2e{d}_{1}1{d}_{n}}{n3}\right)}^{n3}.$(11)
In [4] Grimmett observed that (3) is the generalization of (5). Grone and Merris [5] stated that by the application of arithmeticgeometric 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 ${S}_{n}$, ${K}_{n}$, $G\cong {K}_{1}\vee ({K}_{1}\cup {K}_{n2})$ 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
where ${d}_{i}$ is the degree of the vertex ${v}_{i}$ of G.
Lemma 1 [11]
Lemma 2 [3]
Moreover, ${\lambda}_{1}=2$ if and only if a connected component of G is bipartite and nontrivial.
Lemma 3 [3]
Moreover, the equality holds in (12) if and only if G is a complete graph ${K}_{n}$.
Lemma 4 [12]
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 ${\lambda}_{2}={\lambda}_{3}=\cdots ={\lambda}_{n1}$ if and only if $G\cong {K}_{n}$ or $G\cong {K}_{p,q}$.
Lemma 7 [13]
Moreover, the equality holds in (14) if and only if G is a regular graph.
Lemma 8 [14]
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.
where $b={(\frac{n1{d}_{1}}{n(n2){d}_{1}})}^{1/2}$.
Hence, the result holds. □
for $0\le b\le 1$. Therefore, $f(b)\le f(0)=1$; 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).
Moreover, the equality holds in (16) if and only if G is the complete graph ${K}_{n}$.
for $P\le x\le 2$. Hence, $f(x)$ takes its maximum value at $x=P$ and (16) follows.
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.
Moreover, the equality holds in (17) if and only if $G\cong {K}_{p,q}$.
Moreover, the equality holds in (17) if and only if ${\lambda}_{2}=\cdots ={\lambda}_{n1}$, by Lemma 6, i.e., if and only if $G\cong {K}_{p,q}$. □
Declarations
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.
Authors’ Affiliations
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