On the Number of Spanning Trees of Graphs

Abstract

We establish some bounds for the number of spanning trees of connected graphs in terms of the number of vertices , the number of edges , maximum vertex degree , minimum vertex degree , …first Zagreb index and Randić index .

1. Introduction

Let be a simple connected graph with vertices and edges. Let be the vertex set and the edge set of . If any two vertices and of are adjacent, that is, , then we use the notation . For , the degree of the vertex , denoted by , is the number of the vertices adjacent to . Let , , and be the maximum, the second maximum, and the minimum vertex degree of , respectively.

Let be the first Zagreb index [1] and the general Randić index [2] of the graph , where is a fixed real number. Note that the Randić index is also well studied in the literature. For more details on , see [3, 4].

Let , , and denote the complete graph, the complete bipartite graph, and the star graph of order , respectively. Let be the graph obtained by deleting the edge from the graph and let be the complement of . Let be the vertex-disjoint union of the graphs and . The graph is obtained from by adding all possible edges from vertices of to vertices of ; that is, [5].

The Laplacian matrix of the graph is the matrix , where and are the -adjacency matrix and the diagonal matrix of the vertex degrees of , respectively. The normalized Laplacian matrix of is defined as , where is the matrix which is obtained by taking power of each entry of . The Laplacian eigenvalues and the normalized Laplacian eigenvalues of are the eigenvalues of and , respectively. Let be the Laplacian eigenvalues and the normalized Laplacian eigenvalues of . Note that , , and the multiplicities of these zero eigenvalues are equal to the number of connected components of ; see [6, 7]. For more details on Laplacian and normalized Laplacian eigenvalues, see [6, 8–10].

The number of spanning trees, , of the graph is equal to the total number of distinct spanning subgraphs of that are trees. This quantity is also known as the complexity of and given by the following formula in terms of the Laplacian eigenvalues [5]:

It is well known that the number of spanning trees of is also expressed by the normalized Laplacian eigenvalues as [5, 6]

Now, we give some known upper bounds on :(1)Grimmett [11]: (2)Grone and Merris [12]: (3)Nosal [13]: for -regular graphs, (4)Cvetković et al. (see [5, page 222]): where is the number of edges of ,(5)Das [14]: (6)Zhang [15]: where ,(7)Feng et al. [16]: (8)Li et al. [17]: (9)Bozkurt [18]: where ,(10)Das et al. [19]:

In [11] Grimmet points out that (3) generalizes (5). Grone and Merris [12] observed that, by the application of arithmetic-geometric mean inequality, (4) leads to (3). Das [14] stated that (7) is sharp for or , but (3), (4), (5), and (6) are sharp for only . In [17] Li et al. indicated that (11) is sharp for , , or , but (3) is sharp for only and (7) and (9) are sharp for or . However, Das et al. [19] proved that (11) is not true for . In [15, 16, 18] the authors showed that (8) is better than (3), (9) is better than (7) and (10), and (12) is better than (4). For more bounds and the relations between the number of spanning trees and the structural parameters of graphs such as connectivity, chromatic number, independence number, and clique number, see [17, 19].

We organize this paper in the following way. In Section 2, we give some previously known results which will be needed later. In Section 3, we obtain some bounds for the number of spanning trees of connected graphs in terms of the number of vertices , the number of edges , maximum vertex degree , minimum vertex degree , first Zagreb index , and Randić index . We also showed that some of our results on connected bipartite graphs improve the bounds (9) and (10) for these graphs.

2. Lemmas

In this section, we give some useful lemmas which will be used later. Firstly, we introduce an auxiliary quantity for a graph as where and are the maximum and the minimum vertex degree of , respectively.

The result in the following lemma is also known as Kober’s inequality.

Lemma 1 (see [20]). Let be nonnegative numbers and let be their arithmetic and geometric means, respectively. Then
Moreover, equality in (16) holds if and only if .

Lemma 2 (see [21]). Let be a graph with vertices and normalized Laplacian matrix without isolated vertices. Then

Lemma 3 (see [8]). Let be a graph with vertices and without isolated vertices. Then if and only if is a complete graph .

Lemma 4 (see [8]). Let be a connected graph with vertices. Then if and only if or .

Note that, the Laplacian eigenvalues of a bipartite graph coincide with its signless Laplacian eigenvalues, that is, eigenvalues of the signless Laplacian matrix [9, 10, 22]. Thus, one can arrive at the following result.

Lemma 5 (see [23, 24]). Let be a connected bipartite graph with vertices and let be the maximum vertex degree of . Then with either equalities if and only if is a star graph .

Lemma 6 (see [9]). Let be a graph with vertices. Then , with equality if and only if is disconnected.

Lemma 7 (see [14]). Let be a connected graph with vertices. Then if and only if or or .

3. Main Results

Recently, Das et al. [19] established upper and lower bounds on applying Kober’s inequality to Laplacian eigenvalues of a connected graph . We now consider Kober’s inequality for the normalized Laplacian eigenvalues of in order to present some bounds on .

Theorem 8. Let be a connected graph with vertices, edges, and Randić index . Then
Moreover, equalities in (19) and (20) hold if and only if .

Proof. Taking , , and in Lemma 1, we get
By the proof of Theorem 7 in [19] and Lemma 2, we have
Then, combining (21) with this and (2), we get
This implies that
Hence we obtain the first part of the theorem. Now we suppose that the equalities in (19) and (20) hold. Then, by Lemma 1, we have . Therefore, from Lemma 3, we get that .
Conversely, we can easily see that the equalities in (19) and (20) hold for the complete graph .

We now consider the above theorem for connected bipartite graphs.

Theorem 9. Let be a connected bipartite graph with vertices, edges, and Randić index . Then
Moreover, equalities in (25) hold if and only if .

Proof. Taking , , and in Lemma 1, we have
Since is bipartite, we also have [6]. Then, by Lemma 2, we get
Therefore, combining (26) with this and (2), we arrive at
This implies that
Hence we get the inequalities (25). Now we suppose that the equalities in (25) hold. Then, by Lemma 1, we have . Therefore, by Lemma 4, we conclude that .
Conversely, we can easily see that the equalities in (25) hold for the complete bipartite graph .

We now present the improvement of the results obtained in [16] for bipartite graphs.

Theorem 10. Let be a connected bipartite graph with vertices and edges and let be given by (14). Then with equality if and only if .

Proof. From (1) and Lemmas 5–7, one can prove (30) in a similar way to the proof of Theorem 1.1 in [16].

Remark 11. From Lemma 5, we have . Then by the proof of Theorem 1.1 in [16], one may conclude that (30) improves (9) for bipartite graphs.

Theorem 12. Let be a connected bipartite graph with vertices, edges, and first Zagreb index and let be given by (14). Then with equality if and only if .

Proof. From (1) and Lemmas 5–7, the proof of (31) can be easily given in a similar way to the proof of Theorem 1.2 in [16].

Remark 13. From Lemma 5, we have . Then by the proof of Theorem 1.2 in [16], one may conclude that (31) improves (10) for bipartite graphs.

Remark 14. By using the similar manner in [16], one can easily show that (30) is better than (31). Moreover, if we can obtain a new bound , then we can improve the bounds (30) and (31).

Example 15. Let be a graph with vertex set and edge set
For this graph, is equal to . At rounded three decimal places, the bounds (8), (9), (11), (12), (13), and (19) give , , , , , , , and , respectively. This shows that the bound (19) is the best among the mentioned upper bounds for . But in general sense, they are not comparable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are partially supported by TUBITAK and the Office of Selçuk University Research Project (BAP).