On the Number of Spanning Trees of Graphs

We establish some bounds for the number of spanning trees of connected graphs in terms of the number of vertices (n), the number of edges (m), maximum vertex degree (Δ1), minimum vertex degree (δ), first Zagreb index (M 1), and Randić index (R −1).


Introduction
Let be a simple connected graph with vertices and edges. Let ( ) = {V 1 , V 2 , . . . , V } be the vertex set and ( ) = { 1 , 2 , . . . , } the edge set of . If any two vertices V and V of are adjacent, that is, V V ∈ ( ), then we use the notation V ∼ V . For V ∈ ( ), the degree of the vertex V , denoted by , is the number of the vertices adjacent to V . Let Δ 1 , Δ 2 , and be the maximum, the second maximum, and the minimum vertex degree of , respectively. Let 1 = 1 ( ) = ∑ =1 2 be the first Zagreb index [1] and = ( ) = ∑ V ∼V ( ) the general Randić index [2] of the graph , where ̸ = 0 is a fixed real number. Note that the Randić index −1 = −1 ( ) = ∑ V ∼V 1/ is also well studied in the literature. For more details on −1 , 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 1 ∪ 2 be the vertex-disjoint union of the graphs 1 and 2 . The graph 1 ∨ 2 is obtained from 1 ∪ 2 by adding all possible edges from vertices of 1 to vertices of 2 ; that is, 1 ∨ 2 = 1 ∪ 2 [5].
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 (Δ 1 ), minimum vertex degree ( ), first Zagreb index ( 1 ), and Randić index ( −1 ). We also showed that some of our results on connected bipartite graphs improve the bounds (9) and (10) for these graphs.

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 Δ 1 and are the maximum and the minimum vertex degree of , respectively. The result in the following lemma is also known as Kober's inequality.
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 ≥ 3 vertices and let Δ 1 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 1 ≤ , with equality if and only if is disconnected.

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 −1 . Then
Moreover, equalities in (19) and (20) hold if and only if ≅ .
Proof. Taking = − 1, = 2 , and = 1, 2, . . . , − 1 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 1 = 2 = ⋅ ⋅ ⋅ = −1 . 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 > 2 vertices, edges, and Randić index −1 . Then Moreover, equalities in (25) hold if and only if ≅ , .
We now present the improvement of the results obtained in [16] for bipartite graphs.
Theorem 10. Let be a connected bipartite graph with ≥ 3 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].
Theorem 12. Let be a connected bipartite graph with ≥ 3 vertices, edges, and first Zagreb index 1 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].