Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 294038, 5 pages http://dx.doi.org/10.1155/2014/294038
Research Article On the Number of Spanning Trees of Graphs F. Burcu Bozkurt and DurmuG Bozkurt Department of Mathematics, Science Faculty, Selc¸uk University, Alaeddin Keykubat Campus, 42075 Konya, Turkey Correspondence should be addressed to S¸. Burcu Bozkurt; [email protected] Received 29 August 2013; Accepted 24 December 2013; Published 10 February 2014 Academic Editors: C. D. Fonseca and A. Jaballah Copyright © 2014 S¸. B. Bozkurt and D. Bozkurt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 (Δ 1 ), minimum vertex degree (𝛿), first Zagreb index (𝑀1 ), and Randi´c index (𝑅−1 ).
1. Introduction Let 𝐺 be a simple connected graph with 𝑛 vertices and 𝑚 edges. Let 𝑉(𝐺) = {V1 , V2 , . . . , V𝑛 } be the vertex set and 𝐸(𝐺) = {𝑒1 , 𝑒2 , . . . , 𝑒𝑚 } the edge set of 𝐺. If any two vertices V𝑖 and V𝑗 of 𝐺 are adjacent, that is, V𝑖 V𝑗 ∈ 𝐸(𝐺), then we use the notation V𝑖 ∼ V𝑗 . For V𝑖 ∈ 𝑉(𝐺), the degree of the vertex V𝑖 , denoted by 𝑑𝑖 , is the number of the vertices adjacent to V𝑖 . Let Δ 1 , Δ 2 , and 𝛿 be the maximum, the second maximum, and the minimum vertex degree of 𝐺, respectively. Let 𝑀1 = 𝑀1 (𝐺) = ∑𝑛𝑖=1 𝑑𝑖2 be the first Zagreb index [1] and 𝑅𝛼 = 𝑅𝛼 (𝐺) = ∑V𝑖 ∼V𝑗 (𝑑𝑖 𝑑𝑗 )𝛼 the general Randi´c index [2] of the graph 𝐺, where 𝛼 ≠ 0 is a fixed real number. Note that the Randi´c index 𝑅−1 = 𝑅−1 (𝐺) = ∑V𝑖 ∼V𝑗 1/𝑑𝑖 𝑑𝑗 is also well studied in the literature. For more details on 𝑅−1 , see [3, 4]. Let 𝐾𝑛 , 𝐾𝑝,𝑞 (𝑝 + 𝑞 = 𝑛), and 𝑆𝑛 denote the complete graph, the complete bipartite graph, and the star graph of order 𝑛, respectively. Let 𝐺 − 𝑒 be the graph obtained by deleting the edge 𝑒 from the graph 𝐺 and let 𝐺 be the complement of 𝐺. Let 𝐺1 ∪ 𝐺2 be the vertex-disjoint union of the graphs 𝐺1 and 𝐺2 . The graph 𝐺1 ∨ 𝐺2 is obtained from 𝐺1 ∪𝐺2 by adding all possible edges from vertices of 𝐺1 to vertices of 𝐺2 ; that is, 𝐺1 ∨ 𝐺2 = 𝐺1 ∪ 𝐺2 [5]. The Laplacian matrix of the graph 𝐺 is the matrix 𝐿(𝐺) = 𝐷(𝐺) − 𝐴(𝐺), where 𝐴(𝐺) and 𝐷(𝐺) are the (0, 1)-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 𝜇1 ≥ 𝜇2 ≥ ⋅ ⋅ ⋅ ≥ 𝜇𝑛 be the Laplacian eigenvalues and 𝜆 1 ≥ 𝜆 2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆 𝑛 the normalized Laplacian eigenvalues of 𝐺. Note that 𝜇𝑛 = 0, 𝜆 𝑛 = 0, and the multiplicities of these zero eigenvalues are equal to the number of connected components of 𝐺; see [6, 7]. For more details on Laplacian and normalized Laplacian eigenvalues, see [6, 8–10]. The number of spanning trees, 𝑡(𝐺), of the graph 𝐺 is equal to the total number of distinct spanning subgraphs of 𝐺 that are trees. This quantity is also known as the complexity of 𝐺 and given by the following formula in terms of the Laplacian eigenvalues [5]: 𝑡 (𝐺) =
1 𝑛−1 ∏𝜇 . 𝑛 𝑖=1 𝑖
(1)
It is well known that the number of spanning trees of 𝐺 is also expressed by the normalized Laplacian eigenvalues as [5, 6] 𝑡 (𝐺) = (
∏𝑛𝑖=1 𝑑𝑖 𝑛−1 ) ∏𝜆 𝑖 . 2𝑚 𝑖=1
(2)
Now, we give some known upper bounds on 𝑡(𝐺): (1) Grimmett [11]: 𝑡 (𝐺) ≤
1 2𝑚 𝑛−1 ( ) , 𝑛 𝑛−1
(3)
(2) Grone and Merris [12]: 𝑛
𝑡 (𝐺) ≤ (
𝑛 𝑛−1 ∏𝑖=1 𝑑𝑖 ) ( ), 𝑛−1 2𝑚
(4)
2
The Scientific World Journal (3) Nosal [13]: for 𝑟-regular graphs, 𝑡 (𝐺) ≤ 𝑛𝑛−2 (
𝑟 𝑛−1 ) , 𝑛−1
(5)
(4) Cvetkovi´c et al. (see [5, page 222]): 2 𝑚 𝑡 (𝐺) ≤ 𝑛𝑛−2 (1 − ) , 𝑛
(6)
where 𝑚 is the number of edges of 𝐺, (5) Das [14]: 𝑡 (𝐺) ≤ (
2𝑚 − Δ 1 − 1 𝑛−2 ) , 𝑛−2
(7)
(6) Zhang [15]: 1 2𝑚 𝑛−1 𝑡 (𝐺) ≤ (1 + (𝑛 − 2) 𝑎) (1 − 𝑎)𝑛−2 ( ) , 𝑛 𝑛−1
(8)
where 𝑎 = ((𝑛(𝑛 − 1) − 2𝑚)/2𝑚𝑛(𝑛 − 2))1/2 ,
2. Lemmas
(7) Feng et al. [16]: Δ +1 2𝑚 − Δ 1 − 1 𝑛−2 )( ) , 𝑡 (𝐺) ≤ ( 1 𝑛 𝑛−2 2
(9)
(𝑛−2)/2
𝑀 + 2𝑚 − (Δ 1 + 1) 𝑡 (𝐺) ≤ ( 1 ) 𝑛−2
,
(10)
(8) Li et al. [17]: 𝑡 (𝐺) ≤ 𝛿(
2𝑚 − Δ 1 − 1 − 𝛿 𝑛−3 ) , 𝑛−3
(11)
𝑡 (𝐺) ≤ (1 + (𝑛 − 2) 𝑏) (1 − 𝑏)
𝑛
𝑛 𝑛−1 ∏𝑖=1 𝑑𝑖 ( ) ( ), 𝑛−1 2𝑚 (12) 1/2
where 𝑏 = ((𝑛 − 1 − Δ 1 )/𝑛(𝑛 − 2)Δ1 )
,
2𝑚 − Δ 1 − 𝛿 𝑛−2 𝑛 𝑛−1 1 Δ 1 𝛿( ) ( ) , 2𝑚 𝑛−2 𝑛−1
𝑡 (𝐺) ≤
1 4𝑚 − (𝑀1 + 2𝑚) ] [ 𝑛 (𝑛 − 1) (𝑛 − 2)
1 2 [Δ 1 + 𝛿 + √(Δ 1 − 𝛿) + 4Δ 1 ] , 2
(14)
where Δ 1 and 𝛿 are the maximum and the minimum vertex degree of 𝐺, respectively. The result in the following lemma is also known as Kober’s inequality.
1 𝑁 𝛽 = ∑𝑥𝑖 , 𝑁 𝑖=1
𝑁
1/𝑁
𝛾 = (∏𝑥𝑖 )
(15)
𝑖=1
be their arithmetic and geometric means, respectively. Then
(16)
1 2 𝑛−3 (Δ 1 + Δ 2 + 1) (2𝑚 − Δ 1 − Δ 2 − 1) , 𝑡 (𝐺) < 𝑛−3 4𝑛(𝑛 − 3) 2
𝛼=
2 2 1 1 ∑(√𝑥𝑖 − √𝑥𝑗 ) ≤ 𝛽 − 𝛾 ≤ ∑(√𝑥𝑖 − √𝑥𝑗 ) . 𝑁 (𝑁 − 1) 𝑖
Research Article On the Number of Spanning Trees of Graphs F. Burcu Bozkurt and DurmuG Bozkurt Department of Mathematics, Science Faculty, Selc¸uk University, Alaeddin Keykubat Campus, 42075 Konya, Turkey Correspondence should be addressed to S¸. Burcu Bozkurt; [email protected] Received 29 August 2013; Accepted 24 December 2013; Published 10 February 2014 Academic Editors: C. D. Fonseca and A. Jaballah Copyright © 2014 S¸. B. Bozkurt and D. Bozkurt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 (Δ 1 ), minimum vertex degree (𝛿), first Zagreb index (𝑀1 ), and Randi´c index (𝑅−1 ).
1. Introduction Let 𝐺 be a simple connected graph with 𝑛 vertices and 𝑚 edges. Let 𝑉(𝐺) = {V1 , V2 , . . . , V𝑛 } be the vertex set and 𝐸(𝐺) = {𝑒1 , 𝑒2 , . . . , 𝑒𝑚 } the edge set of 𝐺. If any two vertices V𝑖 and V𝑗 of 𝐺 are adjacent, that is, V𝑖 V𝑗 ∈ 𝐸(𝐺), then we use the notation V𝑖 ∼ V𝑗 . For V𝑖 ∈ 𝑉(𝐺), the degree of the vertex V𝑖 , denoted by 𝑑𝑖 , is the number of the vertices adjacent to V𝑖 . Let Δ 1 , Δ 2 , and 𝛿 be the maximum, the second maximum, and the minimum vertex degree of 𝐺, respectively. Let 𝑀1 = 𝑀1 (𝐺) = ∑𝑛𝑖=1 𝑑𝑖2 be the first Zagreb index [1] and 𝑅𝛼 = 𝑅𝛼 (𝐺) = ∑V𝑖 ∼V𝑗 (𝑑𝑖 𝑑𝑗 )𝛼 the general Randi´c index [2] of the graph 𝐺, where 𝛼 ≠ 0 is a fixed real number. Note that the Randi´c index 𝑅−1 = 𝑅−1 (𝐺) = ∑V𝑖 ∼V𝑗 1/𝑑𝑖 𝑑𝑗 is also well studied in the literature. For more details on 𝑅−1 , see [3, 4]. Let 𝐾𝑛 , 𝐾𝑝,𝑞 (𝑝 + 𝑞 = 𝑛), and 𝑆𝑛 denote the complete graph, the complete bipartite graph, and the star graph of order 𝑛, respectively. Let 𝐺 − 𝑒 be the graph obtained by deleting the edge 𝑒 from the graph 𝐺 and let 𝐺 be the complement of 𝐺. Let 𝐺1 ∪ 𝐺2 be the vertex-disjoint union of the graphs 𝐺1 and 𝐺2 . The graph 𝐺1 ∨ 𝐺2 is obtained from 𝐺1 ∪𝐺2 by adding all possible edges from vertices of 𝐺1 to vertices of 𝐺2 ; that is, 𝐺1 ∨ 𝐺2 = 𝐺1 ∪ 𝐺2 [5]. The Laplacian matrix of the graph 𝐺 is the matrix 𝐿(𝐺) = 𝐷(𝐺) − 𝐴(𝐺), where 𝐴(𝐺) and 𝐷(𝐺) are the (0, 1)-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 𝜇1 ≥ 𝜇2 ≥ ⋅ ⋅ ⋅ ≥ 𝜇𝑛 be the Laplacian eigenvalues and 𝜆 1 ≥ 𝜆 2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆 𝑛 the normalized Laplacian eigenvalues of 𝐺. Note that 𝜇𝑛 = 0, 𝜆 𝑛 = 0, and the multiplicities of these zero eigenvalues are equal to the number of connected components of 𝐺; see [6, 7]. For more details on Laplacian and normalized Laplacian eigenvalues, see [6, 8–10]. The number of spanning trees, 𝑡(𝐺), of the graph 𝐺 is equal to the total number of distinct spanning subgraphs of 𝐺 that are trees. This quantity is also known as the complexity of 𝐺 and given by the following formula in terms of the Laplacian eigenvalues [5]: 𝑡 (𝐺) =
1 𝑛−1 ∏𝜇 . 𝑛 𝑖=1 𝑖
(1)
It is well known that the number of spanning trees of 𝐺 is also expressed by the normalized Laplacian eigenvalues as [5, 6] 𝑡 (𝐺) = (
∏𝑛𝑖=1 𝑑𝑖 𝑛−1 ) ∏𝜆 𝑖 . 2𝑚 𝑖=1
(2)
Now, we give some known upper bounds on 𝑡(𝐺): (1) Grimmett [11]: 𝑡 (𝐺) ≤
1 2𝑚 𝑛−1 ( ) , 𝑛 𝑛−1
(3)
(2) Grone and Merris [12]: 𝑛
𝑡 (𝐺) ≤ (
𝑛 𝑛−1 ∏𝑖=1 𝑑𝑖 ) ( ), 𝑛−1 2𝑚
(4)
2
The Scientific World Journal (3) Nosal [13]: for 𝑟-regular graphs, 𝑡 (𝐺) ≤ 𝑛𝑛−2 (
𝑟 𝑛−1 ) , 𝑛−1
(5)
(4) Cvetkovi´c et al. (see [5, page 222]): 2 𝑚 𝑡 (𝐺) ≤ 𝑛𝑛−2 (1 − ) , 𝑛
(6)
where 𝑚 is the number of edges of 𝐺, (5) Das [14]: 𝑡 (𝐺) ≤ (
2𝑚 − Δ 1 − 1 𝑛−2 ) , 𝑛−2
(7)
(6) Zhang [15]: 1 2𝑚 𝑛−1 𝑡 (𝐺) ≤ (1 + (𝑛 − 2) 𝑎) (1 − 𝑎)𝑛−2 ( ) , 𝑛 𝑛−1
(8)
where 𝑎 = ((𝑛(𝑛 − 1) − 2𝑚)/2𝑚𝑛(𝑛 − 2))1/2 ,
2. Lemmas
(7) Feng et al. [16]: Δ +1 2𝑚 − Δ 1 − 1 𝑛−2 )( ) , 𝑡 (𝐺) ≤ ( 1 𝑛 𝑛−2 2
(9)
(𝑛−2)/2
𝑀 + 2𝑚 − (Δ 1 + 1) 𝑡 (𝐺) ≤ ( 1 ) 𝑛−2
,
(10)
(8) Li et al. [17]: 𝑡 (𝐺) ≤ 𝛿(
2𝑚 − Δ 1 − 1 − 𝛿 𝑛−3 ) , 𝑛−3
(11)
𝑡 (𝐺) ≤ (1 + (𝑛 − 2) 𝑏) (1 − 𝑏)
𝑛
𝑛 𝑛−1 ∏𝑖=1 𝑑𝑖 ( ) ( ), 𝑛−1 2𝑚 (12) 1/2
where 𝑏 = ((𝑛 − 1 − Δ 1 )/𝑛(𝑛 − 2)Δ1 )
,
2𝑚 − Δ 1 − 𝛿 𝑛−2 𝑛 𝑛−1 1 Δ 1 𝛿( ) ( ) , 2𝑚 𝑛−2 𝑛−1
𝑡 (𝐺) ≤
1 4𝑚 − (𝑀1 + 2𝑚) ] [ 𝑛 (𝑛 − 1) (𝑛 − 2)
1 2 [Δ 1 + 𝛿 + √(Δ 1 − 𝛿) + 4Δ 1 ] , 2
(14)
where Δ 1 and 𝛿 are the maximum and the minimum vertex degree of 𝐺, respectively. The result in the following lemma is also known as Kober’s inequality.
1 𝑁 𝛽 = ∑𝑥𝑖 , 𝑁 𝑖=1
𝑁
1/𝑁
𝛾 = (∏𝑥𝑖 )
(15)
𝑖=1
be their arithmetic and geometric means, respectively. Then
(16)
1 2 𝑛−3 (Δ 1 + Δ 2 + 1) (2𝑚 − Δ 1 − Δ 2 − 1) , 𝑡 (𝐺) < 𝑛−3 4𝑛(𝑛 − 3) 2
𝛼=
2 2 1 1 ∑(√𝑥𝑖 − √𝑥𝑗 ) ≤ 𝛽 − 𝛾 ≤ ∑(√𝑥𝑖 − √𝑥𝑗 ) . 𝑁 (𝑁 − 1) 𝑖
