The Number of Spanning Trees in the Composition Graphs

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 613685, 5 pages http://dx.doi.org/10.1155/2014/613685

Research Article The Number of Spanning Trees in the Composition Graphs Feng Li1,2,3 1

College of Computer Science, Qinghai Normal University, Xi’ning 810003, China Institute of Information and System Sciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China 3 Ministry of Education Key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an 710049, China 2

Correspondence should be addressed to Feng Li; [email protected] Received 4 October 2013; Accepted 10 February 2014; Published 19 March 2014 Academic Editor: J. J. Judice Copyright © 2014 Feng Li. 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. Using the composition of some existing smaller graphs to construct some large graphs, the number of spanning trees and the Laplacian eigenvalues of such large graphs are also closely related to those of the corresponding smaller ones. By using tools from linear algebra and matrix theory, we establish closed formulae for the number of spanning trees of the composition of two graphs with one of them being an arbitrary complete 3-partite graph and the other being an arbitrary graph. Our results extend some of the previous work, which depend on the structural parameters such as the number of vertices and eigenvalues of the small graphs only.

1. Introduction In reliable network synthesis, given the class Ω(𝑛, 𝑚) of all connected graphs with 𝑛 vertices and 𝑚 edges, it is very important to seek graphs (known as the 𝑡-optimal graphs) with the most number of spanning trees, so the number of spanning trees is closely connected to reliable network design [1, 2]. When using a probabilistic graph to model a communication network, the reliability of a network can be expressed as a function of the number of connected spanning subgraphs (spanning trees) of different orders. Thus, the number of spanning trees of a graph describing a network is one of the most natural characteristics for its reliability, and deriving closed formulae of the number of spanning trees for various graphs has attracted the attention of a lot of researchers [3–5]. It is well known that the number of spanning trees of some specific family of graphs can be given explicitly, which include the complete graph 𝐾𝑛 , the path 𝑃𝑛 , the cycle 𝐶𝑛 , the wheel 𝑊𝑛 , and the M¨obius ladders; “almost-complete” graphs, the threshold graphs, and some multicomplete/star related graphs can also be obtained [6–9]. Mathematic structures can be properly understood if one has a grasp of their symmetries; it also helps to know whether they can be constructed from smaller constituents,

since many large graphs (networks) are usually composed from some existing smaller graphs (networks) through graph operations (say, product [10]). In this paper, we are mainly concerned with the number of spanning trees of the composition of two graphs. Let 𝐺1 and 𝐺2 be two simple graphs; the composition graphs of 𝐺1 and 𝐺2 , denoted by 𝐺1 ⊙ 𝐺2 , are a graph with vertex set 𝑉(𝐺1 ) × 𝑉(𝐺2 ), and there is an edge between (𝑢1 , 𝑢2 ) and (V1 , V2 ) if there is an edge connecting 𝑢1 and V1 in 𝐺1 , or if 𝑢1 = V1 and there is an edge connecting 𝑢2 and V2 in 𝐺2 . Sometimes it is also called lexicographic product. The topological parameters and some properties of such large graphs (networks) are associated strongly with those of the corresponding smaller ones. Since the composition (lexicographic product) of two graphs is noncommutative, for example see Figure 1. The structure of 𝐾𝑙1 ,𝑙2 ,...,𝑙𝑠 ⊙ 𝐺 is different from the structure of 𝐺 ⊙ 𝐾𝑙1 ,𝑙2 ,...,𝑙𝑠 . However, the number of the spanning trees of the composition graphs 𝐺 ⊙ 𝐾𝑙1 ,𝑙2 ,...,𝑙𝑠 is got in [11]. In this paper, we enumerate spanning trees of the composition graphs with one of them being an arbitrary complete 3-partite graph 𝐾𝑠1 ,𝑠2 ,𝑠3 and the other an arbitrary graph 𝐺. The number of the spanning trees of the composition graphs 𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺 depends only on the number of vertices and eigenvalues of small graphs only.

2

Mathematical Problems in Engineering

P2

P3

P3 ⊙ P2

P2 ⊙ P3

Figure 1

A major open problem which still remains is to devise a technique that would derive close formulae for 𝑡(𝐺1 ⊙ 𝐺2 ) and 𝑡(𝐺2 ⊙ 𝐺1 ), where 𝐺1 and 𝐺2 are arbitrary graphs, or one of them is an arbitrary regular graph. Fiedler [12] gave the eigenvalues of nonnegative symmetric matrices where he obtains the description of the eigenspace of some matrix operations. This actually has been applied for some graph product in [13]. There may be some relations between the techniques used in this note. For more details on the composition graphs and matrix operations the reader is referred to [14–16].

2. Preliminaries We start with fixing some notations. Throughout the paper, let 𝐺 = (𝑉, 𝐸) be a simple graph with vertex set 𝑉 = {V1 , V2 , . . . , V𝑛 } and edge set 𝐸 = {𝑒1 , 𝑒2 , . . . , 𝑒𝑚 }. Let 𝐴(𝐺) = (ℎ𝑖𝑗 )𝑛×𝑛 be the (0, 1)-adjacency matrix of 𝐺, and let 𝐷(𝐺) = diag(𝑑1 , 𝑑2 , . . . , 𝑑𝑛 ) be the diagonal matrix with 𝑑𝑖 being the degree of the 𝑖th vertex of 𝐺. The Laplacian matrix of 𝐺 is defined to be 𝐿(𝐺) = 𝐷(𝐺) − 𝐴(𝐺), and the corresponding characteristic polynomial of 𝐿(𝐺) is denoted by 𝑃(𝐺, 𝜆). Since the matrix 𝐿(𝐺) is symmetric, all its eigenvalues are real. We assume without loss of generality that they are arranged in the nondecreasing order; that is, 0 = 𝜆 1 ≤ 𝜆 2 ≤ ⋅ ⋅ ⋅ ≤ 𝜆 𝑛 . The number of spanning trees of 𝐺 is denoted by 𝑡(𝐺). For other terminologies and notations which are not defined here, the reader is referred to [17]. Lemma 1 (see [17, 18]). Let 0 = 𝜆 1 ≤ 𝜆 2 ≤ ⋅ ⋅ ⋅ ≤ 𝜆 𝑛 be the eigenvalues of the Laplacian matrix of the graph 𝐺; then 󵄨󵄨1 ℎ ⋅ ⋅ ⋅ ℎ1𝑛 󵄨󵄨󵄨 󵄨󵄨 12 󵄨󵄨1 𝜆 − 𝑑 ⋅ ⋅ ⋅ ℎ 󵄨󵄨󵄨 󵄨󵄨 2 2𝑛 󵄨󵄨 ∏ (𝜆 − 𝜆 𝑖 (𝐺)) = 󵄨󵄨󵄨 .. . .. 󵄨󵄨󵄨󵄨 := 𝐵 (𝐺, 𝜆) . (1) . 󵄨 󵄨 . . . 󵄨󵄨 𝑖=2 󵄨󵄨 󵄨 󵄨󵄨1 ℎ ⋅ ⋅ ⋅ 𝜆 − 𝑑𝑛 󵄨󵄨󵄨 󵄨 𝑛2

Theorem 3. Let 𝐺 be a simple graph with 𝑛 vertices. Let 𝜆 𝑖 , 𝑖 = 1, 2, . . . , 𝑛, be the Laplacian eigenvalues of 𝐿(𝐺) which are arranged in the nondecreasing order. Then, the eigenvalues of 𝐿(𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙𝐺) are 0, {𝑛𝑤}2 , {𝑛(𝑤−𝑠𝑖 )}(𝑠𝑖 −1) , and {𝜆 𝑘 (𝐺)+𝑛(𝑤− 𝑠𝑖 )}(𝑛−1)𝑠𝑖 , where the exponents denote the multiplicities of the eigenvalues, for 𝑖 = 1, 2, 3, 𝑘 = 2, 3, . . . , 𝑛, and 𝑤 = ∑3𝑗=1 𝑠𝑗 . Proof. Let 𝐿(𝐺) be the Laplacian matrix of the graph 𝐺. Let 𝑢1 , 𝑢2 , . . . , 𝑢𝑠1 , . . . , 𝑢𝑠1 +𝑠2 , . . . , 𝑢𝑤 (𝑤 = ∑3𝑖=1 𝑠𝑖 ) be the labels of the graph 𝐾𝑠1 ,𝑠2 ,𝑠3 , where 𝑢𝑗 ∼ 𝑢𝑖 if and only if 𝑖 ∈ 𝐴 𝑎 and 𝑗 ∉ 𝐴 𝑎 , or 𝑖 ∉ 𝐴 𝑎 and 𝑗 ∈ 𝐴 𝑎 , where 𝐴 𝑎 = {𝑥 | ∑𝑎−1 𝑖=0 𝑠𝑖 + 1 ≤ 𝑥 ≤ ∑𝑎𝑖=0 𝑠𝑖 } (assume that 𝑠0 = 0), for 𝑎 = 1, 2, 3. The node (𝑢𝑖 , V𝑗 ) of the graph 𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺 is the lexicographic order among vertex sequences; then we can represent the Laplacian matrix 𝐿(𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺) in the following block matrix form: 𝐿 (𝐾𝑠1 ,𝑠2 ,𝑠3

𝐴11 𝐴12 ⋅ ⋅ ⋅ 𝐴1𝑤 [ 𝐴21 𝐴22 ⋅ ⋅ ⋅ 𝐴2𝑤 ] ] [ −[ . , .. .. ] [ .. . . ] 𝑤1 𝑤2 𝑤𝑤 [𝐴 𝐴 ⋅ ⋅ ⋅ 𝐴 ]

Lemma 2 (see [19]). Let 𝐺 be graph on 𝑛 vertices. Let 0 = 𝜆 1 ≤ 𝜆 2 ≤ ⋅ ⋅ ⋅ ≤ 𝜆 𝑛 denote the eigenvalues of 𝐿(𝐺). Then 𝑡 (𝐺) =

1 𝑛 ∏𝜆 (𝐺) . 𝑛 𝑖=2 𝑖

(2)

Let 𝐽0 be the all-zero matrix, and let 𝐽𝑛 be the all-one matrix, of order 𝑛. Now we give the main results of this paper.

(3)

where the element 𝑎𝑗l𝑖𝑘 of the block 𝐴𝑖𝑘 = (𝑎𝑗𝑙𝑖𝑘 )𝑛×𝑛 denotes the adjacency relation of the two nodes (𝑢𝑖 , V𝑗 ) and (𝑢𝑘 , V𝑙 ). By the definition of the composition graphs, (𝑢𝑖 , V𝑗 ) ∼ (𝑢𝑘 , V𝑙 ) if and only if 𝑢𝑖 ∼ 𝑢𝑘 ; or 𝑢𝑖 = 𝑢𝑘 and V𝑗 ∼ V𝑙 . Thus, we have 0 { { { { 1 𝑎𝑗𝑙𝑖𝑘 = { {0 { { {ℎ𝑗𝑙

𝑛

It is well known that the number of spanning trees 𝑡(𝐺) of a given graph 𝐺 can be calculated through Kirchhoff ’s “Matrix-Tree Theorem.” This is one of the first (and most impressive) contributions of spectral theory.

𝐷1 0 ⋅ ⋅ ⋅ 0 0 𝐷2 ⋅ ⋅ ⋅ 0 ] ] .. .. ] .. . . . ] 0 0 ⋅ ⋅ ⋅ 𝐷 𝑤] [

[ [ ⊙ 𝐺) = [ [

𝑖 ≠ 𝑘 𝑖, 𝑘 ∈ 𝐴 𝑎 𝑖 ≠ 𝑘 otherwise 𝑖=𝑘𝑗=𝑙 𝑖 = 𝑘 𝑗 ≠ 𝑙

(𝑎 = 1, 2, 3) .

(4)

That is, 𝑖 ≠ 𝑘, 𝑖, 𝑘 ∈ 𝐴 𝑎 𝐽 { {0 𝐴𝑖𝑘 = {𝐽𝑛 (𝑎 = 1, 2, 3) . (5) 𝑖 ≠ 𝑘 otherwise { {𝐴 (𝐺) 𝑖 = 𝑘 otherwise Similarly, since the sum of the elements of every row (resp., column) of Laplacian matrix 𝐿(𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺) is zero, we have 𝐷 (𝐺) + 𝑛 (𝑤 − 𝑠1 ) 𝐼𝑛 { { 𝐷𝑖 = {𝐷 (𝐺) + 𝑛 (𝑤 − 𝑠2 ) 𝐼𝑛 { {𝐷 (𝐺) + 𝑛 (𝑤 − 𝑠3 ) 𝐼𝑛

1 ≤ 𝑖 ≤ 𝑠1 𝑠1 + 1 ≤ 𝑖 ≤ 𝑠1 + 𝑠2 𝑠1 + 𝑠2 + 1 ≤ 𝑖 ≤ 𝑠1 + 𝑠2 + 𝑠3 . (6)


3

Let 𝐷𝑖 = 𝐷(𝑠𝑖 ), where 𝐷(𝑠𝑖 ) is an 𝑛 × 𝑛 matrix, 𝑖 = 1, 2, . . . , 𝑤. Then we get

𝐿 (𝐾𝑠1 ,𝑠2 ,𝑠3

󵄨 󵄨󵄨 𝐶 󵄨󵄨 1 −𝐽𝑠1 ×𝑠2 −𝐽𝑠1 ×𝑠3 󵄨󵄨󵄨 󵄨 󵄨󵄨−𝐽 ⊙ 𝐺) = 󵄨󵄨 𝑠2 ×𝑠1 𝐶2 −𝐽𝑠2 ×𝑠3 󵄨󵄨󵄨 , 󵄨󵄨 󵄨󵄨 󵄨󵄨−𝐽𝑠3 ×𝑠1 −𝐽𝑠3 ×𝑠2 𝐶3 󵄨󵄨𝑤×𝑤

3

𝑤 = ∑𝑠𝑖 , 𝑖=1

(7)

where 𝐵∗ 𝐵 (𝐺, 𝜆 − 𝑛 (𝑤 − 𝑠1 )) 0 ⋅⋅⋅ 0 [ ] ) − 𝐿 ⋅ ⋅ ⋅ 0 𝐶 𝜆 (𝑠 (𝐺) 1 [ ] ] =[ .. .. .. [ ] [ ] . . . 𝐶 0 ⋅ ⋅ ⋅ 𝜆 (𝑠1 ) − 𝐿 (𝐺)]𝑠1 ×𝑠2 [

1 [1 [ 𝐶 = [ .. [. [1

where 𝐽𝑠𝑖 ×𝑠𝑗 is an 𝑠𝑖 by 𝑠𝑗 block matrix with each block being equal to 𝐽𝑛 , 𝑖, 𝑗 = 1, 2, 3, and 𝐷 (𝑠𝑖 ) − 𝐴 (𝐺) 0 0 ] , 0 𝐷 (𝑠𝑖 ) − 𝐴 (𝐺) 0 𝐶𝑖 = [ 0 0 𝐷 (𝑠 ) − 𝐴 (𝐺) 𝑖 [ ]𝑠𝑖 ×𝑠𝑖 𝑖 = 1, 2, 3. (8) So the characteristic polynomial of the Laplacian matrix is 𝑃 (𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺, 𝜆) = det (𝜆𝐼𝑝×𝑝 − 𝐿 (𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺)) 󵄨󵄨 𝐵 𝐽 󵄨󵄨 󵄨󵄨 1 𝑠1 ×𝑠2 𝐽𝑠1 ×𝑠3 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨𝐽𝑠2 ×𝑠1 𝐵2 𝐽𝑠2 ×𝑠3 󵄨󵄨󵄨 , 󵄨󵄨 󵄨 󵄨󵄨𝐽𝑠 ×𝑠 𝐽𝑠 ×𝑠 𝐵3 󵄨󵄨󵄨 󵄨 3 1 3 2 󵄨𝑤×𝑤

(9) 𝑛

𝑤 = ∑𝑠𝑖 , 𝑖=1

(10)

0 ⋅⋅⋅ 0 ⋅⋅⋅ .. .

0 0] ] .. ] . .]

0 ⋅ ⋅ ⋅ 0]

(14)

In order to compute the determinant in (13), we start by subtracting the first column from the (𝑛𝑠1 + 1)th column to the last column, getting 󵄨󵄨 𝐵∗ 0 0 󵄨󵄨󵄨󵄨 3 󵄨󵄨 󵄨 󵄨󵄨𝐽 𝑃 (𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺, 𝜆) = 𝜆󵄨󵄨 𝑠2 ×𝑠1 𝐴 2 0 󵄨󵄨󵄨 , 𝑤 = ∑𝑠𝑖 , 󵄨 󵄨󵄨 𝑖=1 󵄨󵄨𝐽𝑠3 ×𝑠1 𝐽𝑠3 ×𝑠2 𝐴 3 󵄨󵄨󵄨𝑤×𝑤 (15) where 𝐴𝑗 [ =[ [

𝜆 (𝑠𝑗 ) − 𝐿 (𝐺) − 𝐽𝑛 −𝐽𝑛

[

⋅⋅⋅

−𝐽𝑛

𝜆 (𝑠𝑗 ) − 𝐿 (𝐺) − 𝐽𝑛 ⋅ ⋅ ⋅

−𝐽𝑛

−𝐽𝑛

. . .

. . .

−𝐽𝑛

−𝐽𝑛

] ], ]

. . .

⋅ ⋅ ⋅ 𝜆 (𝑠𝑗 ) − 𝐿 (𝐺) − 𝐽𝑛 ] 𝑠 ×𝑠 𝑗 𝑗

(16) for 𝑗 = 2, 3. It follows from (15) that

where 𝜆 (𝑠𝑖 ) − 𝐿 (𝐺) 0 0 ] , 0 𝜆 (𝑠𝑖 ) − 𝐿 (𝐺) 0 𝐵𝑖 = [ 0 0 𝜆 (𝑠𝑖 ) − 𝐿 (𝐺)]𝑠 ×𝑠 [ 𝑖 𝑖 (11) and 𝜆(𝑠𝑖 ) = (𝜆 − 𝑛(𝑤 − 𝑠𝑖 ))𝐼𝑛 , for 𝑖 = 1, 2, 3. In (10), according to the property of the Laplacian matrix, add all the columns to the first column; then every element of the first column is equal to 𝜆. Extract 𝜆; then every element of the first column is equal to 1. Using the notation 𝐵(𝐺, 𝜆) from Lemma 1, we have 𝑃 (𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺, 𝜆) = det (𝜆𝐼𝑤×𝑤 − 𝐿 (𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺)) 󵄨󵄨 𝐵∗ 𝐽 󵄨󵄨 𝑠1 ×𝑠2 𝐽𝑠1 ×𝑠3 󵄨󵄨 󵄨 = 𝜆󵄨󵄨𝐽𝑠2 ×𝑠1 𝐵2 𝐽𝑠2 ×𝑠3 󵄨󵄨 󵄨󵄨𝐽𝑠 ×𝑠 𝐽𝑠 ×𝑠 𝐵3 󵄨 3 1 3 2

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 , 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑤×𝑤 3

𝑤 = ∑𝑠𝑖 , 𝑖=1

(12)

(13)

𝑃 (𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺, 𝜆) = 𝜆𝐵 (𝐺, 𝜆 − 𝑛 (𝑤 − 𝑠1 ))

(17)

× [det ((𝜆 − 𝑛 (𝑤 − 𝑠1 )) 𝐼𝑛 − 𝐿 (𝐺))] where

󵄨󵄨𝐴 0 󵄨󵄨 󵄨 󵄨󵄨 . 𝐹 = 󵄨󵄨󵄨 2 󵄨 󵄨󵄨 0 𝐴 3 󵄨󵄨󵄨(𝑤−𝑠1 )×(𝑤−𝑠1 )

𝑠1 −1

× 𝐹,

(18)

Let 𝐻𝑖 = det(𝐴 𝑖 ), 𝑖 = 2, 3. According to the property of the determinant, we have 3

𝐹 = 𝐻2 × 𝐻3 = ∏𝐻𝑖 ,

(19)

𝑖=2

where 𝐻𝑖 󵄨󵄨 󵄨󵄨𝜆 (𝑠𝑖 ) − 𝐿 (𝐺) − 𝐽𝑛 −𝐽𝑛 ⋅⋅⋅ −𝐽𝑛 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 −𝐽𝑛 𝜆 (𝑠𝑖 ) − 𝐿 (𝐺) − 𝐽𝑛 ⋅ ⋅ ⋅ −𝐽𝑛 󵄨󵄨 󵄨󵄨 󵄨󵄨 , = 󵄨󵄨󵄨 .. .. .. 󵄨󵄨 󵄨󵄨 . . . 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 −𝐽𝑛 −𝐽𝑛 ⋅ ⋅ ⋅ 𝜆 (𝑠𝑖 ) − 𝐿 (𝐺) − 𝐽𝑛 󵄨󵄨𝑠 ×𝑠 𝑖 𝑖 (20) for 𝑖 = 2, 3.

4


Adding all the columns to the first column and extracting the term 𝜆 − 𝑛 ∑3𝑖=1 𝑠𝑖 , let 𝑋 1

ℎ12 − 1

[1

ℎ𝑛2 − 1

⋅⋅⋅

ℎ1𝑛 − 1 ] ℎ2𝑛 − 1 ] ] , .. ] . ⋅ ⋅ ⋅ 𝜆 − 𝑛 (𝑤 − 𝑠𝑖 ) − 𝑑𝑛 − 1]𝑛×𝑛

[1 𝜆 − 𝑛 (𝑤 − 𝑠𝑖 ) − 𝑑2 − 1 ⋅ ⋅ ⋅ [ = [. .. [ .. .

1 −1 ⋅ ⋅ ⋅ −1 [1 −1 ⋅ ⋅ ⋅ −1] [ ] 𝑌 = [ .. .. .. ] [. . . ] [1 −1 ⋅ ⋅ ⋅ −1]𝑛×𝑛

Then we get the eigenvalues of 𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺 from above equality immediately. This completes the proof. Theorem 4. Let 𝐺 be a graph with Laplacian eigenvalues 0 = 𝜆 1 ≤ 𝜆 2 ≤ ⋅ ⋅ ⋅ ≤ 𝜆 𝑛 . Then the number of spanning trees of the composition graphs of the complete 3-partite graph 𝐾𝑠1 ,𝑠2 ,𝑠3 and the graph 𝐺 is equal to 3

𝑠 −1

𝑡 (𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺) = 𝑛𝑤 ⋅ ∏[𝑛 (𝑤 − 𝑠𝑖 )] 𝑖 𝑖=1

3

(27)

𝑛

× ∏ ∏[𝜆 𝑘 + 𝑛 (𝑤 − 𝑠𝑖 )]

(𝑛−1)𝑠𝑖

,

𝑖=1 𝑘=2

(21)

then 󵄨󵄨𝑋 −𝐽 ⋅ ⋅ ⋅ −𝐽 󵄨󵄨 󵄨󵄨 𝑛 𝑛󵄨 󵄨󵄨 𝑌 𝑍 ⋅ ⋅ ⋅ −𝐽 󵄨󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 𝐻𝑖 = (𝜆 − 𝑛∑𝑠𝑖 ) 󵄨󵄨󵄨 .. .. .. 󵄨󵄨󵄨󵄨 , 󵄨 󵄨󵄨 . . . 󵄨󵄨 𝑖=1 󵄨󵄨 󵄨 󵄨󵄨 𝑌 −𝐽𝑛 ⋅ ⋅ ⋅ 𝑍 󵄨󵄨󵄨𝑠 ×𝑠 𝑖 𝑖

where 𝑤 = ∑3𝑗=1 𝑠𝑗 is the order of 𝐾𝑠1 ,𝑠2 ,𝑠3 . Proof. It follows directly from Lemma 2 and Theorem 3.

3

(22)

where 𝑍 = 𝜆 − 𝑛(𝑤 − 𝑠𝑖 )𝐼𝑛 − 𝐿(𝐺) − 𝐽𝑛 , for 𝑖 = 2, 3. Adding the first column to all the other columns, we get 󵄨󵄨𝐵 (𝐺, 𝜆 − 𝑛 (𝑤 − 𝑠 )) 0 ⋅⋅⋅ 0 󵄨󵄨󵄨󵄨 󵄨󵄨 𝑖 󵄨󵄨 0 󵄨󵄨󵄨󵄨 𝐶 𝑍 + 𝐽𝑛 ⋅ ⋅ ⋅ 󵄨󵄨 𝐻𝑖 = (𝜆 − 𝑛𝑤) 󵄨󵄨󵄨 . . .. 󵄨󵄨󵄨󵄨 .. .. 󵄨󵄨 . 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 𝐶 0 ⋅ ⋅ ⋅ 𝑍 + 𝐽𝑛 󵄨󵄨󵄨𝑠 ×𝑠 󵄨 𝑖 𝑖 (23) = (𝜆 − 𝑛𝑤) ⋅ 𝐵 (𝐺, 𝜆 − 𝑛 (𝑤 − 𝑠𝑖 ))

(24)

𝑠 −1

⋅ [det (𝜆 − 𝑛 (𝑤 − 𝑠𝑖 ) 𝐼𝑛 − 𝐿 (𝐺))] 𝑖 ,

for 𝑖 = 2, 3. From Lemma 1 and the property of the determinant, notice that 𝐵(𝐺, 𝜆) = ∏𝑛𝑖=2 (𝜆 − 𝜆 𝑖 (𝐺)), det(𝜆𝐼𝑛 − 𝐿(𝐺)) = 𝜆∏𝑛𝑖=2 (𝜆 − 𝜆 𝑖 (𝐺)); substituting these into equality (24), we get the value of the determinant 𝐻𝑖 : 𝑛

3. Some Corollaries As direct consequences, we give some corollaries of the above Theorem 4. Corollary 1 (see [19]). The number of spanning trees of the complete 3-partite graph 𝐾𝑠1 ,𝑠2 ,𝑠3 is equal to 3

where 𝑤 = ∑3𝑗=1 𝑠𝑗 is the order of 𝐾𝑠1 ,𝑠2 ,𝑠3 . Proof. According to Theorem 4, let 𝐺 = 𝐾1 , and note that 𝑡(𝐾𝑠1 ,𝑠2 ,𝑠3 ) = 𝑡(𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐾1 ), 𝑛 = 1, and 𝜆(𝐾1 ) = 0. The corollary follows immediately from Theorem 4. Corollary 2 (see [18]). One has the following formulae: 𝑡 (𝐾𝑚 ⊙ 𝐾𝑛 ) = (𝑚𝑛)𝑚𝑛−2 , 𝑝

𝑖=2

𝑠𝑖 −1

𝑛

× [(𝜆 − 𝑛 (𝑤 − 𝑠𝑖 )) ⋅ ∏ [𝜆 − 𝑛 (𝑤 − 𝑠𝑖 ) − 𝜆 𝑖 (𝐺)]]

,

𝑡 (𝐾𝑚 ⊙ 𝐾𝑝,𝑞 ) 2𝑚−2

(25) × {[(𝑝 + 𝑞) (𝑚 − 1) + 𝑝]

for 𝑖 = 2, 3. Combining (17), (19), and (25), we obtain

𝑞−1

𝑝−1 𝑚

[(𝑝 + q) (𝑚 − 1) + 𝑞]

} . (29)

Proof. Note that 𝜆 𝑖 (𝐾𝑛 ) = (0, (𝑛 − 1)𝑛−1 ), 𝑖 = 1, 2, . . . , 𝑛. Then the corollary follows immediately from Theorem 4.

𝑠 −1

𝑃 (𝐾𝑠1 ,𝑠2 ,𝑠3 ⊙ 𝐺, 𝜆) =𝜆(𝜆 − 𝑛𝑤)2 ∏(𝜆 − 𝑛 (𝑤 − 𝑠𝑖 )) 𝑖 𝑖=1

𝑛

𝑞 𝑛−1

= [𝑚 (𝑝 + 𝑞)]

𝑖=2

3

(28)

𝑖=1

𝑡 (𝐾𝑝,𝑞 ⊙ 𝐾𝑛 ) = 𝑛𝑛(𝑝+𝑞)−2 𝑞𝑝−1 𝑝𝑞−1 [(𝑞 + 1) (𝑝 + 1) ]

𝐻𝑖 = (𝜆 − 𝑛𝑤) ⋅ ∏ (𝜆 − 𝑛 (𝑤 − 𝑠𝑖 ) − 𝜆 𝑖 (𝐺))

𝑛

𝑠 −1

𝑡 (𝐾𝑠1 ,𝑠2 ,𝑠3 ) = 𝑤∏(𝑤 − 𝑠𝑖 ) 𝑖 ,

𝑠

× ∏ ∏[𝜆 − 𝑛 (𝑤 − 𝑠𝑖 ) − 𝜆 𝑘 (𝐺)] 𝑖 . 𝑖=1 𝑘=2

(26)

Finally, as direct consequences of Theorem 4, we give closed formulae for the number of spanning trees of some new family of graphs.


5

Corollary 3. Let 𝐾𝑛 be a complete graph, 𝐶𝑛 a cycle with 𝑛 vertices, and 𝜆 the nonzero eigenvalues of 𝐾𝑛 ; then the following results can easily be obtained: 𝑡 (𝐾𝑛 ⊙ 𝐾𝑠,𝑠,𝑠 ) = (𝑛)3𝑛−4 ⋅ (3𝑠)3𝑛−2 ⋅ (3𝑛𝑠 − 𝑠)3(𝑛−1)(𝑠−1) , 3

𝑡 (𝐶𝑛 ⊙ 𝐾𝑠1 ,𝑠2 ,𝑠3 ) = 𝑛32𝑛 𝑤3𝑛−2 ∏(3𝑤 − 𝑠𝑖 )

𝑛(𝑠𝑖 −1)

.

(30)

𝑖=1

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The author is grateful to the two anonymous referees for many friendly and helpful suggestions for improving the presentation of this paper. Project is supported by the National Natural Science Foundation of China (no. 70531030 and no. 10641003), National 973 (no. 2007CB311002 and no. 2013CB329404).

References [1] F. T. Boesch and H. Prodinger, “Spanning tree formulas and chebyshev polynomials,” Graphs and Combinatorics, vol. 2, no. 1, pp. 191–200, 1986. [2] F. T. Boesch, A. Satyanarayana, and C. L. Suffel, “A survey of some network reliability analysis and synthesis results,” Networks, vol. 54, no. 2, pp. 99–107, 2009. [3] A. Cayley, “A theorem on trees,” Quarterly Journal of Mathematics, vol. 23, pp. 276–278, 1889. [4] L. Petingi, F. Boesch, and C. Suffel, “On the characterization of graphs with maximum number of spanning trees,” Discrete Mathematics, vol. 179, no. 1–3, pp. 155–166, 1998. [5] Y. Zhang, X. Yong, and M. J. Golin, “The number of spanning trees in circulant graphs,” Discrete Mathematics, vol. 223, no. 1– 3, pp. 337–350, 2000. [6] K.-L. Chung and W.-M. Yan, “On the number of spanning trees of a multi-complete/star related graph,” Information Processing Letters, vol. 76, no. 3, pp. 113–119, 2000. [7] B. Gilbert and W. Myrvold, “Maximizing spanning trees in almost complete graphs,” Networks, vol. 30, no. 2, pp. 97–104, 1997. [8] P. L. Hammer and A. K. Kelmans, “Laplacian spectra and spanning trees of threshold graphs,” Discrete Applied Mathematics, vol. 65, no. 1–3, pp. 255–273, 1996. [9] A. K. Kelmans and V. M. Chelnokov, “A certain polynomial of a graph and graphs with an extremal number of trees,” Journal of Combinatorial Theory B, vol. 16, no. 3, pp. 197–214, 1974. [10] V. Yegnanarayanan, P. R. Thiripurasundari, and T. Padmavathy, “On some graph operations and related applications,” Electronic Notes in Discrete Mathematics, vol. 33, pp. 123–130, 2009. [11] F. Li, Z. B. Xu, H. X. Zhao, and W. Wang, “On the number of spanning trees of the lexicographic product of netwoks,” Scientia Sinica Informationis, vol. 42, pp. 949–959, 2012. [12] M. Fiedler, “Eigenvalues of nonnegative symmetric matrices,” Linear Algebra and Its Applications, vol. 9, pp. 119–142, 1974.

[13] D. M. Cardoso, I. Gutman, E. A. Martins, and M. Robbiano, “A generalization of fiedler’s lemma and some applications,” Linear and Multilinear Algebra, vol. 59, no. 8, pp. 929–942, 2011. [14] Z. B. Xu, F. Li, and H. X. Zhao, “On the vertex-forwarding index of the lexicographic product of graphs,” Scientia Sinica Informationis. In press. [15] F. Li, W. Wang, Z. Xu, and H. Zhao, “Some results on the lexicographic product of vertex-transitive graphs,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1924–1926, 2011. [16] W. Wang, F. Li, H. Lu, and Z. Xu, “Graphs determined by their generalized characteristic polynomials,” Linear Algebra and Its Applications, vol. 434, no. 5, pp. 1378–1387, 2011. [17] N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, Mass, USA, 2nd edition, 1993. [18] Z. Huang and X. Li, “On the number of spanning trees of some composite graphs,” Acta Mathematics Scientia, vol. 15, no. 3, pp. 259–268, 1995 (Chinese). [19] J. Moon, Enumerating Labelled Trees, Graph Theory and Theoretical Physics, Academic Press, New York, NY, USA, 1967.

Advances in

Operations Research Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Advances in

Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Applied Mathematics

Algebra

Hindawi Publishing Corporation http://www.hindawi.com


Volume 2014

Journal of

Probability and Statistics Volume 2014

The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com


Volume 2014

International Journal of

Differential Equations Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Submit your manuscripts at http://www.hindawi.com International Journal of

Advances in

Combinatorics Hindawi Publishing Corporation http://www.hindawi.com

Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of Mathematics and Mathematical Sciences


Journal of

Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014


Volume 2014

Volume 2014


Volume 2014

Discrete Mathematics

Journal of

Volume 2014


Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014


Volume 2014


Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization



Volume 2014

Volume 2014