Linear Algebra and its Applications 565 (2019) 225–238
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Linear Algebra and its Applications www.elsevier.com/locate/laa
A relation between the signless Laplacian spectral radius of complete multipartite graphs and majorization Mohammad Reza Oboudi Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran
a r t i c l e
i n f o
Article history: Received 4 September 2018 Accepted 12 December 2018 Available online 17 December 2018 Submitted by R. Brualdi MSC: 05C31 05C50 15A18 Keywords: Complete multipartite graphs Signless Laplacian matrix Signless Laplacian spectral radius Majorization
a b s t r a c t Let G be a simple graph with vertices v1 , . . . , vn . Let A(G) be the adjacency matrix of G and D(G) be the diagonal matrix (d1 , . . . , dn ), where di is the degree of vertex vi , for i = 1, . . . , n. The matrix Q(G) = D(G) + A(G) is called the signless Laplacian matrix of G. By the signless Laplacian spectral radius of G, denoted by q(G), we mean the largest eigenvalue of Q(G). Let X = (m1 , . . . , mt ) and Y = (n1 , . . . , nt ), where m1 ≥ · · · ≥ mt ≥ 1 and n1 ≥ · · · ≥ nt ≥ 1 are inteY and ger. We say X majorizes j j let X M Y , if for every j, 1 ≤ j ≤ t − 1, i=1 mi ≥ i=1 ni with equality if j = t. In this paper we find a relation between the majorization and the signless Laplacian spectral radius of complete multipartite graphs. We show that if (m1 , . . . , mt ) M (n1 , . . . , nt ) and (m1 , . . . , mt ) = (n1 , . . . , nt ) then q(Kn1 ,...,nt ) > q(Km1 ,...,mt ), where Kn1 ,...,nt is the complete multipartite graph with t parts of size n1 , . . . , nt . Using the above relation we find that among all complete multipartite graphs with n vertices and t ≥ 3 parts, the split graphs have the minimum signless Laplacian spectral radius and the Turán graphs have the maximum signless Laplacian spectral radius. Finally we obtain that for every positive integers t ≥ 2 and n1 , . . . , nt , n + 2t − 4 + n2 + (n − t)(4t − 8) ≤ q(Kn1 ,...,nt ) 2 3n − 2(a + b) + n2 − 4n(a + b) + 4(a − b)2 + 8abt , ≤ 2
E-mail addresses:
[email protected],
[email protected]. https://doi.org/10.1016/j.laa.2018.12.012 0024-3795/© 2018 Elsevier Inc. All rights reserved.
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where n = n1 + · · · + nt , a = nt and b = nt . In addition we investigate the equality in both sides. © 2018 Elsevier Inc. All rights reserved.
1. Introduction Throughout this paper we will consider only simple graphs (finite and undirected, without loops and multiple edges). Let G = (V, E) be a simple graph. The order of G denotes the number of vertices of G. For two vertices u and v by e = uv we mean the edge e between u and v. For every vertex v ∈ V (G), the degree of v is the number of edges incident with v and is denoted by degG (v). By Δ(G) we mean the maximum degree of vertices of G. Let v ∈ V (G). By G \ v we mean the graph that obtained from G by removing v. The complement of G, denoted by G, is the graph on the same vertices such that two distinct vertices of G are adjacent if and only if they are not adjacent in G. For two disjoint graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ), the disjoint union of G1 and G2 denoted by G1 ∪ G2 is the graph with vertex set V1 ∪ V2 and edge set E1 ∪ E2 . The graph rG denotes the disjoint union of r copies of G. The join of G1 and G2 that is denoted by G1 ∨ G2 is the graph with vertex set V (G1 ) ∪ V (G2 ) and the edge set E(G1 ) ∪ E(G2 ) ∪ {uv : u ∈ V (G1 ) and v ∈ V (G2 )}. For a graph G, a clique C of G is a subset of vertices of G such that every two distinct vertices in C are adjacent. An independent set S of G is a subset of vertices of G such that there is no edge between every two vertices of S. The edgeless graph (empty graph), the complete graph, the cycle, and the path of order n, are denoted by Kn , Kn , Cn and Pn , respectively. Let t and n1 , . . . , nt be some positive integers. By Kn1 ,...,nt we mean the complete multipartite graph with parts size n1 , . . . , nt . In particular the complete bipartite graph with part sizes m and n denoted by Km,n . Let G be a graph with vertex set {v1 , . . . , vn }. The adjacency matrix of G, A(G) = [aij ], is the n ×n matrix such that aij = 1 if vi and vj are adjacent, and aij = 0, otherwise. Let D(G) be the diagonal matrix (d1 , . . . , dn ), where di is the degree of vertex vi , for i = 1, . . . , n. The matrix Q(G) = D(G) + A(G) is called the signless Laplacian matrix of G. The matrices A(G) and Q(G) are symmetric, so all of the eigenvalues of A(G) and Q(G) are real. By the eigenvalues of G we mean those of its adjacency matrix. We denote the eigenvalues of G by λ1 (G) ≥ · · · ≥ λn (G). By the Spec(G) we mean the multiset {λ1 (G), . . . , λn (G)}. By the spectral radius of G, denoted by λ(G), we mean the largest eigenvalue of G. In other words λ(G) = λ1 (G). It is well known that |λi (G)| ≤ λ1 (G), for i = 1, . . . , n. The characteristic polynomial of G that is denoted by P (G, λ) is det(λI − A(G)), where I is n × n identity matrix. In fact P (G, λ) = Πni=1 (λ − λi (G)). Similarly, by the signless Laplacian eigenvalues of G we mean those of its signless Laplacian matrix. We denote the signless Laplacian eigenvalues of G by q1 (G) ≥ · · · ≥ qn (G). It is well
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known that all of the signless Laplacian eigenvalues of G are non-negative (in fact Q(G) is a positive semi-definite matrix). In other words, qn (G) ≥ 0. By the signless Laplacian spectral radius of G, denoted by q(G), we mean the largest signless Laplacian eigenvalue of G. In other words q(G) = q1 (G). The signless Laplacian characteristic polynomial of G that is denoted by PQ (G, λ) is det(λI − Q(G)), where I is n × n identity matrix. In fact PQ (G, x) = Πni=1 (x −qi (G)). By the SpecQ (G) we mean the multiset {q1 (G), . . . , qn (G)}. We note that there are some other kinds of matrices associated to graphs, such as Laplacian matrix and distance matrix. In addition, the concepts Laplacian spectral radius and distance spectral radius of graphs have been studied frequently. Many papers are devoted to the study of characteristic polynomial, the spectra of adjacency matrix, the spectra of Laplacian matrix and the spectra of signless Laplacian matrix of graphs. In particular, the spectral radius, the Laplacian spectral radius and the signless Laplacian spectral radius of graphs have always been of great interest to researchers in algebraic graph theory. See [2], [4–8], [10–17], [19–25] and references therein. Let X = (x1 , . . . , xk ) and Y = (y1 , . . . , yk ), where x1 ≥ · · · ≥ xk and y1 ≥ · · · ≥ yk j are real. We say X majorizes Y and let X M Y , if for every j, 1 ≤ j ≤ k, i=1 xi ≥ j i=1 yi , with equality if j = k. We let X M Y when X = Y and X M Y . The name majorization appeared first in 1959 by Hardy, Littlewood and Polya. The theory of majorization is very useful in so many diverse fields. For more details on this concept we refer to [1] and [9]. In this paper we study the signless Laplacian spectral radius of complete multipartite graphs. We find a relation between the majorization and the signless Laplacian spectral radius of complete multipartite graphs. More precisely, let X = (m1 , . . . , mt ) and Y = (n1 , . . . , nt ), where m1 ≥ · · · ≥ mt ≥ 1 and n1 ≥ · · · ≥ nt ≥ 1 are integer. We show that if (m1 , . . . , mt ) M (n1 , . . . , nt ) and (m1 , . . . , mt ) = (n1 , . . . , nt ) then q(Kn1 ,...,nt ) > q(Km1 ,...,mt ). Using the above inequality we find that among all complete multipartite graphs with n vertices and t ≥ 3 parts, the split graphs have the minimum signless Laplacian spectral radius and the Turán graphs have the maximum signless Laplacian spectral radius. We recall that the split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Also the Turán graph with n vertices and t parts is a complete multipartite graph formed by partitioning a set of n vertices into t subsets, with sizes as equal as possible, and connecting two vertices by an edge if and only if they belong to different subsets. 2. Signless Laplacian eigenvalues of complete multipartite graphs In this section we study and compare the signless Laplacian spectral radius of complete multipartite graphs. One can compute the signless Laplacian characteristic polynomial of complete multipartite graphs by the next result.
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Theorem 1. [25] Let t ≥ 2 and n1 , . . . , nt be some positive integers. Let n = n1 + · · · + nt . Then PQ (Kn1 ,...,nt , x) =
t
(x − n + ni )ni −1
i=1
t
(x − n + 2ni ) −
t
i=1
ni
i=1
t
(x − n + 2nj ) .
j=1, j=i
Now we obtain a lower bound for the signless Laplacian spectral radius of complete multipartite graphs. Theorem 2. Let t ≥ 2 and n1 , . . . , nt be some positive integers. Then q(Kn1 ,...,nt ) ≥ n1 + · · · + nt . Moreover the equality holds if and only if t = 2. Proof. Let n = n1 + · · · + nt . Using Theorem 1 we obtain that PQ (Kn1 ,...,nt , x) =
t i=1 (x
t 1 − i=1
− n + ni )ni −1 (x − n + 2ni )
ni x−n+2ni
.
(1)
Using Equation (1), it is not hard to obtain the signless Laplacian eigenvalues of Kn1 ,n2 . In fact, SpecQ (Kn1 ,n2 ) = n1 + n2 , n1 , . . . , n1 , n2 , . . . , n2 , 0 .
n2 −1
n1 −1
Hence q(Kn1 ,n2 ) = n1 + n2 . So the theorem is proved for t = 2. Therefore to complete the proof it suffices to show that for every t ≥ 3, q(Kn1 ,...,nt ) > n1 + · · · + nt . Assume that t ≥ 3. Putting x = n = n1 + · · · + nt in PQ (Kn1 ,...,nt , x) (in Equation (1)) we obtain that PQ (Kn1 ,...,nt , n) = 2t (1 − 2t )
t i=1
nni i .
(2)
This shows that PQ (Kn1 ,...,nt , n) < 0, since t ≥ 3. On the other hand the leading part of PQ (Kn1 ,...,nt , x) is xn . Thus there exists a real number y0 > n such that for every z ≥ y0 , PQ (Kn1 ,...,nt , z) > 0. So PQ (Kn1 ,...,nt , n) PQ (Kn1 ,...,nt , y0 ) < 0. Therefore by the Bolzano’s intermediate value theorem for continuous functions we find that Pd (Kn1 ,...,nt , x) has at least one real root in the interval (n, y0 ). This shows that q(Kn1 ,...,nt ) > n. The proof is complete. 2 Now we obtain one of the main results of the paper. For every function f from R to R with at least one real root and with finite number real roots, by λ(f ) we mean the largest real root of f .
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Theorem 3. Let t ≥ 3 and n1 , . . . , nt be some positive integers. If n1 ≥ n2 + 2, then q(Kn1 −1,n2 +1,n3 ,...,nt ) > q(Kn1 ,n2 ,n3 ,...,nt ). Proof. Assume that n1 ≥ n2 + 2. Since n2 ≥ 1, n1 ≥ 3. Let G = Kn1 ,n2 ,n3 ,...,nt and H = Kn1 −1,n2 +1,n3 ,...,nt . We show that q(H) > q(G). Let n = n1 + · · · + nt . Assume that f (x) = (x − n + n1 )n1 −1 (x − n + n2 )n2 −1 (x − n + 2n1 ) (x − n + 2n2 ) ×
t
(x − n + ni )ni −1 (x − n + 2ni )
i=3
and g(x) = (x − n + n1 − 1)n1 −2 (x − n + n2 + 1)n2 (x − n + 2n1 − 2) (x − n + 2n2 + 2) ×
t
(x − n + ni )ni −1 (x − n + 2ni ).
i=3
Let n1 n2 ni φ(x) = 1 − − − x − n + 2n1 x − n + 2n2 i=3 x − n + 2ni t
and n2 + 1 n1 − 1 ni − − . x − n + 2n1 − 2 x − n + 2n2 + 2 i=3 x − n + 2ni t
ψ(x) = 1 −
Using Theorem 1 (or Equation (1)) we obtain that PQ (G, x) = f (x) φ(x)
and
PQ (H, x) = g(x) ψ(x).
(3)
One can easily check that for every x, (x − n + n1 − 1)(x − n + n2 + 1) − (x − n + n1 )(x − n + n2 ) = n1 − n2 − 1 > 0 and (x − n + 2n1 − 2)(x − n + 2n2 + 2) − (x − n + 2n1 )(x − n + 2n2 ) = 4(n1 − n2 − 1) > 0. Hence we conclude that, for every x ≥ n, (x − n + n1 − 1)(x − n + n2 + 1) > (x − n + n1 )(x − n + n2 ) > 0
(4)
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and (x − n + 2n1 − 2)(x − n + 2n2 + 2) > (x − n + 2n1 )(x − n + 2n2 ) > 0. Claim. For every x > n,
n1 −1 x−n+2n1 −2
+
n2 +1 x−n+2n2 +2
>
n1 x−n+2n1
+
n2 x−n+2n2
(5)
> 0.
Proof of the claim. First note that for every x ≥ n, x − n + 2n1 − 2 > 0 (since n1 ≥ 3), x − n + 2n2 + 2 > 0, x − n + 2n1 > 0, and x − n + 2n2 > 0. So for every x ≥ n, n1 n2 x−n+2n1 + x−n+2n2 > 0. Let K(x) =
n2 + 1 n1 n1 − 1 n2 + − − x − n + 2n1 − 2 x − n + 2n2 + 2 x − n + 2n1 x − n + 2n2
and L(x) = (x − n + 2n1 − 2)(x − n + 2n2 + 2)(x − n + 2n1 )(x − n + 2n2 ) K(x). Since for every x ≥ n, (x − n + 2n1 − 2)(x − n + 2n2 + 2)(x − n + 2n1 )(x − n + 2n2 ) > 0, to complete the proof of the claim it suffices to show that for every x > n, L(x) > 0. One can see that L(x) = 4(n1 − n2 − 1)(x − n)(x − n + n1 + n2 ). This shows that for every x > n, L(x) > 0. Thus the claim is proved. We note that if n1 −1 n2 +1 n1 n2 x = n, then x−n+2n + x−n+2n = x−n+2n + x−n+2n = 1 > 0. 1 −2 2 +2 1 2 Now we continue the proof of theorem. The above claim implies that for every x > n, φ(x) > ψ(x).
(6)
It is clear that for every x ≥ n, f (x) > 0 and g(x) > 0 (note that n1 ≥ 3). On the other hand since t ≥ 3, by Theorem 2 we find that q(G) > n and q(H) > n. Thus by Equation (3) we conclude that q(G) = λ(φ) and q(H) = λ(ψ). Also the Equation (3) implies that for every x ≥ λ(ψ), φ(x) > ψ(x).
(7)
Since the leading part of PQ (H, x) is xn , for sufficiently large y we have PQ (H, y) > 0. In fact for every x > q(H), PQ (H, x) > 0. Since q(H) > n and for every x > n we have g(x) > 0, by Equation (3) we conclude that for every x > q(H), ψ(x) > 0. In other words for every x > λ(ψ), ψ(x) > 0. Now we show that λ(ψ) > λ(φ). First note that by Equation (7), φ(λ(ψ)) > ψ(λ(ψ)) = 0. So φ(λ(ψ)) > 0. On the other hand, since for every x > λ(ψ), ψ(x) > 0, by Equation (7) we obtain that for every x > λ(ψ), φ(x) > ψ(x) > 0. Combining these
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facts we obtain that for every x ≥ λ(ψ), φ(x) > 0. This shows that λ(φ) < λ(ψ). In other words, q(H) > q(G). The proof is complete. 2 3. Majorization and the signless Laplacian spectral radius of complete multipartite graphs Let n ≥ 1 be an integer and Bn = {(z1 , . . . , zn ) ∈ Zn : z1 ≥ z2 ≥ · · · ≥ zn }. Let X = (x1 , . . . , xn ) ∈ Bn and Y = (y1 , . . . , yn ) ∈ Bn . By X Y we mean that X = Y or there exists 1 ≤ j ≤ n such that x1 = y1 , . . . , xj−1 = yj−1 and xj > yj . In fact, this ordering is the lexicographic order on Bn . We let X Y if and only if X Y and X = Y . We recall that X M Y if and only if j i=1
xi ≥
j
yi , for j = 1, . . . , n − 1 and
i=1
n i=1
xi =
n
yi .
i=1
Also we let X M Y if and only if X M Y and X = Y . Remark 1. Let X, Y ∈ Bn . If X M Y , then X Y . By contradiction, if Y X, then there exists j ∈ {1, . . . , n} such that x1 = y1 , . . . , xj−1 = yj−1 and yj > xj . Hence j j i=1 yj > i=1 xj , a contradiction. Note that the converse is not valid. For example (7, 2, 2) (5, 5, 1), but (7, 2, 2) M (5, 5, 1). For every j, k ∈ {1, . . . , n}, let ej = (0, . . . , 0, 1, 0, . . . , 0) and ej,k = ek − ej . Let
j−1
n−j
X, Y ∈ Bn . We say X is convertible to Y if there is a sequence, say X = Y0
Y1 · · · Yt = Y , in Bn , such that for every i ∈ {1, . . . , t}, Yi = Yi−1 − eji ,ki (equivalently Yi−1 = Yi + eji ,ki ) for some ki > ji . This operation is a special kind of Robin Hood operation, elementary Robin Hood operation, in which money is taken from a relatively rich individual l and given to the individual whose wealth is immediately below that of individual l in the ranking of wealth, see page 11 of [1]. Also it is a special kind of T-transforms of integers (transfer), see the chapter 5 (the fourth section) of [9]. For example X = (9, 9, 6, 6) is convertible to Y = (10, 8, 7, 5), since X = (9, 9, 6, 6) (9, 9, 7, 5) (10, 8, 7, 5) and (9, 9, 7, 5) = (9, 9, 6, 6) − e3,4 and (10, 8, 7, 5) = (9, 9, 7, 5) − e1,2 . As an another example one can easily see that (8, 8, 4) is not convertible to (10, 5, 5). Majorization transfers in more general sets are considered in [3]. The following theorem states the sufficient and necessary condition for the convertibility. Also see Lemma 1 (page 195) of [9]. Theorem 4. [18] Let X, Y ∈ Bn . Then X is convertible to Y if and only if Y M X. Remark 2. [15] Let X, Y ∈ Bn . Assume that X is convertible to Y . Then there is a sequence X = Y0 Y1 · · · Yt = Y in Bn , such that for every i ∈ {1, . . . , t},
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Yi = Yi−1 − eji ,ki for some ki > ji . So Yi is convertible to Yi+1 , for i = 0, 1, . . . , t − 1. Hence by Theorem 4, for i = 0, 1, . . . , t − 1 we obtain that Yi M Yi+1 . Thus X = Y0 M Y1 M · · · M Yt = Y . Using Theorem 4 and Remarks 1 and 2 we obtain the following result. Theorem 5. Let X, Y ∈ Bn and X = Y . Then Y M X if and only if there exists a sequence such as X = Y0 ≺M Y1 ≺M · · · ≺M Yt = Y , where for i = 0, 1, . . . , t we have Yi ∈ Bn . Let n1 , m1 , . . . , nk , mk be some real numbers. By {n1 , . . . , nk } M {m1 , . . . , mk } we mean that there exist two permutations π and σ on the set {1, . . . , k} such that nπ(1) ≥ · · · ≥ nπ(k) , mσ(1) ≥ · · · ≥ mσ(k) and (nπ(1) , . . . , nπ(k) ) M (mσ(1) , . . . , mσ(k) ). Theorem 6. [15] Let n1 , m1 , . . . , nk , mk and x1 , . . . , xt be some real numbers. Then {n1 , . . . , nk } M {m1 , . . . , mk } ⇐⇒ {n1 , . . . , nk , x1 , . . . , xt } M {m1 , . . . , mk , x1 , . . . , xt }. Now we obtain a relation between the signless Laplacian spectral radius of complete multipartite graphs with respect to majorization. Theorem 7. Let t ≥ 3 and m1 , . . . , mt and n1 , . . . , nt be some positive integers such that m1 ≥ · · · ≥ mt and n1 ≥ · · · ≥ nt . If (m1 , . . . , mt ) M (n1 , . . . , nt ), then q(Kn1 ,...,nt ) > q(Km1 ,...,mt ). Proof. Clearly if (m1 , . . . , mt ) = (n1 , . . . , nt ), then Km1 ,...,mt ∼ = Kn1 ,...,nt . Now assume that (m1 , . . . , mt ) M (n1 , . . . , nt ). In other words (m1 , . . . , mt ) majorizes (n1 , . . . , nt ) and (m1 , . . . , mt ) = (n1 , . . . , nt ). We show that q(Kn1 ,...,nt ) > q(Km1 ,...,mt ). Let X = (m1 , . . . , mt ) and Y = (n1 , . . . , nt ). Since X majorizes Y , by Theorem 5 there is a sequence X0 , X1 , . . . , Xh in Bt such that X = X0 M X1 M · · · M Xh−1 M Xh = Y,
(8)
where for i = 1, . . . , h we have Xi = Xi−1 + (0, . . . , 0, −1, 0, . . . , 0, 1, 0, . . . , 0) such that
αi −1
βi −1
1 ≤ αi , βi ≤ t and αi + βi ≤ t. Consider an arbitrary integer j ∈ {1, . . . , h}. Assume that Xj−1 = (w1 , . . . , wt ) and Xj = (z1 , . . . , zt ). We claim that q(Kz1 ,...,zt ) > q(Kw1 ,...,wt ). Without losing the generality suppose that αj = βj = 1. It means that w1 ≥ w2 + 1 and Xj = Xj−1 + (−1, 1, 0, . . . , 0). Hence Xj−1 = (w1 , w2 , w3 , . . . , wt ) and Xj = (w1 − 1,
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w2 +1, w3 , . . . , wt ). Since Xj−1 M Xj , we obtain that w1 ≥ w2 +2. Hence by Theorem 3 we conclude that q(Kw1 −1,w2 +2,w3 ,...,wt ) > q(Kw1 ,w2 ,w3 ,...,wt ). Thus the claim is proved. Now by the above claim and Equation (8) we find that q(KY ) = q(KXh ) > q(KXh−1 ) > · · · > q(KX1 ) > q(KX0 ) = q(KX ), where by KXi (i = 0, . . . , h) we mean Ky1 ,...,yt if Xi = (y1 , . . . , yt ). Therefore q(KY ) > q(KX ). In other words q(Kn1 ,...,nt ) > q(Km1 ,...,mt ). 2 4. The minimum and maximum value of signless Laplacian spectral radius of complete multipartite graphs In this section by applying the previous theorems we find the minimum and maximum value of signless Laplacian spectral radius of complete multipartite graphs. For every integer t ≥ 1, let Ct = {(n1 , . . . , nt ) ∈ Zt : n1 ≥ n2 ≥ · · · ≥ nt ≥ 1}, where Z is the set of all integers. It is easy to see that (Ct , M ) is a partially ordered set. In the next theorem we determine the maximal and the minimal elements of the poset (Ct , M ). A more general of this result has been obtained in [6] by different proof. Theorem 8. Let (Ct , M ) as mentioned above. Then for every (n1 , . . . , nt ) ∈ Ct we have n n n n (n − t + 1, 1, . . . , 1) M (n1 , . . . , nt ) M ( , . . . , , , . . . , ),
t t t
t t−1 r
s
where n = n1 + · · · + nt , r = n − t nt and s = t − r. Proof. First we note that the fairly distribution of n objects in t dishes is that the size of r dishes is nt and the size of other dishes (s dishes) is nt for some r and s. To obtain r and s it suffices to solve the following system r+s=t
and
n n r + s = n. t t
It is easy to check that r = n − t nt . So s = t − r. Let (n1 , . . . , nt ) ∈ Ct and n = n1 + · · · + nt . Hence n1 ≥ · · · ≥ nt ≥ 1. Thus for every k ∈ {1, . . . , t}, n = n1 + · · · + nk + nk+1 + · · · + nt ≥ n1 + · · · + nk + t − k. Hence n − t + k ≥ n1 + · · · + nk . This shows that (n − t + 1, 1, . . . , 1) M (n1 , . . . , nt ).
t−1
To prove the other part, first note that tn1 ≥ n1 +· · ·+nt ≥ tnt . That is tn1 ≥ n ≥ tnt . Hence n1 ≥ nt and nt ≤ nt . We complete the proof by induction on t. For t = 1 there is nothing to prove. Thus assume that t ≥ 2. Let x = n1 − nt . So x ≥ 0 and
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n2 + n3 + · · · + nt−1 + (nt + x) = n − nt .
(9)
Let r = n − t nt and s = t − r. It is not hard to see that the fairly distribution of n − nt objects in t − 1 dishes is that the size of r − 1 dishes is nt and the size of other dishes (s dishes) is nt . Therefore by Equation (9) and the induction hypothesis we conclude that n n n n {n2 , n3 , . . . , nt−1 , nt + x} M { , . . . , , , . . . , }. t t t
t s
r−1
Thus by Theorem 6, n n n n n {n2 , n3 , . . . , nt−1 , nt + x, } M { , . . . , , , . . . , }. t t t t
t r
(10)
s
Since nt ≤ nt , one can see that {n1 , nt } M {n1 − x, nt + x}. Thus by Theorem 6, {n1 , n2 , . . . , nt−1 , nt } M {n1 − x, n2 , . . . , nt−1 , nt + x}. Thus by Equation (10), since n1 − x = nt , we obtain that n n n n {n1 , n2 , . . . , nt } M { , . . . , , , . . . , }. t t t
t r
s
The proof is complete. 2 Let t and n, where n ≥ t ≥ 2, be some positive integers. By T (n, t) we mean the Turán graph with n vertices and t parts. In other words, T (n, t) = K n , . . . , n , n , . . . , n ,
t t t t r
s
where r = n −t nt and s = t −r. Also by S(n, t) we mean the split graph Kt−1 ∨Kn−t+1 . In fact S(n, t) = Kn−t+1,1, . . . , 1 .
t−1
Using Theorems 7 and 8 we find that among all complete multipartite graphs with n vertices and t ≥ 3 parts, the split graphs have the minimum signless Laplacian spectral radius and the Turán graphs have the maximum signless Laplacian spectral radius. More precisely we obtain the following result.
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Theorem 9. Let t ≥ 3 and n1 , . . . , nt be some positive integers. Let n = n1 + · · · + nt . Then q(T (n, t)) ≥ q(Kn1 ,...,nt ) ≥ q(S(n, t)). Moreover in the left side the equality holds if and only if Kn1 ,...,nt ∼ = T (n, t) and in the right side the equality holds if and only if Kn1 ,...,nt ∼ = S(n, t). Proof. Without losing the generality assume that n1 ≥ · · · ≥ nt . By Theorem 8, n n n n (n − t + 1, 1, . . . , 1) M (n1 , . . . , nt ) M ( , . . . , , , . . . , ),
t t t t t−1 r
(11)
s
where r = n − t nt and s = t − r. We consider the following cases: n n 1) Assume that (n1 , . . . , nt ) = (n − t + 1, 1, . . . , 1) and (n1 , . . . , nt ) = ( , . . . , ,
t t t−1 r
n n , . . . , ). Thus by Theorem 7 and Equation (11) we obtain that
t t s
q(T (n, t)) = q(K n n n n ) > q(Kn1 ,...,nt ) > q(Kn−t+1,1, . . . , 1 ) , . . . , , , . . . ,
t−1
t t t t r
s
= q(S(n, t)). n n n n 2) If (n1 , . . . , nt ) = (n − t + 1, 1, . . . , 1) and (n1 , . . . , nt ) = ( , . . . , , , . . . , )
t t t
t t−1
r
s
n n n n or (n1 , . . . , nt ) = (n −t +1, 1, . . . , 1) and (n1 , . . . , nt ) = ( , . . . , , , . . . , ),
t t t t t−1 r
s
then similar to the previous case by Theorem 7 and Equation (11) the result follows. n n n n 3) Assume that (n1 , . . . , nt ) = (n − t + 1, 1, . . . , 1) = ( , . . . , , , . . . , ). It
t t t t t−1 r
s
is not hard to see that n = t and (n1 , n2 , . . . , nt ) = (1, 1, . . . , 1) or n = t + 1 and (n1 , n2 , . . . , nt ) = (2, 1, . . . , 1). If (n1 , n2 , . . . , nt ) = (1, 1, . . . , 1) or (n1 , n2 , . . . , nt ) = (2, 1, . . . , 1), there is nothing to prove. So the proof is complete. 2 Theorem 10. Let t ≥ 2 and n1 ≥ · · · ≥ nt be some positive integers. Let n = n1 +· · ·+nt . Then q(Kn1 ,...,nt ) ≥
n + 2t − 4 +
n2 + (n − t)(4t − 8) . 2
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Moreover the equality holds if and only if t = 2 or t ≥ 3 and (n1 , . . . , nt ) = (n − t + 1, 1, . . . , 1).
t−1
Proof. Clearly n ≥ t. For every integer n and t where n ≥ t ≥ 2, let α(n, t) = n+2t−4+
n2 +(n−t)(4t−8) . 2
It is easy to check that α(n, t) ≥ n and the equality holds if and only if t = 2. First assume that t = 2. By Theorem 2, q(Kn1 ,n2 ) = n = α(n, 2). Now let t ≥ 3. By Theorem 9 we have q(Kn1 ,...,nt ) ≥ q(S(n, t)), and the equality holds if and only if Kn1 ,...,nt ∼ = S(n, t). We note that if r, s and a1 ≥ · · · ≥ ar and b1 ≥ · · · ≥ bs are some positive integers, then Ka1 ,...,ar ∼ = Kb1 ,...,bs if and only if r = s and a1 = b1 , . . . , ar = br . Therefore to complete the proof it suffices to show that q(S(n, t)) = α(n, t). Using Theorem 1 we obtain that PQ (S(n, t), x) = (x − t + 1)n−t (x − n + 2)t−2 x2 + (4 − n − 2t)x + 2t2 − 6t + 4 . Hence SpecQ (S(n, t)) =
t − 1, . . . , t − 1, n − 2, . . . , n − 2,
n−t
n + 2t − 4 ±
t−2
n2 + (n − t)(4t − 8) . 2
This shows that q(S(n, t)) = α(n, t). The proof is complete.
2
Remark 3. Let t ≥ 2 and n1 ≥ · · · ≥ nt be some positive integers and n = n1 + · · · + nt . Using Theorem 1 we can find the signless Laplacian characteristic polynomial of Turán graph T (n, t). Let a = nt and b = nt . In fact PQ (T (n, t), x) = (x − n + a)r(a−1) (x − n + b)s(b−1) (x − n + 2a)r−1 (x − n + 2b)s−1 f (x), where f (x) = x2 + (−3n + 2a + 2b)x + 2n2 − 2n(a + b) − 2abt + 4ab. We note that the roots of f (x) are 3n − 2(a + b) ±
n2 − 4n(a + b) + 4(a − b)2 + 8abt . 2
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Similar to Theorem 10, by Remark 3 one can prove the following result. So we omit its proof. Theorem 11. Let t ≥ 2 and n1 ≥ · · · ≥ nt be some positive integers and n = n1 + · · · + nt . Let a = nt and b = nt . Then q(Kn1 ,...,nt ) ≤
3n − 2(a + b) +
n2 − 4n(a + b) + 4(a − b)2 + 8abt . 2
Moreover the equality holds if and only if t = 2 or t ≥ 3 and (n1 , . . . , nt ) = (a, . . . , a,
r
b, . . . , b), where r = n − t nt and r + s = t.
s
We finish the paper by the following remark. Remark 4. One may ask about the relation between the majorization and the Laplacian spectral radius of complete multipartite graphs. We note that in this case there in nothing to prove, since we can compute the Laplacian spectrum of complete multipartite graphs easily. It is well known that by knowing the Laplacian spectrum of a graph G one can obtain the Laplacian spectrum of the complement of G. On the other hand Kn1 ,...,nt = Kn1 ∪ · · · ∪ Knt and the Laplacian spectrum of the complete graph Ks is {s, . . . , s, 0}.
s−1
Therefore we can determine the Laplacian spectrum of Kn1 ,...,nt . In fact Spec(Kn1 ,...,nt ) =
n, . . . , n, n − n1 , . . . , n − n1 , . . . , n − nt , . . . , n − nt , 0 ,
t−1
n1 −1
nt −1
where n = n1 + · · · + nt . In particular, the Laplacian spectral radius of Kn1 ,...,nt is n1 + · · · + nt . Acknowledgement The author is grateful to the referee for helpful comments. References [1] B.C. Arnold, Majorization and the Lorenz Order: A Brief Introduction, Lecture Notes in Statistics, vol. 43, Springer-Verlag, 1987. [2] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, New York, 2012. [3] R.A. Brualdi, G. Dahl, Majorization for partially ordered sets, Discrete Math. 313 (2013) 2592–2601. [4] D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Theory and Application, Academic Press, Inc., New York, 1980. [5] D. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Spectra, London Mathematical Society Student Texts, vol. 75, Cambridge University Press, Cambridge, 2010.
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