Xihe Li, Ligong Wang and Shangyuan Zhang

Transcription

1 Indian J. Pre Appl. Math., 49(1): , March 2018 c Indian National Science Academy DOI: /s THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF SOME STRONGLY CONNECTED DIGRAPHS 1 Xihe Li, Ligong Wang and Shangyan Zhang Department of Applied Mathematics, School of Science, Northwestern Polytechnical Uniersity, Xi an, Shaanxi , P. R. China s: lgwangmath@163.com; lxhdhr@163.com; @qq.com (Receied 3 Noember 2016; accepted 1 Jne 2017) Let G be a strongly connected digraph and Q( G) be the signless Laplacian matrix of G. The spectral radis of Q( G) is called the signless Lapliacian spectral radis of G. Let 1 -digraph and 2 -digraph be two kinds of generalized strongly connected -digraphs and let θ 1 -digraph and θ 2 -digraph be two kinds of generalized strongly connected θ-digraphs. In this paper, we determine the niqe digraph which attains the maximm(or minimm) signless Laplacian spectral radis among all 1 -digraphs and θ 1 -digraphs. Frthermore, we characterize the extremal digraph which achiees the maximm signless Laplacian spectral radis among 2 -digraphs and θ 2 -digraphs, respectiely. Key words : The signless Laplacian spectral radis; 1 -digraph; 2 -digraph; θ 1 -digraph; θ 2 -digraph. 1. Introdction Throghot this article, all digraphs are finite simple strongly connected digraphs, i.e., withot loops and mltiple arcs. Let G = (V ( G), E( G)) be a digraph with ertex 1 This work was spported by the National Natral Science Fondation of China (No ) and the National College Stdents Innoation and Entreprenership Training Program (No ).

2 114 XIHE LI, LIGONG WANG AND SHANGYUAN ZHANG set V ( G) and arc set E( G). Two ertices are called adjacent if they are connected by an arc. If there is an arc from to, we indicate this by writing (, ), call the head of (, ), and the tail of (, ), respectiely. For any ertex, let N + G () = { V ( G) (, ) E( G)} and N G () = { V ( G) (, ) E( G)} denote the sets of ot-neighbors and in-neighbors of, respectiely. Let d + G () = N + G () and d G () = N G () denote the ot-degree and in-degree of in G. The digraph G is called strongly connected if for eery pair of ertices, V ( G), there exist a directed path from to and a directed path from to. Let P l and C l denote the directed path and directed cycle on l ertices, respectiely. Sppose P k = k, we call 1 the initial ertex and k the terminal ertex of the directed path P k, respectiely. Let A( G) = (a ij ) n n denote the adjacency matrix of a digraph G, where a ij = 1 if ( i, j ) E( G) and a ij = 0 otherwise. Let D( G) be the diagonal matrix with ot-degrees of the ertices of G. Then the matrix Q( G) = D( G) + A( G) is called the signless Laplacian matrix of Q( G), and let q( G) denote its signless Laplacian spectral radis, the largest modls of an eigenale of Q( G). The polynomial φ( G, λ) = det(λi Q( G)) is defined as the characteristic polynomial with respect to the signless Laplacian matrix Q( G). The collection of eigenales of Q( G) together with mltiplicates is called the Q-spectrm of G. There are many articles on the signless Laplacian spectrm of ndirected graphs [1, 2, 12, 14, 16]. There are also many articles on the topic of adjacency spectrm [3, 6, 9], and the Laplacian spectrm [8, 13]. For additional remarks on this topic we refer the reader to see two excellent sreys [10] and [11]. Howeer, there is not mch known abot digraphs. The matrix Q( G) is nonnegatie and irredcible when G is strongly connected. It follows from the Perron-Frobenis Theorem [7] that q( G) is an eigenale of the signless Laplacian matrix Q( G) and there is a positie nit eigenector corresponding to q( G). The positie nit eigenector corresponding to q( G) is called the Perron ector of Q( G). In [15], Xi and Wang characterized the extremal digraphs which achiee the maximm and minimm signless Laplacian spectral radis among strongly connected bicyclic digraphs. In [4], Go and Li characterized the extremal digraphs which achiee the maximm and minimm adjacency spectral radis among two kinds of generalized strongly connected bicyclic digraphs. In or paper, we mainly generalize

3 THE SIGNLESS LAPLACIAN SPECTRAL RADIUS 115 their reslts and obtain some reslts abot the signless Laplacian spectral radis of generalized and θ-digraphs. The rest of this paper is organized as follow. In Section 2, we characterize the extremal digraphs which attain the maximm and minimm signless Laplacian spectral radis among all 1 -digraphs. At the same time, we also obtain the maximm signless Laplacian spectral radis among 2 -digraphs. In Section 3, we characterize the extremal digraphs which attain the maximm and minimm signless Laplacian spectral radis among all θ 1 -digraphs. Under some conditions, we also obtain the maximm signless Laplacian spectral radis among all θ 2 -digraphs. In Section 4, we determine the extremal digraphs which attain the maximm and minimm signless Laplacian spectral radis among all 1 and θ 1 -digraphs. 2. The signless Laplacian Spectral Radis of 1 -digraphs and 2 -digraphs A 1 -digraph is a graph consisting of m (m 2) directed cycles with jst a ertex in common (as shown in Figre 1). We se 1 (k 1, k 2,..., k m ) to denote the 1 - digraph sch that m i=1 k i + 1 = n. Withot loss of generality, let 1 k i k i+1 for i = 1, 2,..., m 1. In this section, we first proe that 1 (1,..., 1, n m) is the niqe digraph which attains the maximm signless Laplacian spectral radis and 1 (a 1, a 2,..., a m ) sch that a i = n 1 and a m j = n 1 for any i {1, 2,..., m (n m 1 m n 1 n 1 )} and j {m (n 1 m ) + 1,..., m}, is the niqe digraph which m m attains the minimm signless Laplacian spectral radis among all 1 (k 1, k 2,..., k m )- digraphs on n ertices. Lemma 2.1 [5]. Let A be a nonnegatie irredcible matrix with the largest eigenale ϱ(a) and row sms s 1, s 2,..., s n, then min s i ϱ(a) max s i. 1 i n 1 i n Moreoer, one of the eqalities holds if and only if the row sms of A are all eqal. Lemma 2.2 For any p, q {1, 2,..., m}, if 2 k p k q, then we hae q( 1 (k 1, k 2,..., k p 1, k p, k p+1,..., k q 1, k q, k q+1,..., k m )) <

4 116 XIHE LI, LIGONG WANG AND SHANGYUAN ZHANG C k ,k 2,1 2 C k ,1 mk, m 1,k m,1 1 C k m+ 1 Figre 1: The digraph 1 (k 1, k 2,..., k m ). q( 1 (k 1, k 2,..., k p 1, k p 1, k p+1,..., k q 1, k q + 1, k q+1,..., k m )). Proof : Let G = 1 (k 1, k 2,..., k p 1, k p, k p+1,..., k q 1, k q, k q+1,..., k m ) and G = 1 (k 1, k 2,..., k p 1, k p 1, k p+1,..., k q 1, k q +1, k q+1,..., k m ) in the following. Sppose that x = (x, x 1,1,..., x 1,k1 ; x 2,1,..., x 2,k2 ;... ; x m,1,..., x m,km ) is the Perron ector of Q( G ) corresponding to q( G ), where x corresponds to, x i,j corresponds to i,j (i = 1, 2,..., m and j = 1, 2,..., k i ), respectiely. Since Q( G ) x = q( G ) x, it is not difficlt to see that q( G )x 1,i1 = x 1,i1 + x 1,i1 +1, i 1 = 1, 2,..., k 1 1, q( G )x 2,i2 = x 2,i2 + x 2,i2 +1, i 2 = 1, 2,..., k 2 1,. q( G )x m,im = x m,im + x m,im +1, i m = 1, 2,..., k m 1, q( G )x = mx + x 1,1 + x 2, x m,1, q( G )x j,kj = x j,kj + x, j = 1, 2,..., m. Then we hae x j,kj = (q( G ) 1) kj 1 x j,1, j = 1, 2,..., m. Frthermore, x = (q( G ) 1) k j x j,1, j = 1, 2,..., m.

5 THE SIGNLESS LAPLACIAN SPECTRAL RADIUS 117 Ths, we hae (q( G ) m)(q( G ) 1) n 1 x m,1 = m (q( G ) 1) n 1 k j x m,1. By Perron-Frobenis Theorem, we hae x m,1 > 0, therefore Similarly, we hae (q( G ) m)(q( G ) 1) n 1 = (q( G ) m)(q( G ) 1) n 1 = m j=1 m (q( G ) 1) n 1 k j. j=1 j=1,j p and q (q( G ) 1) n 1 k j +(q( G ) 1) n k p + (q( G ) 1) n k q 2. Let f(x) = (x m)(x 1) n 1 m j=1 (x 1)n 1 k j and g(x) = (x m)(x 1) n 1 m j=1,j p and q (x 1)n 1 k j (x 1) n k p (x 1) n k q 2. It is easy to see that q( G ) and q( G ) are the largest roots of f(x) = 0 and g(x) = 0, respectiely. f(x) g(x) = ((x 1) n k p 1 (x 1) n k q 2 )(x 2) > 0, for all x > 2. Since the minimm row sm of Q( G ) is 2, and the row sms of Q( G ) are not all eqal, then by Lemma 2.1, we hae q( G ) > 2. Therefore we hae q( G ) < q( G ). Using the aboe Lemma, we immediately obtain: Theorem 2.3 Among all 1 (k 1, k 2,..., k m )-digraphs of order n, the digraph 1 (1,..., 1, n m) is the niqe digraph which achiees the maximm signless Laplacian spectral radis, and 1 (a 1, a 2,..., a m ) sch that a i = n 1 and a m j = n 1 for m any i {1, 2,..., m (n 1 m n 1 n 1 )} and j {m (n 1 m ) + 1,..., m}, m m is the niqe digraph which achiees the minimm signless Laplacian spectral radis. Now, let 2 -digraph be a strongly connected tricyclic digraph of order n(= p + q + s + 2) (as shown in Figre 2), denoted by 2 (p, q, s), which obtained from three directed cycles C p+1, C q+2 and C s+1 by identifying a ertex of C p+1 with a ertex of C q+2 and identifying a ertex of C s+1 with a ertex of C q+2. Withot loss of generality, we assme that 1 p s. In the following, we will proe that the digraph 2 (1, 0, n 3) is the niqe digraph which achiees the maximm spectral radis among all 2 (p, q, s)- digraph of order n(= p + q + s) with q p s.

6 118 XIHE LI, LIGONG WANG AND SHANGYUAN ZHANG C p + 1 C q+ 2 C s + 1 1, p 1 1,2 1,p 1,1 2,t 2, t 1 2, t + 1 2, t 2 2,q 2,1 3,1 3,s 3,2 3, s 1 Figre 2: The digraph 2 (p, q, s). Lemma 2.4 If 1 p s, and p 2 then q( 2 (p, q, s)) < q( 2 (p 1, q, s + 1)). Proof : Let G = 2 (p, q, s) and G = 2 (p 1, q, s + 1) in the following. Sppose that x = (x, x, x 1,1,..., x 1,p, x 2,1,..., x 2,q, x 3,1,..., x 3,s ) is the Perron ector of Q( G ) corresponding to q( G ), where x corresponds to, x corresponds to and x i,j corresponds to i,j (i = 1, j = 1, 2,..., p; i = 2, j = 1, 2,..., q; i = 3, j = 1, 2,..., s), respectiely. Since Q( G ) x = q( G ) x, it is not difficlt to see that q( G )x = 2x + x 1,1 + x 2,t, q( G )x = 2x + x 2,1 + x 3,1, q( G )x 1,i1 = x 1,i1 + x 1,i1 +1, i 1 = 1, 2,..., p 1, q( G )x 1,p = x 1,p + x, q( G )x 2,i2 = x 2,i2 + x 2,i2 +1, i 2 = 1, 2,..., t 2, q( G )x 2,t 1 = x 2,t 1 + x, q( G )x 2,i2 = x 2,i2 + x 2,i2 +1, i 2 = t, t + 1,..., q 1, q( G )x 2,q = x 2,q + x, q( G )x 3,i3 = x 3,i3 + x 3,i3 +1, i 3 = 1, 2,..., s 1, q( G )x 3,s = x 3,s + x.

7 THE SIGNLESS LAPLACIAN SPECTRAL RADIUS 119 Then we hae x 3,s = (q( G ) 1)x 3,s 1 = (q( G ) 1) 2 x 3,s 2 = = (q( G ) 1) s 1 x 3,1, x = (q( G ) 1) s x 3,1, x 1,p = (q( G ) 1)x 1,p 1 = (q( G ) 1) 2 x 1,p 2 = = (q( G ) 1) p 1 x 1,1, x = (q( G ) 1) p x 1,1, x 2,q = (q( G ) 1)x 2,q 1 = (q( G ) 1) 2 x 2,q 2 = = (q( G ) 1) q t x 2,t, x 2,t 1 = (q( G ) 1)x 2,t 2 = (q( G ) 1) 2 x 2,t 3 = = (q( G ) 1) t 2 x 2,1, x = (q( G ) 1)x 2,t 1 = (q( G ) 1) t 1 x 2,1, x = (q( G ) 1)x 2,q = (q( G ) 1) q t+1 x 2,t. Frthermore, x 2,1 = (q( G ) 1) (t 1) x = (q( G ) 1) t (x + x 1,1 + x 2,t ) = (q( G ) 1) t x + (q( G ) 1) t x 1,1 + (q( G ) 1) t x 2,t = (q( G ) 1) 1 x 2,1 + (q( G ) 1) t p x + (q( G ) 1) t q+t 1 x = (q( G ) 1) 1 x 2,1 + (q( G ) 1) 1 p x 2,1 + (q( G ) 1) q 1 x. Then we get Then x 2,1 = (q( G ) 1) q 1 1 (q( G ) 1) 1 (q( G ) 1) p 1 x. q( G )x = 2x + x 2,1 + x 3,1 = (q( G ) 1) q 1 2x + 1 (q( G ) 1) 1 (q( x + (q( G ) 1) s x. G ) 1) p 1 Ths we dedce that (q( G ) 2) 2 (q( G ) 1) n 2 x (q( G ) 2)(q( G ) 1) q+s x

8 120 XIHE LI, LIGONG WANG AND SHANGYUAN ZHANG = (q( G ) 1) p+q+1 x (q( G ) 1) p+q x (q( G ) 1) q x + (q( G ) 1) p+s x. By Perron-Frobenis Theorem, we hae x > 0, therefore (q( G ) 2) 2 (q( G ) 1) n 2 (q( G ) 2)(q( G ) 1) q+s = (q( G ) 1) p+q+1 (q( G ) 1) p+q (q( G ) 1) q + (q( G ) 1) p+s. Similarly, we hae (q( G ) 2) 2 (q( G ) 1) n 2 (q( G ) 2)(q( G ) 1) q+s+1 = (q( G ) 1) p+q (q( G ) 1) p+q 1 (q( G ) 1) q + (q( G ) 1) p+s. Let f(x) = (x 2) 2 (x 1) n 2 (x 2)(x 1) q+s (x 1) p+q+1 + (x 1) p+q + (x 1) q (x 1) p+s and g(x) = (x 2) 2 (x 1) n 2 (x 2)(x 1) q+s+1 (x 1) p+q + (x 1) p+q 1 + (x 1) q (x 1) p+s. It s easy to see that q( G ) is the largest real root of f(x) = 0 and q( G ) is the largest real root of g(x) = 0. Since f(x) g(x) = (x 2) 2 ((x 1) q+s (x 1) p+q 1 ) > 0 for all x > 1. Since the minimm row sm of Q( 2 (p, q, s)) is 2, and the row sms of Q( 2 (p, q, s)) are not all eqal, then by Lemma 2.1, we hae q( 2 (p, q, s)) > 1. Then we hae q( 2 (p, q, s)) < q( 2 (p 1, q, s + 1)). Lemma 2.5 If 1 q p s, then q( 2 (p, q, s)) < q( 2 (p + 1, q 1, s)). Proof : Let G = 2 (p, q, s) and G = 2 (p + 1, q 1, s) in the following. Sppose that x = (x, x, x 1,1,..., x 1,p, x 2,1,..., x 2,q, x 3,1,..., x 3,s ) is the Perron ector of Q( G ) corresponding to q( G ), where x corresponds to, x corresponds to and x i,j corresponds to ij (i = 1, j = 1, 2,..., p; i = 2, j = 1, 2,..., q; i = 3, j = 1, 2,..., s), respectiely. Similar to the proof of Lemma 2.4, we hae (q( G ) 2) 2 (q( G ) 1) n 2 (q( G ) 2)(q( G ) 1) q+s = (q( G ) 1) p+q+1 (q( G ) 1) p+q (q( G ) 1) q + (q( G ) 1) p+s. and (q( G ) 2) 2 (q( G ) 1) n 2 (q( G ) 2)(q( G ) 1) q+s 1

9 THE SIGNLESS LAPLACIAN SPECTRAL RADIUS 121 = (q( G ) 1) p+q+1 (q( G ) 1) p+q (q( G ) 1) q 1 + (q( G ) 1) p+s+1. Let f(x) = (x 2) 2 (x 1) n 2 (x 2)(x 1) q+s (x 1) p+q+1 + (x 1) p+q + (x 1) q (x 1) p+s and g(x) = (x 2) 2 (x 1) n 2 (x 2)(x 1) q+s 1 (x 1) p+q+1 + (x 1) p+q + (x 1) q 1 (x 1) p+s+1. It s easy to see that q( G ) is the largest real root of f(x) = 0 and q( G ) is the largest real root of g(x) = 0. Since f(x) g(x) = (x 2)((x 1) q+s 1 + (x 1) q 1 + (x 1) p+s (x 1) q+s ) > 0 for all x > 2 when 1 q p s. Since the minimm row sm of Q( 2 (p, q, s)) is 2, and the row sms of Q( 2 (p, q, s)) are not all eqal, then by Lemma 2.1, we hae q( 2 (p, q, s)) > 2. Then we hae q( 2 (p, q, s)) < q( 2 (p + 1, q 1, s)). Similar to the proof of Lemma 2.5, we can obtain the following reslt. Lemma 2.6 If 1 q p s, then q( 2 (p, q, s)) < q( 2 (p, q 1, s + 1)). Using the aboe Lemmas, we immediately obtain the following theorem. Theorem 2.7 Among all 2 (p, q, s)-digraphs of order n with q p s, the digraph 2 (1, 0, n 3) is the niqe digraph which achiees the maximm signless Laplacian spectral radis. Problem 2.8 Among all 2 (p, q, s)-digraphs of order n(= p + q + s + 2) with 1 p s, which digraph achiees the maximm (or minimm) signless Laplacian spectral radis? 3. The signless Laplacian spectral radis of θ 1 -digraphs and θ 2 -digraphs A θ-graph is a graph consisting of three paths which hae the same end-ertices. In [4], the athors defined the generalized strongly connected θ-digraph as follow. For conenience, we se the abbreiation θ-digraph for the generalized strongly connected θ-digraph. The θ-digraph consists of s + t (s 2, t 1 and st = m) directed paths P k1 +2, P k2 +2,..., P ks +2 and P l1 +2, P l2 +2,..., P lt +2 sch that the initial ertex of P k1 +2, P k2 +2,..., P ks+2 is the terminal ertex of P l1 +2, P l2 +2,..., P lt+2, and the initial ertex of P l1 +2, P l2 +2,..., P lt +2 is the terminal ertex of P k1 +2, P k2 +2,..., P ks +2 (as shown in Figre 3), denoted by θ(k 1, k 2,..., k s ; l 1, l 2,..., l t ) sch that s i=1 k i +

10 122 XIHE LI, LIGONG WANG AND SHANGYUAN ZHANG t j=1 l j + 2 = n. Withot loss of generality, let k i k i+1 for i = 1, 2,..., s 1, and l j l j+1 for j = 1, 2,..., t 1. In particlarly, we write θ(k 1, k 2,..., k s ; l 1, l 2,..., l t ) as θ 1 (k 1, k 2,..., k m ; l 1 )(as shown in Figre 4) when s 2, t = 1, and st = m. In the following, we first proe that θ 1 (0, 1,..., 1, n m; 0) is the niqe digraph which achiees the maximm signless Laplacian spectral radis and θ 1 (0, 1,..., 1; n m 1) is the niqe digraph which achiees the minimm signless Laplacian spectral radis among all θ 1 (k 1, k 2,..., k m ; l 1 )- digraphs for fixed n. 1,1 1,2 1,k1 s,1 s,2 sk, s 1,l 1 1, l1 1 1,1 tl, t tl, t -1 t,1 Figre 3: The digraph θ(k 1, k 2,..., k s ; l 1, l 2,..., l t ). 1,1 2,1 m,1 1,2 2,2 m,2 1,k 1 2,k 2 mk, m 1,l 1 1, l1-1 1,1 Figre 4: The digraph θ 1 (k 1, k 2,..., k m ; l 1 ). Lemma 3.1 For any p, q {1, 2,..., m}, if 1 k p k q, then we hae q( θ 1 (k 1, k 2,..., k p 1, k p, k p+1,..., k q 1, k q, k q+1,..., k m ; l 1 )) < q( θ 1 (k 1, k 2,..., k p 1, k p 1, k p+1,..., k q 1, k q + 1, k q+1,..., k m ; l 1 )).

11 THE SIGNLESS LAPLACIAN SPECTRAL RADIUS 123 Proof : Let G 1 = θ 1 (k 1, k 2,..., k p 1, k p, k p+1,..., k q 1, k q, k q+1,..., k m ; l 1 ) and G 2 = θ 1 (k 1, k 2,..., k p 1, k p 1, k p+1,..., k q 1, k q + 1, k q+1,..., k m ; l 1 ) in the following. Sppose that x = (x, x, x 1,1,..., x 1,k1 ; x 2,1,..., x 2,k2 ;... ; x m,1,..., x m,km ; x l1,1,..., x l1,l 1 ) is the Perron ector of G 1 corresponding to q( G 1 ), where x and x correspond to and, x i,j corresponds to i,j (i = 1, 2,..., m and j = 1, 2,..., k i ), x l1,j corresponds to 1,j (j = 1, 2,..., l 1 ), respectiely. Since Q( G 1 ) x = q( G 1 ) x, we hae q( G 1 )x 1,i1 = x 1,i1 + x 1,i1 +1, i 1 = 1, 2,..., k 1 1, q( G 1 )x 2,i2 = x 2,i2 + x 2,i2 +1, i 2 = 1, 2,..., k 2 1,. q( G 1 )x m,im = x m,im + x m,im+1, i m = 1, 2,..., k m 1, q( G 1 )x 1,k1 = x 1,k1 + x, q( G 1 )x 2,k2 = x 2,k2 + x,. q( G 1 )x m,km = x m,km + x, q( G 1 )x = mx + x 1,1 + x 2, x m,1, q( G 1 )x = x + x l1,1 q( G 1 )x l1,j = x l1,j + x l1,j+1, j = 1, 2,..., l 1 1, q( G 1 )x l1,l 1 = x l1,l 1 + x. Then we get x j,kj = (q( G 1 ) 1) kj 1 x j,1, j = 1, 2,..., m, x l1,l 1 = (q( G 1 ) 1) l1 1 x l1,1, x = (q( G 1 ) 1) k j x j,1, j = 1, 2,..., m. Frthermore x = (q( G 1 ) 1) l 1 x l1,1 = (q( G 1 ) 1) l1+1 x = (q( G 1 ) 1) l 1+k m +1 x m,1.

12 124 XIHE LI, LIGONG WANG AND SHANGYUAN ZHANG Ths we dedce that (q( G 1 ) m)(q( G 1 ) 1) n 1 x m,1 = By Perron-Frobenis Theorem, we hae x m,1 > 0, therefore Similarly, we hae (q( G 1 ) m)(q( G 1 ) 1) n 1 = (q( G 2 ) m)(q( G 2 ) 1) n 1 = m i=1,i p and q m (q( G 1 ) 1) n 2 l 1 k i x m,1. i=1 m (q( G 1 ) 1) n 2 l 1 k i. i=1 (q( G 2 ) 1) n 2 l 1 k i +(q( G 2 ) 1) n 1 l 1 k p + (q( G 2 ) 1) n 3 l 1 k q. Let f(x) = (x m)(x 1) n 1 m i=1 (x 1)n 2 l 1 k i and g(x) = (x m)(x 1) n 1 m i=1,i p and q (x 1)n 2 l 1 k i (x 1) n 1 l 1 k p (x 1) n 3 l 1 k q. It is easy to see that q( G 1 ) and q( G 2 ) are the largest roots of f(x) = 0 and g(x) = 0, respectiely. f(x) g(x) = ((x 1) n 2 l 1 k p (x 1) n 3 l 1 k q )(x 2) > 0, for all x > 2. Since the minimm row sm of Q( G 1 ) is 2, and the row sms of Q( G 1 ) are not all eqal, then we hae q( G 1 ) > 2. Then we hae q( G 2 ) > q( G 1 ). Similar to the proof of Lemma 3.1, we can obtain the following lemma. Lemma 3.2 For any p {1, 2,..., m}, we hae q( θ 1 (k 1, k 2,..., k p 1, k p, k p+1,..., k m ; l 1 )) < q( θ 1 (k 1, k 2,..., k p 1, k p + 1, k p+1,..., k m ; l 1 1)). Combining Lemmas 3.1 and 3.2, we hae the following theorem. Theorem 3.3 Among all θ 1 (k 1, k 2,..., k m ; l 1 )-digraphs of order n, θ 1 (0, 1,..., 1, n m; 0) is the niqe digraph which achiees the maximm signless Laplacian spectral radis and θ 1 (0, 1,..., 1; n m 1) is the niqe digraph which achiees the minimm signless Laplacian spectral radis. Now, when s 2, t 1, and st = m, we write θ(k 1, k 2,..., k s ; l 1, l 2,..., l t ) as θ 2 (k 1, k 2,..., k m ; l 1, l 2,..., l t ). In the following, we shall proe that θ 2 (0, 1,..., 1; 0, 1,..., 1, n

13 THE SIGNLESS LAPLACIAN SPECTRAL RADIUS 125 s t+1) is the niqe digraph which achiees the maximm signless Laplacian spectral radis among all θ 2 (k 1, k 2,..., k m ; l 1, l 2,..., l t ) -digraphs for fixed n. Similar to the proof of Lemma 3.1, we obtain the following reslts. Lemma 3.4 For any p, q {1, 2,..., s}, if 1 k p k q, then we hae q( θ 2 (k 1, k 2,..., k p 1, k p, k p+1,..., k q 1, k q, k q+1,..., k s ; l 1, l 2,..., l t )) < q( θ 2 (k 1, k 2,..., k p 1, k p 1, k p+1,..., k q 1, k q + 1, k q+1,..., k s ; l 1, l 2,..., l t )). Lemma 3.5 For any p, q {1, 2,..., t}, if 1 l p l q, then we hae q( θ 2 (k 1, k 2,..., k s ; l 1, l 2,..., l p 1, l p, l p+1,..., l q 1, l q, l q+1,..., l t )) < q( θ 2 (k 1, k 2,..., k s ; l 1, l 2,..., l p 1, l p 1, l p+1,..., l q 1, l q + 1, l q+1,..., l t )). Lemma 3.6 If s t, k i = l i (i = 1, 2,..., s 1) and k s l t, then q( θ 2 (k 1, k 2,..., k s 1; l 1, l 2,..., l t + 1)) > q( θ 2 (k 1, k 2,..., k s ; l 1, l 2,..., l t )). Combining Lemmas 3.4, 3.5 and 3.6, we hae the following theorem. Theorem 3.7 If s t and s i=1 k i t j=1 l j, then among all θ 2 (k 1, k 2,..., k m ; l 1, l 2,..., l t )-digraphs, the digraph θ 2 (0, 1,..., 1; 0, 1,..., 1, n s t + 1) is the niqe digraph which achiees the maximm signless Laplacian spectral radis. 4. The Signless Laplacian Spectral Radis of 1 -digraphs and θ 1 -digraphs In the following, we will discss the digraph which achiees the maximm and minimm signless Laplacian spectral radis among all 1 -digraphs and θ 1 -digraphs. The following well-known theorem can be fond in [5]. Theorem 4.1 [5]. Let G = (V ( G), E( G)) be a simple digraph on n ertices,,, w distinct ertices of V ( G), (, ) E( G) and x = (x 1, x 2,..., x n ) be the niqe positie nit eigenector corresponding to the signless Laplacian spectral radis of q( G), where x i corresponds to the ertex i. Let H = G {(, )}+{(, w)} (Noting

14 126 XIHE LI, LIGONG WANG AND SHANGYUAN ZHANG that if (, w) E( G), then H has mltiple arc (, w)). If x w x, then q( H ) q( G). Frthermore, if H is strongly connected and x w x, then q( H ) > q( G). Lemma 4.2 For any θ 1 (k 1, k 2,..., k m ; l 1 )- digraph, there exists a 1 (k 1, k 2,..., k m 1, k m + l 1 + 1)- digraph sch that q( θ 1 (k 1, k 2,..., k m ; l 1 )) < q( 1 (k 1, k 2,..., k m 1, k m + l 1 + 1)). Proof : By lemma 3.1, we know that x = (q( θ 1 (k 1, k 2,..., k m ; l 1 )) 1) l1+1 x > x. It is easy to see that 1 (k 1, k 2,..., k m 1, k m + l 1 + 1) = θ 1 (k 1, k 2,..., k m ; l 1 ) {( 1k1, ), ( 2k2, ),..., ( m 1km 1, )}+{( 1k1, ), ( 2k2, ),..., ( m 1km 1, )}. Ths we hae q( θ 1 (k 1, k 2,..., k m ; l 1 )) < q( 1 (k 1, k 2,..., k m 1, k m + l 1 + 1)). Lemma 4.3 For any 1 (k 1, k 2,..., k m 1, k m )- digraph, there exists a θ 1 (k 1, k 2,..., k m 1, k m 1; 0)- digraph sch that q( θ 1 (k 1, k 2,..., k m 1, k m 1; 0)) < q( 1 (k 1, k 2,..., k m 1, k m )). Proof : By lemma 3.1, we know that x = (q( θ 1 (k 1, k 2,..., k m ; l 1 )) 1) l1+1 x > x. It is easy to see that 1 (k 1, k 2,..., k m 1, k m ) = θ 1 (k 1, k 2,..., k m 1, k m 1; 0) {( 1k1, ), ( 2k2, ),..., ( m 1km 1, )}+{( 1k1, ), ( 2k2, ),..., ( m 1km 1, )}. Ths we hae q( θ 1 (k 1, k 2,..., k m 1, k m 1; 0)) < q( 1 (k 1, k 2,..., k m 1, k m )). From Lemmas 4.2 and 4.3, we know that the digraph which achiees the maximm signless Laplacian spectral radis among all 1 -digraphs and θ 1 -digraphs mst be in -digraphs, and the digraph which achiees the minimm signless Laplacian spectral radis among all 1 -digraphs and θ 1 -digraphs mst be in θ 1 -digraphs. Combining Theorems 2.3 and 3.3, Lemmas 4.2 and 4.3, we can immediately get the following theorem. Theorem 4.4 Among all 1 -digraphs and θ 1 -digraphs of order n, the digraph 1 (1, 1,..., 1, n m) is the digraph which achiees the maximm signless Laplacian spectral radis and the digraph θ 1 (0, 1,..., 1; n m 1) the niqe digraph which achiees the minimm signless Laplacian spectral radis. References 1. C. J. B and J. Zho, Starlike trees whose maximm degree exceed 4 are determined by their Q-spectra, Linear Algebra Appl., 436 (2012),

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