Maxima of the signless Laplacian spectral radius for planar graphs

Electronic Journal of Linear Algebra Volume 0 Volume 0 (2015) Article 51 2015 Maxima of the signless Laplacian spectral radius for planar graphs Guanglong Yu Yancheng Teachers University, yglong01@16.com Jianyong Wong Linyi University Shu-guang Guo Yancheng Teachers University Follow this and additional works at: http://repository.uwyo.edu/ela Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Yu, Guanglong; Wong, Jianyong; and Guo, Shu-guang. (2015), "Maxima of the signless Laplacian spectral radius for planar graphs", Electronic Journal of Linear Algebra, Volume 0, pp. 795-811. DOI: https://doi.org/10.1001/1081-810.202 This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact scholcom@uwyo.edu.

MAXIMA OF THE SIGNLESS LAPLACIAN SPECTRAL RADIUS FOR PLANAR GRAPHS GUANGLONG YU, JIANYONG WANG, AND SHU-GUANG GUO Abstract. The signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In this paper, it is proved that the graph K 2 P has the maximal signless Laplacian spectral radius among all planar graphs of order n 456. Key words. Signless Laplacian, Spectral radius, Planar graph. AMS subject classifications. 05C50. 1. Introduction. Recently, the signless Laplacian has attracted the attention of researchers (see [ 6, 9]). Some results on the signless Laplacian spectrum have been reported since 2005 and a new spectral theory called Q-theory is being developed by many researchers. Simultaneously, the application of the Q-theory has been extensively explored [7, 8, 16]. Schwenk and Wilson initiated the study of the eigenvalues of planar graphs [12]. In [2], D. Cao and A. Vince conjectured that K 2 P has the maximum spectral radius among all planar graphs of order n, where denotes the join of two graphs obtainedfromtheunionofthesetwographsbyjoiningeachvertexofthefirstgraphto each vertex of the second graph. The conjecture is still open. With the development of the Q-theory, a natural question is: What about the maximum signless Laplacian spectral radius of planar graphs? By some comparisons in [9], it seems plausible that K 2 P also has the maximal signless Laplacian spectral radius among planar graphs. In this paper, we confirm that among planar graphs with order n 456, K 2 P has the maximal signless Laplacian spectral radius. The layout of this paper is as follows. Section 2 gives some notations and some needed lemmas. In Section, our results are presented. Received by the editors November 25, 2014. Accepted for publication on September 15, 2015. Handling Editor: Bryan L. Shader. Department of Mathematics, Yancheng Teachers University, Yancheng, 224002, Jiangsu, P.R. China (yglong01@16.com). Supported by NSFC (1127115, 11171290, 11201417), Jiangsu Qing Lan Project (2014A), and JSNSFC (BK20151295). Department of Mathematics, Linyi University, Linyi, 276005, Shandong, P.R. China. 795

796 Guanglong Yu, Jianyong Wang, and Shu-guang Guo 2. Preliminaries. All graphs considered in this paper are undirected and simple, i.e., no loops or multiple edges are allowed. Denote by G = G[V(G), E(G)] a graph with vertex set V(G) and edge set E(G). The number of vertices, resp., edges, of G is denoted by n = V(G), resp., m(g) = E(G). Recall that given a graph G, Q(G) = D(G) + A(G) is the signless Laplacian matrix of G, where D(G) = diag(d 1,d 2,...,d n )withd i = d G (v i )beingthedegreeofvertexv i (1 i n), and A(G) being the adjacency matrix of G. The signless Laplacian spectral radius of G, denoted by q(g), is the largest eigenvalue of Q(G). For a connected graph G of order n, the Perron eigenvector of Q(G) is the unit (with respect to the Euclidean norm) positive eigenvector corresponding to q(g); the standard eigenvector of Q(G) is the positive eigenvector X = (x 1,x 2,...,x n ) T R n corresponding to q(g) n satisfying = 1. i=1 Denote by K n, C n, P n a complete graph, a cycle and a path of order n, respectively. For a graph G, if there is no ambiguity, we use d(v) instead of d G (v), use δ or δ(g) to denote the minimum vertex degree, use or (G) to denote the largest vertex degree, and use or (G) to denote the second largest vertex degree. In a graph, the notation v i v j denotes that vertex v i is adjacent to v j. Denote by K s,t a complete bipartite graph with one part of size s and another part of size t. In a graph G, for a vertex u V(G), let N G (u) denote the neighbor set of u, and let N G [u] = {u} N G (u). G(u) = G[N G [u]], G (u) = G[N G (u)] denote the subgraphs induced by N G [u], N G (u), respectively. The reader is referred to[1, 10] for the facts about planar and outer-planar graphs. A graph which can be drawn in the plane in such a way that edges meet only at points corresponding to their common ends is called a planar graph, and such a drawing is called a planar embedding of the graph. A simple planar graph is (edge) maximal if no edge can be added to the graph without violating planarity. In the planar embedding of a maximal planar graph G of order n, each face is triangle. For a planar graph G of order n, we have m(g) n 6 with equality if and only if it is maximal. In a maximal planar graph G of order n 4, δ(g). A graph G is outer-planar if it has a planar embedding, called standard embedding, in which all vertices lie on the boundary of its outer face. A simple outer-planar graph is (edge) maximal if no edge can be added to the graph without violating outer-planarity. In a standard embedding of a maximal outer-planar graph G of order n, the boundary of the outer face is a Hamiltonian cycle (a cycle contains all vertices) of G, and each of the other faces is triangle. For an outer-planar graph G, we have m(g) 2n with equality if and only if it is maximal. In a maximal planar graph G of order n 4 and for a vertex u V(G), we have that G (u) is an outer-planar graph, and G(u) = u G (u). From a nonmaximal planar graph G, by inserting edges to G, a maximal planar graph G can be obtained. From spectral graph theory, for a graph

Maxima of the Signless Laplacian Spectral Radius for Planar Graphs 797 G, it is known that q(g+e) > q(g) if e / E(G). Consequently, when we consider the maxima of the signless Laplacian spectral radius among planar graphs, it suffices to consider the maximal planar graphs directly. Note that for n = 1,2,,4, a maximal planar graph G of order n is isomorphic to K n. Their signless Laplacian spectral radii can be easily determined by some computations. As a result, to consider the maxima of the signless Laplacian spectral radius among planar graphs of order n, we pay more attentions to those of order n 5. Next we introduce some needed lemmas. Lemma 2.1. [1] Let u be a vertex of a maximal outer-planar graph on n 2 vertices. Then v ud(v) n+d(u) 4. Lemma 2.2. [11] Let G be a graph. Then { q(g) max u V(G) d G (u)+ 1 d G (u) } d G (v). v u Lemma 2.. [5] Let G be a connected graph containing at least one edge. Then q(g) +1 with equality if and only if G = K 1,n 1.. Main results. Lemma.1. Let G be a maximal planar graph of order n. Then { q(g) max d G (u)+2+ n 9 }. u V(G) d G (u) Proof. Let u V(G), N G (u) = {v 1,v 2,...,v t }, and V 1 = V(G)\N G [u]. For 1 i t, let α i = d G (u)(v i ). Note that m(g (u)) = E(G (u)) = 1 t α i. 2 Between N G (u) and V 1, there are n 6 1 2 [ d G (v) = d G (u)+ n 6 1 2 v u i=1 t α i d G (u) edges. Consequently, i=1 ] t α i d G (u) + i=1 t α i = n 6+ 1 2 i=1 t α i. Since G (u) is an outer-planar graph, m(g (u)) 2d G (u). As a result, v ud G (v) n 9+2d G (u), and thus, d G (u)+ 1 d G (u) v u d G (v) d G (u)+2+ n 9 d G (u). i=1

798 Guanglong Yu, Jianyong Wang, and Shu-guang Guo { By Lemma 2.2, q(g) max d G (u)+2+ n 9 }. u V (G) d G (u) Remark 1. Let f(x) = x+2+ n 9. It can be checked that f(x) is convex x when n 4. Let G be a maximal planar graph of order n with largest degree (G). As discussed in section 2, we know that if n 4, then d G (u) for any vertex u V(G). Thus, d G (u)+ 1 { d G (v) max 5+ n 9, (G)+2+ n 9 }. d G (u) (G) v u Moreover, if n 6 and (G) n, then q(g) n+2. v 1 vn v 5 v 4 v Fig..1. H n. v 2 Let H 1 = K 1, H 2 = K 2, and H n = k 2 P for n (see Fig..1). Lemma.2. If n 5, then q(h n ) > n+2. Proof. Let X = (x 1,x 2,...,x n ) T R n be the standard eigenvector of Q(H n ). By symmetry, x 1 = x 2, x = x n. By Lemma 2., q(h n ) n. (.1) Note that q(h n )x 1 = (n 1)x 1 +x 2 + i= x 1 = n = ()x 1 +1. Thus, i= 1 q(h n ) n+2. n n Note that q(h n ) = 6 x n +2()x 1 2(x +x n ). Thus, n i= As a result, (.2) = 2()x 1 4x q(h n ) 6 i= and 1 = n i=1 = 2()x 1 4x q(h n ) 6 x = 2() (q(h n) n)(q(h n ) 6). 4(q(H n ) n+2) +x 1 +x 2.

Maxima of the Signless Laplacian Spectral Radius for Planar Graphs 799 Note that q(h n )x = x +x 1 +x 2 +x 4. Then (q(h n ) 2)(x 1 x ) = (n 4)x 1 + n. The fact n 5 implies that x 1 > x. Combining with (.1) and (.2), we get (.) 2() (q(h n ) n)(q(h n ) 6) 4(q(H n ) n+2) < i=5 1 q(h n ) n+2. Simplifying (.), we get q 2 (H n ) (6+n)q(H n )+4n+8 > 0. It follows that q(h n ) > n+2. FromRemark1andLemma.2, weseethattoconsiderthemaximaofthesignless Laplacian spectral radius among planar graphs of order n 5, it suffices to consider those with maximum degree n 1 or. Lemma.. [14] Let A be an irreducible nonnegative square real matrix of order n and spectral radius ρ. If there exists a nonnegative real vector y 0 and a real coefficient polynomial function f such that f(a)y ry (r R), then f(ρ) r. Lemma.4. Let 1 k 12 be an integer number, and let G be a maximal planar graph of order n 115, where d G (v 1 ) = (G) =, for i = 2,,...,k +1, n 6 +1 d G(v i ) n 61, and for k+2 i n, d G (v i ) < n 6 +1. Then q(g). Proof. Let X = (x 1,x 2,x,...,x n ) T R n be a positive vector, where 1, i = 1; 1 = k, 2 i k +1; n k 1, k +2 i n. For v 1, we have ()x 1 + v j v 1 (n k 2) +1+ < n+2. x 1 n k 1 For v i (k +2 i n), we have d G (v i ) + v j v i d G (v i )+ k+1 j=1 xj+(d G (v i ) k 1) n k 1 n k 1 n+2, d(v i ) k +1; d G (v i )+ dg (v i ) j=1 n k 1 d G (v i )+ 1+d G (v i ) 1 k n k 1 < n+2, d G (v i ) k.

800 Guanglong Yu, Jianyong Wang, and Shu-guang Guo (.4) For v i (2 i k +1), since n 115 and 1 k 12, we have d G (v i ) > k. Thus, d G (v i ) + v j v i = (d G(v i ) 1) + + v j v i ( 1+ k Let f(k) = n k 1 with respect to k, we get d G (v i ) 1+ = ( 1+ ) d G (v i ) k+1 j=1 + (dg(vi) k) n k 1 1 ) k d G (v i ) n k 1 k k 2 +2k 1. n k 1 k 2 +2k 1. Taking derivation of f(k) n k 1 f (k) = (n k 1)(2+d G(v i ) 8k)+kd G (v i ) k 2 (n k 1) 2. Since d G (v i ) k and n 115, we get f (k) > 0. This implies that f(k) is monotone increasing with respect to k. Since n 115, k 12, and d G (v i ) n 61, we conclude that ( 1+ ) k d G (v i ) n k 1 Thus, from (.4), we get d G(v i ) + v j v i k 2 +2k 1 < n+2. n k 1 < n+2. By the above discussion, we get Q(G)X (n+2)x. The proof is now completed by applying Lemma.. Lemma.5. Let G be a maximal planar graph of order n 80 with d G (v 1 ) = (G) =, and (G) n 62. Then q(g) n+2. Proof. Supposed G (v 2 ) = (G). LetX = (x 1,x 2,x,...,x n ) T R n beapositive vector, where 1, i = 1; = 1, i = 2;, i n. For v 1, we have ()x 1 + v j v 1 x 1 +1+ (n ) < n+2. Next, there are two cases to consider.

Maxima of the Signless Laplacian Spectral Radius for Planar Graphs 801 v 2j v f 1 v f v 2j 1 vf+1 v 2j+1 v n v 2s v n v 21 v 2t v 2s+1 v v 2 v n 1 Fig..2. d G (v 2j ). v v 2 v n 1 Fig... d G (v f ). Case 1. v 2 N G (v 1 ). Suppose N G (v 1 ) = {v 2,v,...,v,v n 1 }. Without loss of generality suppose that v 1 is in the outer face of G v 1. Then v n is in one of the inner faces of G v 1 (see Fig..2). For v 2, since d G (v 2 ), we have d G (v 2 )x 2 + v j v 2 x 2 d G (v 2 )+1+ (d G(v 2 ) 1) n+2. Denote by C v1 = v 2 v v v n 1 v 2 the Hamiltonian cycle in G v 1. Suppose that v i s (2 i n 1) are distributed along the clockwise direction on C v1 and suppose N G v1 (v 2 ) = {v 21,v 22,...,v 2t }, where for 1 i t 1, 2 i < 2 i+1, v 21 = v, v 2t = v n 1 (see Fig..2). For 1 j t, suppose there are l j 1 vertices between v 2j 1 and v 2j along the clockwise direction on C v1, where if j = 1, we let v 20 = v 2. Along the clockwise direction on C v1, suppose there are l t vertices between v 2t and v 2. For each v 2j (1 j t, see Fig..2), noting that l j 1 +l j n d G (v 2 ) and d G (v 2 ) n 62, we have and d G (v 2j ) l j 1 +l j +5 n+2 d G (v 2 ) 64, d G (v 2j )x 2j + v k v 2j x k x 2j d G (v 2j )+ 2+ (d G(v 2j ) 2) n+2. For each v f (N G (v 1 )\{v 2,v 21,v 22,...,v 2t }), then along the clockwise direction on C v1, there exists 0 s t such that v f is between v 2s and v 2s+1, where v 2t+1 = v 2 (see Fig..). Note that l s n d G (v 2 ). Then d G (v f ) l s + n d G (v 2 ) 62, and thus, d G (v f )x f + v k v f x k x f d G (v f )+ 2+ (d G(v f ) 2) n+2.

802 Guanglong Yu, Jianyong Wang, and Shu-guang Guo Note that v n is in one of the inner faces of G v 1. Suppose that in G v 1, v n is in a face v 2 v 2z v 2z+1v 2z+2 v 2z+1 v 2 (see Fig..4). Note that l z n d G (v 2 ) and d G (v n ) l z +. Then d G (v n ) n d G (v 2 ) 62, and d G (v n )x n + v k v n x k x n v 2z v v 2 v n v n 1 Fig..4. d G (v n ). v2z+1 d G (v n )+ 1+ (dg(vn) 1) v 22 v 21 v 4 v 2 v < n+2. v n Fig..5. d G (v 2 ). v 2t Case 2. v 2 / N G (v 1 ). Without loss of generality suppose that v 1 is in the outer face of G v 1. Then v 2 is in one of the inner faces of G v 1. Then N G (v 1 ) = {v,v 4,v 5,...,v n 1,v n }. Suppose that C v1 = v v 4 v n 1 v n v is the Hamiltonian cycle in G v 1, v i s ( i n) are distributed along the clockwise direction on C v1, and suppose N G v1 (v 2 ) = {v 21,v 22,...,v 2t }, where for 1 i t 1, 2 i < 2 i+1 (see Fig..5). For 2 j t, along the clockwise direction on C v1, suppose there are l j 1 vertices between v 2j 1 and v 2j. Along the clockwise direction on C v1, suppose that there are l t vertices between v 2t and v 21. For each v 2j (1 j t), by an argument similar to Case 1, we have d G (v 2j ) 64, and d G (v 2j )x 2j + v k v 2j x k x 2j d G (v 2j )+ 2+ (d G(v 2j ) 2) n+2. By an argument similar to Case 1, for each v i (N G (v 1 )\{v 21,v 22,...,v 2t }), we have d G (v i ) n d G (v 2 ) 62, and d G (v i ) + v k v i x k d G (v i )+ 1+ (dg(vi) 1) < n+2. For v 2, since d G (v 2 ), we have d G (v 2 )x 2 + v k v 2 x k x 2 d G (v 2 )+ d G(v 2 ) n+1.

Maxima of the Signless Laplacian Spectral Radius for Planar Graphs 80 By the above discussion, we get Q(G)X (n+2)x. The proof is now completed by applying Lemma.. Lemma.6. Let G be a maximal planar graph of order n 4, where d(v 1 ) = (G) =, and for 2 i n, d G (v i ) < 1+ n 6. Then q(g). Proof. Let X = (x 1,x 2,x,...,x n ) T R n be a positive vector, where For v 1, we have = 1, i = 1; 4 n 1, 2 i n. ()x 1 + v j v 1 x 1 + 4() n 1 < n+2. For v i (2 i n), we have d G (v i ) + v j v i d G (v i )+ 1+ 4(dG(vi) 1) n 1 4 n 1 < n+2. By the above discussion, we get Q(G)X (n + 2)X. Applying Lemma. completes the proof. Theorem.7. Let G be a maximal planar graph of order n 80 with (G) =. Then q(g) n+2. Proof. This theorem follows from Lemmas.4.6. Lemma.8. Let 1 k 1 be an integer number, and let G be a maximal planar graph of order n 91, where d G (v 1 ) = (G) = n 1, for i = 2,,...,k +1, n 7 + 19 7 d(v i) n 75, and for k+2 i n, d G (v i ) < n 7 + 19 7. Then q(g) n+2. Proof. Let X = (x 1,x 2,x,...,x n ) T R n be a positive vector, where 1, i = 1; 2 = k, 2 i k +1; 7 (n k 1), k +2 i n. For v 1, we have (n 1)x 1 + v j v 1 x 1 = n+2.

804 Guanglong Yu, Jianyong Wang, and Shu-guang Guo For v i (k +2 i n), we have d G (v i ) + v j v i d G (v i )+ d G (v i )+ 5 +7(d G(v i) k 1) (n k 1) 7 (n k 1) 5 7 (n k 1) n+2, d G (v i ) k +1; < n+2, d G (v i ) k. For v i (2 i k +1), since n 91 and 1 k 1, we have d G (v i ) > k. Thus, d G (v i ) + v j v i = (d G(v i ) 1) + + v j v i (.5) d G (v i ) 1+ = d G (v i ) 1+ k+1 j=1 + 7(dG(vi) k) (n k 1) 2 k 5 + 7(dG(vi) k) (n k 1) 2 k = d G (v i ) 1+ 5 7 2 k + 2 k(d G(v i ) k). n k 1 As the proof of Lemma.4, since n 91 and k < d G (v i ) n 75, we can prove that Thus, (.5) implies d G(v i ) + v j v i d G (v i ) 1+ 5 7 2 k + 2 k(d G(v i ) k) n+2. n k 1 n+2. By the above discussion, we get Q(G)X (n + 2)X. Applying Lemma. completes the proof. Lemma.9. Let G be a maximal planar graph of order n 6, where d(v 1 ) = (G) = n 1, and for 2 i n, d G (v i ) < n 7 + 19 7. Then q(g) n+2. Proof. Let X = (x 1,x 2,x,...,x n ) T R n be a positive vector, where 1, i = 1; = n 1, 2 i n. For v 1, we have (n 1)x 1 + v j v 1 x 1 = n+2. For v i (2 i n), we have d G (v i ) + v j v i d G (v i )+ 1+ (dg(vi) 1) n 1 n 1 n+2.

Maxima of the Signless Laplacian Spectral Radius for Planar Graphs 805 By the above discussion, we get Q(G)X (n+2)x. The proof is now completed by applying Lemma.. Lemma.10. Let G be a maximal planar graph of order n 461 with d G (v 1 ) = (G) = n 1 and n 81 (G) n 4. Then q(g) n+2. Proof. Without loss of generality suppose that d G (v 2 ) = (G). Let X = (x 1,x 2,x,...,x n ) T R n be a positive vector, where = 1, i = 1; 4 7, i = 2; 17 7(), i n. For v 1, we have (n 1)x 1 + v j v 1 x 1 = n+2. For v 2, since d G (v 2 ) n 4, we have d G (v 2 )x 2 + v j v 2 x 2 d G (v 2 )+ 1+ 17(d G(v 2) 1) 7() 4 7 < n+2. Without loss of generality suppose that v 1 is in the outer face of G v 1, C v1 = v 2 v v n 1 v n v 2 is the Hamiltonian cycle in G v 1, v i s (2 i n) are distributed along the clockwise direction on C v1, and suppose N G v1 (v 2 ) = {v 21,v 22,...,v 2t }, where for 1 i t 1, 2 i < 2 i+1, v 21 = v, v 2t = v n. On C v1, along the clockwise direction, for 1 j t, suppose that there are l j 1 vertices between v 2j 1 and v 2j, where if j = 1, we let v 20 = v 2. Along the clockwise direction on C v1, suppose that there are l t vertices between v 2t and v 2. Foreachv 2j (1 j t), notingthatl j 1 +l j d G (v 2 )andd G (v 2 ) n 81, we have and d G (v 2j ) l j 1 +l j +4 n+2 d G (v 2 ) 8 d G (v 2j )x 2j + v k v 2j x k x 2j d G (v 2j )+ 11 7 + 17(dG(v2 j ) 2) 7() 17 7() n+2.

806 Guanglong Yu, Jianyong Wang, and Shu-guang Guo Foreachv f (N G (v 1 )\{v 2,v 21,v 22,...,v 2t }), alongthe clockwisedirection, there exists 0 s t such that v f is between v 2s and v 2s+1 on C v1. Then d G (v f ) l s +2 n d G (v 2 ) 81 and d G (v f )x f + v k v f x k x f d G (v f )+ 11 7 + 17(dG(v f) 2) 7() 17 7() n+2. By the above discussion, we get Q(G)X (n + 2)X. Applying Lemma. completes the proof. Lemma.11. Let G be a maximal planar graph of order n 15 with d G (v 1 ) = (G) = n 1. (i) If (G) =, then q(g) < q(h n ); (ii) If (G) = n, then q(g) < q(h n ). vk 1 v k 2 v 4 v v 2 v k v k+1 v n 1 v n v k v k+1 v k 1 v 4 v v 2 v n v k+2 v n 1 D 1 D 2 vk 1 v k+1 v k 2 v k+2 v 4 v v k v 2 v n v n 1 v k+2 v k+1 v k v k 1 v 4 v v l 2 vl 1 v l v 2 v n v l+1 v n 1 D Fig..6. D 1 D 4. D 4 Proof. Without loss of generality suppose that d G (v 2 ) = (G), v 1 is in the outer face of G v 1, and suppose that C v1 = v 2 v v n 1 v n v 2 is the Hamiltonian cycle in G v 1 (see Fig..6). (i) d G (v 2 ) = and v k / N G (v 2 ) (4 k n 1). Then G v 1 = D1 (see Fig..6). For convenience, we suppose G v 1 = D 1. By Lemma 2., we get that q(g) > 15.

Maxima of the Signless Laplacian Spectral Radius for Planar Graphs 807 Let X = (x 1,x 2,...,x n ) T R n be the Perron eigenvector of Q(G). Note that (.6) (.7) q(g)x k = x k +x k 1 +x k+1 +x 1, q(g)x 2 = ()x 2 +x 1 +. i n,i k Equations (.6) and (.7) imply that (q(g) )(x 2 x k ) = (n 5)x 2 + i k 2 + k+2 i n. Because n 15, it follows immediately that x 2 > x k. Note that q(g)x k 1 = 5x k 1 +x 1 +x 2 +x k 2 +x k +x k+1, Thus, q(g)x k+1 = 5x k+1 +x 1 +x 2 +x k 1 +x k +x k+2. (.8) q(g)(x k 1 +x k+1 ) = 6(x k 1 +x k+1 )+2(x 1 +x k +x 2 )+x k 2 +x k+2. From (.6) and (.7), we also get that (.9) q(g)(x 2 +x k ) = ()x 2 +x k +2x 1 +2x k 1 +2x k+1 + +. i k 2 k+2 i n From (.8) and (.9), we have (.10) (q(g) 4)[x 2 +x k (x k 1 +x k+1 )] = (n 11)x 2 +(x 2 x k )+ i k + k+ i n. The fact that n 15 and (.10) holding imply that x 2 +x k > x k 1 +x k+1. Let F = G v k 1 v k+1 +v 2 v k. Note the relation between the Rayleigh quotient and the largest eigenvalue of a non-negative real symmetric matrix, and note that X T Q(F)X X T Q(G)X = (x 2 +x k ) 2 (x k 1 +x k+1 ) 2. It follows that when n 15, then q(f) > X T Q(F)X > X T Q(G)X = q(g). Because F = H n, it follows that q(h n ) > q(g). Then (i) follows. (ii) d G (v 2 ) = n. Since v 1 is adjacent to v 2, among v 2,v,...,v n 1,v n, there must be two vertices nonadjacent to v 2. Thus, there are three cases for G, that is,

808 Guanglong Yu, Jianyong Wang, and Shu-guang Guo G = D 2, G = D or G = D 4 (see Fig..6). Without loss of generality suppose that in D 2, neither v k nor v k+1 are adjacent to v 2 ; in D, neither v k 1 nor v k+1 are adjacent to v 2 ; in D 4, neither v k nor v l are adjacent to v 2. By Lemma 2., we know that q(g) > 15. Case 1. G = D 2. For convenience, we suppose that G = D 2. Because d G (v 2 ) = n, it follows that 4 k. Let X = (x 1,x 2,...,x n ) T R n be the Perron eigenvector of Q(G). Note that (.11) q(g)x k+1 = x k+1 +x 1 +x k +x k+2, (.12) q(g)x k = 4x k +x 1 +x k 1 +x k+1 +x k+2. From (.11) and (.12), we get (.1) (q(g) 2)(x k x k+1 ) = x k +x k 1. Since n 15 and (.1) holds, we conclude that x k > x k+1. (.14) Note that q(g)x 2 = (n )x 2 +x 1 + +. i k 1 k+2 i n From (.12) and (.14), we get (.15) q(g)(x 2 +x k ) = (n )x 2 +2x 1 +4x k +2x k 1 +x k+1 +2x k+2 + i k 2 + k+ i n. Note that (.16) q(g)x k 1 = 5x k 1 +x 1 +x 2 +x k 2 +x k +x k+2. From (.11) and (.16), we get that (.17) q(g)(x k 1 +x k+1 ) = 5x k 1 +x k 2 +2x 1 +x 2 +2x k +x k+1 +2x k+2. From (.12) and (.17), we get that (.18) (q(g) 2)(x k 1 +x k+1 x k ) = x 1 +x 2 +x k 2 +2x k 1 +x k+2 > 0.

Maxima of the Signless Laplacian Spectral Radius for Planar Graphs 809 Since n 15 and (.18) holds, we conclude that x k 1 +x k+1 > x k. From (.11) and (.14), we get that (.19) (q(g) 4)(x 2 x k+1 ) = (n 7)x 2+x k 1 +x k+1 x k + i k 2 x i+ k+ i n > 0. Since n 15 and (.19) holds, we conclude that x 2 > x k+1. From (.12) and (.14), we get that (.20) (q(g) 4)(x 2 x k ) = (n 8)x 2 +x 2 x k+1 + + > 0. i k 2 k+ i n Since n 15 and (.20) holds, we conclude that x 2 > x k. From (.14) and (.16), we get that (.21) (q(g) 4)(x 2 x k 1 ) = (n 9)x 2 +x 2 x k + +. Since n 15 and (.21) holds, we conclude that x 2 > x k 1. Note that i k k+ i n (.22) q(g)x k+2 = 6x k+2 +x k+1 +x k +x k 1 +x k+ +x 2 +x 1. From (.14) and (.22), we get that (.2) (q(g) 5)(x 2 x k+2 ) = (n 11)x 2+x 2 x k +x 2 x k+1 + x i+. Since n 15 and (.2) holds, we conclude that x 2 > x k+2. From (.16) and (.22), we get that i k 2 k+4 i n (.24) q(g)(x k 1 +x k+2 ) = 2x 1 +2x 2 +x k 2 +6x k 1 +2x k +x k+1 +7x k+2 +x k+. From (.15) and (.24), we get that (.25) q(g)(x 2 +x k ) q(g)(x k 1 +x k+2 ) = (n 14)x 2 +4x 2 4x k 1 +2x k +5x 2 5x k+2 + i k + k+4 i n. Since n 15 and (.25) holds, we conclude that x 2 +x k > x k 1 +x k+2. Let F = G v k 1 v k+2 +v 2 v k. Note that X T Q(F)X X T Q(G)X = (x 2 +x k ) 2 (x k 1 +x k+2 ) 2.

810 Guanglong Yu, Jianyong Wang, and Shu-guang Guo It follows that when n 15, then q(f) > X T Q(F)X > X T Q(G)X = q(g). By (i), it follows immediately that q(h n ) > q(f) > q(g). Case 2. G = D. For convenience, we suppose that G = D. Because d G (v 2 ) = n, it follows that 5 k. Let F = G v k v k+2 +v 2 v k+1. By an argument similartocase1, it canbeprovedthatq(g) < q(f). By(i), wegetthatq(f) < q(h n ). Then q(g) < q(h n ). Case. G = D 4. For convenience, we suppose that G = D 4. Because d G (v 2 ) = n, it follows that 4 k l 2, l n 1. Let F = G v l 1 v l+1 +v 2 v l. By an argument similar to Case 1, it can be proven that q(g) < q(f). By (i), we get that q(f) < q(h n ). Then q(g) < q(h n ). From the above three cases, (ii) follows. Theorem.12. Let G be a planar graph of order n 456. Then q(g) q(h n ) with equality if and only if G = H n. Proof. This theorem follows from the discussions in Section 2, Lemmas.1,.2,.8.11 and Theorem.7. Remark 2. As for the planar graphs of order n 455, perhaps by computations with computer, one can check and find which ones have the maximal signless Laplacian spectral radius. By some computations and comparisons with computer, for the planargraphsofordern 10, we find H n has the maximal signlesslaplacianspectral radius. Based on this, for the planar graphs of order n 455, we conjecture that the graph H n still has the maximal signless Laplacian spectral radius. Acknowledgment. We offer many thanks to the referees for their kind reviews and helpful suggestions. REFERENCES [1] J. Bondy and U. Murty. Graph Theory. Springer, London, 2008. [2] D. Cao and A. Vince. The spectral radius of a planar graph. Linear Algebra Appl., 187:251 257, 199. [] D. Cvetković, P. Rowlinson, and S. Simić. Signless Laplacians of finite graphs, Linear Algebra Appl., 42:155 171, 2007. [4] D. Cvetković and S. Simić. Towards a spectral theory of graphs based on the signless Laplacian, I. Publications De l institut Mathématique Nouvelle série, 85(99):19, 2009. [5] D. Cvetković and S. Simić. Towards a spectral theory of graphs based on the signless Laplacian, II. Linear Algebra Appl., 42:2257 2272, 2010. [6] D. Cvetković and S. Simić. Towards a spectral theory of graphs based on the signless Laplacian, III. Discrete Math., 4:156 166, 2010.

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