SOME SHARP UPPER BOUNDS ON THE SPECTRAL RADIUS OF GRAPHS. Lihua Feng, Qiao Li and XiaoDong Zhang 1. INTRODUCTION


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1 TAIWANESE JOURNAL OF MATHEMATICS Vol. 11, No. 4, pp , September 007 This paper is available online at SOME SHARP UPPER BOUNDS ON THE SPECTRAL RADIUS OF GRAPHS Lihua Feng, Qiao Li and XiaoDong Zhang Abstract. In this paper, we first give a relation between the adjacency spectral radius and the Qspectral radius of a graph. Then using this result, we further give some new sharp upper bounds on the adjacency spectral radius of a graph in terms of degrees and the average degrees of vertices. Some known results are also obtained. 1. INTRODUCTION Let G =(V,E) be a simple graph with vertex set V and edge set E. For v V, the degree of v is denoted by d v, the average degree of v, denoted by m v, equals to ( uv E(G) d u)/d v. Then d v m v is the degree of v. For two vertex u and v, we use u v to mean that these two vertices are adjacent. Let A(G) be the adjacency matrix of G and D(G) be the diagonal degree matrix of G, respectively. We call the matrix L(G)=D(G) A(G) the Laplacian matrix of G, while call the matrix Q(G) =D(G) +A(G) the Qmatrix of G. We denote the largest eigenvalues of A(G), L(G) and Q(G) by ρ(g), λ(g) and µ(g), respectively, and call them the adjacency spectral radius, the Laplacian spectral radius, the Qspectral radius of G, respectively. If X is a real symmetric matrix, its eigenvalues must be real, and the largest one is denoted in this paper by ρ(x). It follows immediately that if G is a simple graph, then A(G), L(G) and Q(G) are symmetric matrices, moreover A(G) and Q(G) are irreducible nonnegative matrices. Let K = K(G) be a vertexedge incidence matrix of G. Thus Q(G) = D(G)+ A(G) =KK t and K t K =I m + A(L G ), where L G is the line graph of G. Since KK t and K t K have the same nonzero eigenvalues, we can get that µ(g) =+ρ(l G ). Received June 6, 005, accepted October 4, 005. Communicated by Xuding Zhu. 000 Mathematics Subject Classification: 05C50. Key words and phrases: Adjacency matrix, Qmatrix, Line graph, Spectral radius. Supported by National Natural Science Foundation of China (No and No ). 989
2 990 Lihua Feng, Qiao Li and XiaoDong Zhang For other terminologies, we follow [1] and [6]. Recently, the following results on the spectral radius were obtained: (i) [10] Let G be a simple connected graph with n vertices, the maximum degree and the second largest degree, here. If there are p vertices with degree, then (1) ρ(g) 1+ ( +1) +4p( ). The equality holds if and only if G is regular or G = K p H, the join of the two graphs, H is a ( p) regular graph with n p vertices. (ii) [] For any connected graph, we have () ρ(g) m δ(n 1) + (δ 1). The equality holds if and only if G is regular or a star plus copies of K,or a complete graph plus a regular graph with smaller degree of vertices. (iii) [] For any connected graph, we have (3) ρ(g) max du m u. The equality holds if and only if G is a regular graph or a semiregular bipartite graph. Recently, Shu et al. got the following relation between λ(g) and µ(g). Lemma 1.1. [9] For a connected graph G, we have λ(g) µ(g), with equality if and only if G is bipartite. This paper is organized as follows. In Section, we present a relation between ρ(g) and µ(g). In Section 3, we give some sharp upper bounds on the adjacency spectral radius by using the results included in Section.. SOME LEMMAS Lemma.1. [1] Let G be a simple connected graph and L G be the line graph of G, then the spectral radius of L G satisfies ( ) ρ(l G ) max u v du (d u + m u )+d v (d v + m v ),
3 Some Sharp Upper Bounds on the Spectral Radius of Graphs 991 the equality holds if and only if G is regular or semiregular. Lemma.. Let G be a connected graph, then ρ(g) 1 µ(g), and the equality holds if and only if G is regular. Proof. Let X =(x 1,x,..., x n ) t R n be a unit eigenvector that belongs to ρ(g), X =(x 1,x,..., x n) R n be a unit eigenvector that belongs to µ(g). First, we will show that ρ(g) 1 µ(g). Since µ(g) =X t (D + A)X = (x i + x j), while ρ(g) =X t AX = x i x j 1 1 (x i + x j ) (x i + x j) = 1 µ(g). So we get the desired inequality. If the equality holds, all the inequalities above must be equalities. So we have ρ(g) = x i x j = 1 (x i + x j ). Since x i x j x i + x j, we must have x i = x j when v i and v j are adjacent. Since G is connected, we get that X is the multiple of all one vector. From AX = ρx, we get that G must be regular. Conversely, it is easy to check that when G is regular, the equality holds. Remark 1. Since µ(g) =ρ(d + A) =+ρ(l G ), where L G is the line graph of G. So this Lemma gives a relation between the spectral radius of a graph and its line graph, namely, ρ(g) 1 ρ(l G)+1, the equality holds if and only if G is a regular graph.
4 99 Lihua Feng, Qiao Li and XiaoDong Zhang Corollary.3. If G is a bipartite graph with degree sequence d 1,d,,d n. Then 1 λ(g) d n i, where the sum is taken over all the 1 i n. Moreover, the equality holds if and only if G is regular bipartite. 1 Proof. From [5] or [11], we know ρ(g) n d i, and the equality holds if and only if G is a regular graph or a semiregular bipartite graph. Together with Lemma 1.1 and Lemma., we get the result. Lemma.4. polynomial, then [7] Let G be a graph of order n and Q = Q(G), let P be any min v V (G) s v (P (Q)) P (µ) max v V (G) s v (P (Q)), where s v (X) denotes the vth row sum of a matrix X. The equality holds if and only if the row sums of Q are equal. Lemma.5. Let G be a simple graph and be the maximum degree, then µ(g) + +8T, where T = max{d u m u u V } is the maximum degree. Moreover, the equality holds if and only if G is regular. Proof. Since Q = D + A, by a simple calculation, we have s v (Q) =d v and s v (AD) =s v (A )= u v d u = d v m v. Then So we have By Lemma.4, we have and then s v (Q )=s v (D + DA + AD + A ) = s v (D(D + A)) + s v (AD)+s v (A ) = d v s v (Q)+d v m v s v (Q)+d v m v s v (Q Q) max v V (G) d v m v =T. µ µ T 0, µ + +8T.
5 Some Sharp Upper Bounds on the Spectral Radius of Graphs 993 In order to get the equality above, all inequalities in the above should be equalities. That is d v = and d v m v =T hold for any vertex v. So by Lemma.4, G is regular. Conversely, when G is regular, it is easy to check that the equality holds. Remark. From the proof of this Lemma, we can get easily that µ(g) max (G) d u +d u m u, and this is one of the main results in [8]. Corollary.6. Let G be a simple connected graph, then λ(g) 1 ( + +8 ( m (n 1)δ +(δ 1) ) ). Equality holds if and only if G is regular bipartite. Proof. Since d u m u m (n 1)δ +(δ 1), together with Lemma 1.1 and Lemma.5, we can get the result. Lemma.7. Let G be a simple connected graph, then µ(g) max (G) ( d u + d u m u ), and the equality holds if and only if G is regular or semiregular bipartite. Proof. Let x =(x v,v V ) be a unit vector such that µx = Qx, then for any u V, µx u = d u x u + v u x v = v u (x v + x u ). By the CauchySchwarz inequality, we have µ x u d u (x u + x v ) Hence v u = d ux u +d u x u(µ d u )+d u x v. v u µ x u = (d u µ d u )x u + d u v u = (d u µ d u)x u + x v d u m u x u. Thus we have (µ d u µ + d u d um u )x u 0. So there exists one vertex u such that µ d u µ + d u d u m u 0,
6 994 Lihua Feng, Qiao Li and XiaoDong Zhang which implies our result. If the equality holds, then we have for any u V, uv, uw E, x u + x v = x u + x w, that is, for any u V, the neighbors of u have equal eigencomponents. While from µx u = d u x u + v u x v, we have So d v = d w and for any uv E, (µ d u )x u = d u x v, (µ d v )x v = d v x u, (µ d w )x w = d w x u. (a) µ = d u + d v. Since G is connected, there are at most two different degrees in G. If the subgraph G[N(u)] induced by N(u) has at least one edge, then from (a) we have d u = d v and G must be regular. If G[N(u)] contains no edge, then G must be semiregular bipartite. So we get the desired result. Conversely, it is easy to check that when G is either regular or semiregular bipartite, the result holds. ( Corollary.8. Let G be a simple connected graph, then λ max (G) du + ) du m u. Equality holds if and only if G is regular bipartite or semiregular bipartite. 3. MAIN RESULTS AND EXAMPLES In this section, we establish some upper bounds for the adjacency spectral radius of a graph G by the matrix Q(G) and its line graph L G. Finally, we give two examples to illustrate that our results are sometimes better than the known ones in Section 1. Theorem 3.1. Let G be a simple connected graph, then d (4) ρ(g) max u + d u m u (G). The equality holds if and only if G is a regular graph. Proof. From the proof of Lemma.5, we have s v (Q )=s v (D +DA+AD + A )=d v +d vm v, from Lemma.4, we have µ(g) max (G) d u +d u m u, and by Lemma., we get the result.
7 Some Sharp Upper Bounds on the Spectral Radius of Graphs 995 Theorem 3.. Let G be a simple connected graph, then du (d u + m u )+d v (d v + m v ) (5) ρ(g) max u v, the equality holds if and only if G is a regular graph. Proof. By Lemmas.1 and. and Remark 1, we can get the result. Theorem 3.3. Let G be a simple connected graph, then (6) ρ(g) + +8T, 4 where T = max{d u m u u V } is the maximum degree in V. Moreover, the equality holds if and only if G is regular. Proof. By Lemmas. and.5, we can get the desired result. Theorem 3.4. Let G be a simple connected graph, then 1 (7) ρ(g) max (G) (d u + ) d u m u. The equality holds if and only if G is a regular graph. Proof. From Lemma. and Lemma.7, we can get the desired result. Remark 3. It is easy to check that (4), (7) and (3) are incomparable, since d u and m u are incomparable in general. While a simple computation shows that (7) is better than (4). Theorem 3.5. Let G be a simple connected graph, then { 1 (8) ρ(g) max u + d v )+ 4((d ) } (d u d v ) +4m u m v : uv E(G). The equality holds if and only if G is a regular graph. Proof. Consider D 1 QD, by modifying the proof of Theorem.14 in [4] and a similar argument in Lemma.7 for the case when the equality holds, we can get { 1 µ(g) max u + d v )+ ((d ) } (d u d v ) +4m u m v : uv E(G), equality holds if and only if G is regular or semiregular bipartite. From Lemma., we can get the result.
8 996 Lihua Feng, Qiao Li and XiaoDong Zhang Remark 4. It is easy to check that 1 ((d u + d v )+ ) (d u d v ) +4m u m v d u + d v +m u m v d u + d v + m u m v + d u d v. d u + d v So by Lemma., we can get that ρ(g) 1 ((d u + d v )+ ) (d u d v ) 4 +4m u m v 1 ( ) d u + d v +m um v 1 ( d u + d v + m u m v + d u d ) v. d u + d v Even for these two new bounds, they are incomparable with (4), (5), (7). At last, we give two examples to illustrate the results in this section are sometimes better than those cited in Section 1. Let G 1 and G be two graphs as shown in the following figures. G 1 G We summarize all the upper bounds for the spectral radius of graphs considered in this paper as follows: ρ (1) () (3) (4) (5) (6) (7) (8) G G The table above shows in a general sense, these bounds are incomparable except for (4) and (7). (8) is the best of all. ACKNOWLEDGMENT The authors are grateful to the referees for their valuable comments, corrections and suggestions which lead to an improvement of this paper.
9 Some Sharp Upper Bounds on the Spectral Radius of Graphs 997 REFERENCES 1. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, New York, Macmilan Press, D. S. Cao, Bounds on the eigenvalues and chromatic number, Linear Algebra Appl., 70 (1998), K. ch. Das, An improved upper bound for the Laplacian graph eigenvalues, Linear Algebra Appl., 368 (003), K. ch. Das, A characterization on graphs which achieved the upper bound for the largest Laplacian eigenvalue of graphs, Linear Algebra Appl., 376 (004), M. Hofmeister, Spectral radius and degree sequence, Math. Nachr., 139 (1988), R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, H. Q. Liu, M. Lu and F. Tian, On the Laplacian spectral radius of a graph, Linear Algebra Appl., 376 (004), J. S. Li and Y. L. Pan, de Caen s inequality and bound on the largest Laplacian eigenvalue of a graph, Linear Algebra Appl., 38 (001), J. L. Shu, Y. Hong and K. WenRen, A sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph, Linear Algebra Appl., 347 (00), J. L. Shu and Y. R. Wu, Sharp upper bounds on the spectral radius of graphs, Linear Algebra Appl., 377 (004), A. M. Yu, M. Lu and F. Tian, On the spectral radius of graphs, Linear Algebra Appl., 387 (004), X. D. Zhang, Two sharp upper bounds for the Lapalacian eigenvalues, Linear Algebra Appl.,36 (003), Lihua Feng School of Mathematics, Shandong Institute of Business and Technology, Yantai, Shndong 64005, P. R. China Qiao Li and XiaoDong Zhang Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 0040, P. R. China
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