Sharp lower bounds of the least eigenvalue of planar graphs

Linear Algebra and its Applications 296 (1999) 227±232 www.elsevier.com/locate/laa

Sharp lower bounds of the least eigenvalue of planar graphs q Yuan Hong *, Jin-Long Shu

1

Department of Mathematics, East China Normal University, Shanghai 200062, People's Republic of China Received 30 December 1997; accepted 3 June 1999 Submitted by R.A. Brualdi

Abstract Let G be a simple graph with n P 3 vertices and orientable genus g and non-orientable genus h. We de®ne the Euler characteristic v…G† of a graph G by v…G† ˆ maxf2 ÿ 2g; 2 ÿ hg. Let k…G† be the least eigenvalue of the adjacency matrix A of G. In this paper, we obtain the following lower bounds of k…G† p k…G† P ÿ 2…n ÿ v…G††: In particular, if G is the planar graph, then p k…G† P ÿ 2n ÿ 4 the equality holds if and only if G  K2;nÿ2 . Further, we have same result of series± parallel graph. Ó 1999 Elsevier Science Inc. All rights reserved. AMS classi®cation: 05C50; 05C10 Keywords: Planar graph; Least eigenvalue; Genus

q

Research supported by National Natural Science Foundation of China No. 19671029. Corresponding author. E-mail: [email protected] 1 E-mail: [email protected] *

0024-3795/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 9 9 ) 0 0 1 2 9 - 9

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1. Introduction In this paper, all graphs are ®nite undirected graphs without loops and multiple edges. Let G be a graph with n vertices and let the eigenvalues of G be the eigenvalue of the adjacency matrix A of G. Since A is a real symmetric matrix, its eigenvalues must be real and ordered as k1 …G† P k2 …G† P


P kn …G†: In particular, we denote the largest eigenvalue, k1 …G† by q…G†, and the least eigenvalue kn …G† by k…G†. If G is bipartite graph, then ki …G† ˆ ÿknÿi‡1 …G†, i ˆ 1; 2; . . . ; ‰n=2Š. A surface is de®ned as a connected compact topological space which is locally homeomorphic to 2-dimensional Euclidean space R2 . Every surface is topologically equivalent either to Sg obtained from the sphere S0 by adding g ``handles'', for some g P 0 or to Nh obtained from the sphere S0 by adding h ``crosscaps'' for some h > 0. The surfaces appearing in this paper are closed surfaces. A curve in a surface S is the image of a continuous 1-1 map f : ‰0; 1Š ! S. We say that a graph G can be embedded in surface S if G can be represented in S such that the vertices of G are distinct points in S, and each edge uv in G is a curve in S joining the points corresponding to u and v. Moreover, two edges in S do not intersect except possibly at an end. We shall not distinguish the graph and embedding. If a graph G is embedded in a surface S, then the closures of the connected components of S ÿ G are called the faces of G. If each pair of faces meet either on a vertex or on an edge,then we say those faces meet properly. If each face of G is a closed 2-cell and the faces of G all meet properly (allow the possibility that faces do not meet), then we say that G is a polyhedral map. De®ne the orientable genus of G, written as g…G†, to be the least integer t such that G is embedded in St . Similarly de®ne the non-orientable genus of G, written as h…G†, to be the least integer t such that G is embedded in Nt . If G is planar, then we set h…G† ˆ 0. An embedding of G in a surface of genus g…G† is called a minimal embedding. A minimal embedding of a connected graph is a 2-cell embedding. The genus of a graph is a rather important parameter. We say that a graph G is an orientable surface Sg graph if the orientable genus of G is g. Similarly, we say that a graph G is a non-orientable surface Nh graph if the non-orientable genus of G is h. In particular, the projective planar graphs (the planar graph, the torus graphs, the Klein bottle graphs, respectively) have non-orientable genus 1 (orientable genus is 0, orientable genus is 1, non-orientable genus is 2, respectively). We have known that a graph is planar if and only if it contains no subdivision of K5 or K3;3 [1], and a graph is series±parallel graph if and only if it

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contains no subdivision of K4 . So if a graph is series±parallel graph then it must be planar and its orientable genus is 0. Now, we de®ne the Euler characteristic v…S† of a surface S by v…S† ˆ 2 ÿ 2g for the surface S with orientable genus g; v…S† ˆ 2 ÿ h for the surface S with non-orientable genus h: In 1978, Schwenk and Wilson [8] raised a question of what can be said about the eigenvalues of a planar graph. In the past 20 years, this problem has attracted considerable interest and there are some results (see [2,4±6]) In particular, Hong and Jinsong Shi [7] further gave the sharp upper bounds of spectral radius of series±parallel graph with n vertices is not greater than p 1=2 ‡ 2n ÿ 15=4. In this paper, the sharp lower bound of the least eigenvalue k…G† of a planar graph G with n P 3 vertices is given by p k…G† P ÿ 2n ÿ 4 the equality holds if and only if G  K2;nÿ2 , where K2;nÿ2 is complete bipartite graph. However, it is surprise that there are same result for series±parallel graph. In generally, let G be a simple graph with n P 3 vertices and orientable genus g and non-orientable genus h. We de®ne the Euler characteristic of G by v…G† ˆ maxf2 ÿ 2g; 2 ÿ hg. Then the lower bounds of the least eigenvalue k…G† of G as following p k…G† P ÿ 2…n ÿ v…G††: The terminology not de®ned here can be found in [1,9].

2. Lemmas and results Lemma 1 [9]. If G is a connected graph with n P 3 vertices, m edges and girth l P 3, and if G is embeddable in a surface S then m6

l …n ÿ v…S††: lÿ2

Some special cases of this lemma, stated in the next two propositions, are of particular interest. Proposition 1. If G is a connected K3 -free graph with n P 3 vertices, m edges and Euler characteristic v…G†. Then m 6 2…n ÿ v…G††:

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Proposition 2. If G is a connected bipartite graph with n P 3 vertices, m edges and Euler characteristic v…G†. Then m 6 2…n ÿ v…G††: Lemma 2 [3]. If G is a simple connected graph with n P 3 vertices, then there exists a connected bipartite subgraph H of G such that k…G† P k…H † with equality if and only if G  H . Proof. Let X ˆ …x1 ; x2 ; . . . ; xn †t be a unitary eigenvector corresponding to k…G†. Let H 0 be the graph obtained from G by deleting all the edges vi vj such that xi xj P 0. Clearly, H 0 is bipartite with bipartition …X ; Y †. Where X ˆ fvi j xi > 0g and Y ˆ fvj j xj < 0g. Let A…H 0 † ˆ aij …H 0 † be the adjacency matrix of H 0 . Then k…G† ˆ kn …G† ˆ

n X n X iˆ1

ˆ

n X n X iˆ1

jˆ1

jˆ1

X

aij xi xj P

aij xi xj

xi xj <0

aij …H 0 †xi xj P min

jjX jjˆ1

n X n X iˆ1

aij …H 0 †xi xj

jˆ1

ˆ k…H 0 †: If H 0 is connected, then let H  H 0 . Otherwise, there exists a component H of H 0 such that k…H † ˆ k…H 0 †.  Lemma 3. If G is a connected bipartite graph with n vertices and m edges, then p k…G† P ÿ m with equality if and only if G is a complete bipartite graph. Proof. This P inequality follows from the fact that k1 …G† ˆ q…G† ˆ ÿkn …G† ˆ n ÿk…G† and iˆ1 k2i ˆ 2m. If the equality holds if and only if k1 …G† ˆ ÿk…G†, and k2 …G† ˆ


ˆ knÿ1 …G† ˆ 0. This implies that i€ G is a complete bipartite graph since G is connected.  The cardinality of the vertex set of a graph G is denoted by m…G† and the cardinality of the edge set of G is denoted by e…G†. Theorem 1. If G is a graph with n P 3 vertices and m edges, then p k…G† P ÿ 2…n ÿ v…G††:

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Proof. By Lemma 2, we have a connected bipartite subgraph H of G such that k…G† P k…H †: Then, by Proposition 2 and Lemma 3, we get p p p k…H † P ÿ e…H † P ÿ 2…m…H † ÿ v…H †† P ÿ 2…n ÿ v…G††: Hence k…G† P ÿ

p 2…n ÿ v…G††:

This completes the proof.



Theorem 2. If G is a planar graph with n P 3 vertices and m edges, then p k…G† P ÿ 2n ÿ 4 the equality holds if and only if G  K2;nÿ2 . Proof. For any planar graph G, v…G† ˆ 2, using Theorem 1, we have p k…G† P ÿ 2n ÿ 4: p If k…G† ˆ ÿ 2n ÿ 4, by Lemma 2, Proposition 2 and Lemma 3, we get a connected bipartite subgraph H of G such that p p ÿ 2n ÿ 4 ˆ k…G† P k…H † P ÿ e…H † p p P ÿ 2…m…H † ÿ 2† P ÿ 2n ÿ 4: So k…G† ˆ k…H †; e…H † ˆ 2n ÿ 4. Since H is a connected bipartite graph, by Lemma 3, this implies that H is a complete bipartite graph, and G  H  K2;nÿ2 . p Conversly, it is easy to show that k…K2;nÿ2 † ˆ ÿ 2n ÿ 4 and K2;nÿ2 is the planar graph. This completes the proof.  Using Theorem 1, we have the following corollaries. Corollary 1. If G is a series±parallel graph with n P 3 vertices, then p k…G† P ÿ 2n ÿ 4 with equality if and only if G  K2;nÿ2 . Proof. This follows from the fact that graph G is planar, and complete bipartite graph K2;nÿ1 is series±parallel too. 

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Corollary 2. If G is a projective planar graph with n P 3 vertices, then p k…G† P ÿ 2n ÿ 2: Corollary 3. If G is a toroidal graph with n P 3 vertices, then p k…G† P ÿ 2n: Corollary 4. If G is a Klein bottle graph with n P 3 vertices, then p k…G† P ÿ 2n:

Acknowledgements The authors are grateful to the referees for many helpful suggestions which led to an improved version of this paper. References [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan, New York, 1976. [2] D. Cao, A. Vince, The spectral radius of a planar graph, Linear Algebra Appl. 187 (1993) 251± 257. [3] D. Cao, Y. Hong, The distribution of eigenvalues of graphs, Linear Algebra Appl. 216 (1995) 211±224. [4] Y. Hong, A bound on the spectral radius of graphs, Linear Algebra Appl. 108 (1988) 135±139. [5] Y. Hong, On the spectral radius and the genus of graphs, J. Combin. Theory Ser. B 65 (2) (1995) 262±268. [6] Y. Hong, Upper bounds of the spectral radius of graphs in terms of genus, J. Combin. Theory Ser. B 74 (2) (1998) 153±159. [7] Y. Hong, J. Shi, On the spectral radius of graphs without K4 -minors, submitted. [8] A.J. Schwenk, R.J. Wilson, in: L.W. Beineke, R.J. Wilson, (Eds.), On the Eigenvalues of a Graph, Selected Topics in Graph Theory, Academic Press, New York, 1978, pp. 307±336. [9] A.T. White, L.W. Beineke, in: L.W. Beineke, R.J. Wilson, (Eds.), Topological Graph Theory, Selected Topics in Graph Theory, Academic Press, New York, 1978, pp. 15±50.