Applied Mathematics Letters 0 (007 158 16 wwwelseviercom/locate/aml Spectral radii of graphs with given chromatic number Lihua Feng, Qiao Li, Xiao-Dong Zhang Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 0040, PR China Received 30 September 005; accepted 18 November 005 Abstract We consider the set G n, of graphs of order n with the chromatic number In this note, we prove that in G n, the Turán graph T n, has the maximal spectral radius; and P n if =, C n if = 3andn is odd, Cn 1 1 (l if = 3andn is even, K if 4 has the minimal spectral radius Thus we answer a problem raised by Cao [DS Cao, Index function of graphs, J East China Norm Univ Sci Ed 4 (1987 1 8 (in Chinese MR89m:05084] and Hong [Y Hong, Bounds of eigenvalues of graphs, Discrete Math 13 (1993 65 74] in the affirmative c 006 Elsevier Ltd All rights reserved Keywords: Chromatic number; Spectral radius 1 Introduction In this note, we consider undirected simple connected graphs Let G be a graph with vertex set V (G and edge set E(G The eigenvalues of G are the eigenvaluesof its adjacency matrix A(G The largest eigenvalue of A(G is called the spectral radius of G, denoted by ρ(g In general, the spectral radius ρ(a of a nonnegative matrix A is the largest norm of its eigenvalues We refer the reader to [3] for more details in spectral graph theory The chromatic number χ(g of a graph G is the minimum number of colors such that G can be colored in a way such that no two adjacent vertices have the same color The classical result of Broos []saidthatχ(g Δ(G + 1 with equality if and only if G is an odd cycle or a complete graph Wilf [11] andcao[5] further improved this result by using the spectral radius of a graph Let G n, be the set of connected graphs of order n with chromatic number We define the following two functions on the spectral radius: r(g n, = min{ρ(g G G n, }, R(G n, = max{ρ(g G G n, } A natural question to as is: for the given set G n,, how do we describe the graphs that attain the above two numbers? Supported by the National Natural Science Foundation of China (No 10371075 and the project sponsored by SRF for ROCS, SEM Corresponding author E-mail addresses: lihuafeng@sjtueducn (L Feng, qiaoli@sjtueducn (Q Li, xiaodong@sjtueducn (X-D Zhang 0893-9659/$ - see front matter c 006 Elsevier Ltd All rights reserved doi:101016/jaml00511030
L Feng et al / Applied Mathematics Letters 0 (007 158 16 159 First, Cao [6] gave some partial results on r(g n, and conjectured that r(g n, = ρ(k (l if the chromatic number is large enough and +l = n, where K (l is obtained by joining a path of order l to the complete graph K Further, Hong ([8] problem posed the following problem: Let G be a connected graph with n vertices and chromatic number ; we already now that 1 ρ(g n( 1 What is the best possible lower bound? In this note, we investigate the above two problems In Section, we prove that R(G n, = ρ(t n,,wheret n, is the Turán graph with parts whose partition sets differ in size by at most one We also determine r(g n, completely in Section 3 To determine R(G n, Let G be a graph of order n with χ(g = From the definition, G has color classes and each is an independent set Suppose the classes have orders n 1 n n Since the addition of edges will increase the spectral radius, the graphs which achieve R(G n, must be complete -partite graphs Next we prove that of all the complete -partite graphs, the Turán graph T n, has maximal spectral radius Theorem 1 Let G be a complete -partite graph of order n The numbers of the vertices of the parts are ordered as n 1 n n Thenρ(G ρ(t n, with equality if and only if G = T n, Proof The characteristic polynomial of G (see [3, p 74] is λ (1 n n i (λ + n i λ + n i=1 i i=1 Obviously, its largest root is the unique positive solution of 1 i=1 n i λ + n i = 0 We claim that if n 1 n, then ρ(g <ρ(t n, Consider the following equation: f (δ, λ = 1 n 1 1 δ λ + n 1 δ i= n i λ + n i n + δ λ + n + δ, where 0 δ n 1 n is an integer So f (0,ρ(G = 0 Taingthe derivativewith respecttoδ, wehave, forλ>0, d f (δ, λ dδ = λ (λ + n 1 δ λ (λ + n + δ < 0 Hence f (δ, λ is decreasing with respect to δ for λ>0 Thus f (1,ρ(G < 0 This means that if we increase n by one and decrease n 1 by one in G, the spectral radius will increase So we complete the proof Hence, we may present the main result in this section Theorem T n, is the only graph that attains R(G n, Corollary 3 Let T n, be a Turán graph with α(0 α< parts of size d + 1 and α parts of size d, Then ρ(t n, = 1 ( n d 1 + (n + 1 4α(d + 1 n d with the last equality if and only if T n, is regular Proof From Eq (* in Theorem 1, ρ(t n, satisfies α(d + 1 d( α + = 1 λ + d + 1 λ + d Note that n = d + α, wehave λ (n d 1λ (n d α(d + 1 = 0, (*
160 L Feng et al / Applied Mathematics Letters 0 (007 158 16 Fig 1 The tree W n (n 7 which implies the result The last equality holds if and only if α = 0sinceρ(T n, is a strictly decreasing function with respect to α By Theorem 1 and Corollary 3, we get the main result in [7], and the equality case is given as well Corollary 4 Let G be a connected graph of order n and chromatic number Then ρ(g n(1 1 with equality if and only if G is a regular Turán graph 3 To determine r(g n, We consider the graph G uv obtained from the connected graph G by subdividing the edge uv, that is, by replacing uv with edges uw, vw where w is an additional vertex We call the following two types of paths internal paths: (a a sequence of vertices v 0,v 1,,v +1 ( where v 0,v 1,,v are distinct, v +1 = v 0 of degree at least 3, d vi = fori = 1,,, andv i 1 and v i (i = 1,, + 1 are adjacent (b A sequence of distinct vertices v 0,v 1,,v +1 ( 0 such that v i 1 and v i (i = 1,, + 1 are adjacent, d v0 3, d v+1 3andd vi = whenever 1 i Lemma 31 ([4] If uv lies on an internal path of the connected graph G and G W n (see Fig 1, then ρ(g uv <ρ(g Lemma 3 ([10] Suppose G is a nontrivial simple connected graph Let u,vbe two vertices at distance not greater than one, ie, u,v are adjacent or identical For nonnegative integers, l, let G(, l denote the graph obtained from G by adding pendant paths of length and l at u,v, respectively If l 1,then ρ(g(, l > ρ(g( + 1, l 1 AgraphG is said to be -critical if χ(g = and χ(g u = 1foreveryu V (G It is well nown that (see [1] every -chromatic graph contains a -critical subgraph Lemma 33 ([9] Suppose χ(g = 4 Let G be a -critical graph on more than vertices (so G = K Then ( 1 3 E(G + ( V (G 1 Lemma 34 ([6] The spectral radius of K (l, 4, satisfies ( 1 ρ(k (l <1 3 + ( + 1 + 4 Proof We present a proof here for completeness Since K (l contains K as a subgraph, the left hand side of the above inequality is obvious In the following, we denote the characteristic polynomial of P l, the path on l vertices, by f l (λ Then P(K (l,λ = (λ + 1(λ + 1 1 f l (λ (λ + (λ + 1 f l 1 (λ = (λ + 1 [(λ + 1(λ + 1 f l (λ (λ + f l 1 (λ] = (λ + 1 [λ(λ + f l (λ ( 1 f l (λ (λ + f l 1 (λ] = (λ + 1 [(λ + (λf l (λ f l 1 (λ ( 1 f l (λ] = (λ + 1 [(λ + f l+1 (λ ( 1 f l (λ]
L Feng et al / Applied Mathematics Letters 0 (007 158 16 161 Since f l+1 (λ = λf l (λ f l 1 (λ, by the induction hypothesis, the sequence f l+1(λ f l (λ f l+1 (λ f l (λ Hence > λ + λ 4 is strictly decreasing and P(K (l,λ >(λ+ 1 f l (λg(λ, where g(λ = (λ + λ+ λ 4 + 1 Obviously, g(λ is an increasing function with respect to λ Solving the equation g(λ = 0, we have that its largest root is λ 0 = 1 ( 3 + If λ>λ 0, P(K (l ( + 1 + 4 > 3 (l,λ >0andρ(K <λ 0 We complete the proof We now can present the main result in this section Theorem 35 (1 If =,thenp n is the only graph in G n, that attains r(g n, (1 If = 3 and n is odd, then C n is the only graph in G n, that attains r(g n, ( If = 3 and n is even, then Cn 1 1 is the only graph in G n, that attains r(g n,,wherecn 1 1 is obtained from the cycle C n 1 by adding one pendent vertex (3 If 4,thenK (l is the only graph in G n, that attains r(g n, Proof Let G be a connected graph of order n with chromatic number (1 It is easy to see that (1 holds (for example, [3, p 78] ( If = 3, then G must contain an odd cycle C h as a 3-critical subgraph for odd integer h ByLemma 31, we have ρ(c 1 3 >ρ(c1 4 > >ρ(c1 n 1 If n is odd and h = n, the result holds If n is odd and h < n,theng contains Ch 1 as a subgraph since G is connected So we have ρ(g ρ(c 1 h ρ(c1 n 1 >ρ(c1 n >ρ(c n If n is even and h < n, by a similar argument, we have ρ(g ρ(c 1 h ρ(c1 n 1 (3 If 4, we discuss the following two cases (a If G does not contain K as a subgraph, we assume that G contains a -critical subgraph H ;thenρ(g ρ(h Since ρ(h E(H V (H, and in view of Lemma 33,wehave ρ(g ρ(h 1 + It is easy to see that 1 + 3 1 > 1 3 1 ( 3 + ( + 1 + 4 Hence Lemma 34 implies ρ(g >ρ(k (l (b If G contains K as a subgraph, then by repeated use of Lemma 3, we can get the result directly Combining the above two cases, we complete the proof of (3
16 L Feng et al / Applied Mathematics Letters 0 (007 158 16 Acnowledgments The authors are grateful to the referees for their valuable comments, corrections and suggestions which led to improvement of this note References [1] B Bollobás, Extremal Graph Theory, Academic Press, London, 1978 (Chapter 5 [] RL Broos, On coloring the node of a networ, Proc Camb Phil Soc 37 (1941 194 197 [3] D Cvetović, M Doob, H Sachs, Spectra of Graphs, Academic Press, New Yor, 1980 [4] D Cvetović, P Rowlinson, SK Simic, Eigenspaces of Graphs, Cambridge University Press, 1997 [5] DS Cao, Bounds on eigenvalues and chromatic number, Linear Algebra Appl 70 (1998 1 13 [6] DS Cao, Index function of graphs, J East China Norm Univ Sci Ed 4 (1987 1 8 (in Chinese MR89m:05084 [7] C Edwards, C Elphic, Lower bounds for the clique and the chromatic number of graph, Discrete Appl Math 5 (1983 51 64 [8] Y Hong, Bounds of eigenvalues of graphs, Discrete Math 13 (1993 65 74 [9] M Krivelevich, An improved upper bound on the minimal number of edges in color-critical graphs, Electr J Combin 1 (1998 R4 [10] Q Li, KQ Feng, On the largest eigenvalues of graphs, Acta Math Appl Sin (1979 167 175 (in Chinese MR80:05079 [11] HS Wilf, The eigenvalues of a graph and its chromatic number, J London Math Soc 4 (1967 330 33