On the spectral radius, kdegree and the upper bound of energy in a graph


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1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 57 (007) ISSN On the spectral radius, kdegree and the upper bound of energy in a graph Yaoping Hou a, Zhen Tang a and Chingwah Woo b a Department of Mathematics, Hunan Normal University Changsha, Hunan , China b Department of Mathematics, City University of Hong Kong Hong Kong, China (Received November 8, 006) Abstract: Let G be a simple graph. For v V (G), kdegree d k of v is the number of walks of length k of G starting at v. In this paper a lower bounds of the spectral radius of G in terms of the kdegree of vertices is presented and the upper bounds of energy of a connected graph is obtained. 1 Introduction Let G = (V, E) be a finite graph. For v V (G), the degree of v written by d, is the number of edges incident with v. The number of walks of length k of G starting at v is denoted by d k, is also called kdegree of the vertex v and d k is called average kdegree d
2 of the vertex v. Clearly, one has d 0 = 1, d 1 = d, and d k+1 = where N is the set of all neighbors of the vertex v. w N d k (w), In view of the wellknown fact that with j denote the all one vector defined on the set V (G), the vector D k = (d k ) coincides with A k j, where A = A(G) is the adjacency matrix of the graph G. In [3], Dress and Gutman showed an interesting inequality on the number of walks in a graph, and the discussion of the equality holding resulted classification graphs in exactly five classes. A graph G is called regular graph if there exists a constant r such that d = r holds for every v V (G), in which case G is also called rregular. Obviously, this is equivalent with the assertion that Aj = rj holds. Further, a graph G is called a, bsemiregular if {d, d(w)} = {a, b} holds for all edges vw E(G). Clearly, this implies A j = abj. A semiregular graph that is not regular will henceforth be called strictly semiregular. Clearly, a connected strictly semiregular graph must be bipartite. A graph G is called harmonic [3](pseudoregular [13]) if there exists a constants µ such that d = µd holds for every v V (G) in which case G is also called µharmonic, clearly, a graph G is µharmonic if and only if A j = µaj holds.( In other words, A j and Aj are linear dependent.) A graph G is called semiharmonic [3] if there exists a constants µ such that d 3 = µd holds for every v V (G) in which case G is also called µsemiharmonic, clearly, a graph G is µsemiharmonic if and only if A 3 j = µaj holds. (In other words, A 3 j and Aj are linear dependent.) Thus every µharmonic graph is µ semiharmonic. Also every a, bsemiregular graph is absemiharmonic. A semiharmonic graph that is not harmonic will henceforth be called strictly semiharmonic. Finally, a graph G is called (a, b)pseudosemiregular if { d d, d (w) } = {a, b} holds d(w) for all edges vw E(G). A pseudosemiregular graph that is not pseudoregular will henceforth be called strictly pseudosemiregular. Clearly, a connected strictly pseudosemiregular graph must be bipartite. In this paper, we present a lower bound for the spectral radius of a connected graph and give a upper bound for the energy of a graph by using this new lower bound of spectral radius.
3 The spectral radius a graph and kdegree Let G be a connected graph. The spectral radius (largest eigenvalue of A(G)) of G is denoted by ρ(g). In [4], the following theorem was given: Theorem 1 Let G be a connected graph with degree sequence d 1, d,..., d n. Then ρ(g) d, (1) n with equality if and only if G is regular or a semiregular. In [13], the following theorem was given: Theorem Let G be a connected graph. Then ρ(g) d d, () with equality if and only if G is a pseudoregular graph or a strictly pseudosemiregular graph. In this section we prove a more general result which generalizes and unifies the above two results. Lemma 3 Let G be a connected graph and k 0. Then ρ(g) d k+1, (3) d k with equality if and only if A k+ (G)j = ρ (G)A k (G)j. Proof. Since Dk T A D k = A k+1 j, A k+1 j = D k+1, D k+1 = d k+1, thus, by RayleighRitz Theorem, ρ(g) = ρ(a ) DT k A D k D k, D k d k+1, and equality d k holds if and only if D k is a eigenvector of A (G) corresponding to the eigenvalue ρ(g), that is, if and only if A k+ (G)j = ρ (G)A k (G)j. For k = 0, the above lemma becomes Theorem 1, this is because that A (G)j = µj if and only if G is regular or semiregular (see [3]). By Theorem 1 of [3], every graph G for
4 which some integers k, l with 0 l < k and some constant µ such that d k = µd l for all v V (G) (equivalently, A k j = µa l j ) is semiharmonic, and even harmonic in case k l is odd.) Therefore, for k 1, ρ(g) d k+1 d k and the equality holds if and only if G is a semiharmonic graph. In the next lemma we prove that strictly semiharmonic connected graphs and strictly pseudosemiregular connected graphs are the same. Lemma 4 Let G be a connected graph. Then following are equivalent: (1). G is strictly semiharmonic. (). G is strictly pseudosemiregular. Proof. Let G be a strictly semiharmonic graph. Then G must have a vertex v 0 of the smallest average degree, say a = d (v 0 ) d(v 0 ). Let w 0 be the vertex that is adjacent to v 0 and has the largest average degree among the neighbors of v 0, say b = d (w 0 ) d(w 0 ). Then d 3 (w 0 ) = u N(v 0) d (u) d (v 0 ) d(v 0 ) d (w 0 ) = abd(w 0 ), with equality holding if and only if all neighbors of w 0 have average degree a. d 3 (v 0 ) = u N(w 0) d (u) d (w 0 ) d(w 0 ) d (v 0 ) = abd(v 0 ), with equality holding if and only if all neighbors of v 0 have average degree b. Since G is strictly semiharmonic, a b and d 3 = µd holds for all v V (G), all neighbors of v 0 must have average degree b and all neighbors of w 0 must have average degree a. Since G is connected, it follows immediately that G is (bipartite) strictly pseudosemiregular. If G is strictly pseudosemiregular and let V = V 1 V be a bipartition of V (G). Set a = d (u) d(u) (u V 1) and b = d d (v V ) and a b. Hence G is not harmonic. For all u V 1, we have d 3 (u) = v N(u) d = v N(u) d d d = bd (u) = abd(u) and similarly, d 3 = abd for all v V. Thus d 3 = abd for all v V and hence G is a strictly absemiharmonic graph.
5 Combining lemmas 3 and 4 we have the main result which generalizes Theorem. Theorem 5 Let G be a connected graph and k 1. Then ρ(g) d k+1 d k, with equality if and only if G is pseudoregular or strictly pseudosemiregular. Theorem 6 Let G be a connected graph and f(k) = k+1, k 0. (4) Then f(k) is an increasing sequence and d k lim f(k) = ρ(g). n Proof. We prove the monotone increasing of f(k) by showing that f(k + 1) f(k). Note that the inner product A k j, A k j = (A k j) T A k j = d k. By CauchySchwarz inequality, we have A k j, A k j A k+ j, A k+ j A k j, A k+ j = A k+1 j, A k+1 j. That is, d k+ d k+1 Hence, f(k + 1) f(k) for k 0. d k+1 d k. Since the sequence f(k) is an monotonically increasing sequence and has a upper bound ρ(g), the limit lim k f(k) must exist. In order to prove the limit it suffices to prove lim k f(k) = ρ(g). Let ρ(g) = λ 1 > λ λ n be all eigenvalues of A(G) and X 1, X,, X n be unit eigenvectors corresponding these eigenvalues of A(G). Then X 1, X,, X n consist of a orthonormal basis of R n. Assume that j = n i=1 θ ix i, then θ i = j T X i, i = 1,,..., n. D In order to prove lim f(k) = ρ(g) it suffices to prove that k k Dk, D k approaches a unit eigenvector corresponding the eigenvalue ρ (G) of A (G) when k. Note that D k Dk, D k = A k j A k j, A k j = A k j and A k j = A k d k n θ i X i = i=1 n i=1 θ i λ k i X i,
6 we have d k = n i=1 θ i λ 4k i. If G is nonbipartite, since θ 1 > 0 and λ 1 > λ i for all i =, 3,..., n, then the vector θ i λ k i X i D k approaches X 1 if i = 1, 0 if i > 1. Thus the eigenvector Dk, D k n i=1 θ i λ4k i approaches X 1 and the result follows. If G is bipartite, since θ 1 > 0 and λ 1 > λ i for all i =, 3,..., n 1, λ n = λ 1, then the θ i λ k i X i θ 1 X 1 θ vector approaches n X if i = 1, 0 if 1 < i < n, n if i = n. θ 1 + θn θ 1 + θn n i=1 θ i λ4k i D k Thus the eigenvector Dk, D k approaches θ 1X 1 + θ n X n, which is a unit eigenvector θ 1 + θn corresponding the eigenvalue ρ (G) of A (G) and the result follows. 3 Upper bounds for the energy of a graph The energy of a graph G, denoted by E(G), is defined as E(G) = n i=1 λ i, where λ 1 λ λ n are the eigenvalues of the adjacency matrix of G. This concept was introduced by I. Gutman and is studied intensively in mathematical chemistry, since it can be used to approximate the total πelectron energy of a molecule(see [5] ). In 1971 McClelland discovered the first upper bound for E(G) as follows: E(G) mn. (5) Since then, numerous other bounds for E(G) were found (see [1,,6 10,1,14 16]). Recently, Yu et. al [14] proved the following result: Theorem 7 Let G be a nonempty graph with n vertices and m edges. Then ) (G) E(G) d (G) d + (G) (n 1) (m d. (6) (G) d Equality holds if and only if one of the following statements holds: (1) G = n K ; () G = K n ; (3) G is a nonbipartite connected µpseudoregular graph with three distinct eigenvalues m µ µ,, m µ, where µ > m. n 1 n 1 n
7 Remark In fact, the graph appears in (3) of above Theorem must be noncompleted m ( strongly regular graph with two nontrivial eigenvalues both with absolute value m n ), n 1 for details see the proof of the next Theorem. The following is the main results of this section which generalize the above result. Since general case is not substantially different, we concentrate on the particular case of connected graph for the sake of readability. Theorem 8 Let G be a connected graph with n(n ) vertices and m edges. Then (G) E(G) d k+1 (G) d k + (G) (n 1) (m d k+1 ) (G) d k. (7) Equality holds if and only if G is the complete graph K n or G is a strongly regular graph m ( with two nontrivial eigenvalues both with absolute value m n ). n 1 Proof. Let λ 1 λ λ n be the eigenvalues of G. By the CauchyScharz inequality we have E(G) = n n λ i = λ 1 + λ i i=1 i= n (n 1) λ i (n = 1)(m λ 1). i= Now since the function F (x) = x+ (n 1)(m x ) decreases on the interval m/n x m. By Theorem 6 we have (G) λ 1 f(k) f(0) = d n Hence F (λ 1 ) F ( (G) d k+1 /d k ) and m n. (G) E(G) d k+1 (G) d k + (G) (n 1) (m d k+1 ) (G) d k. (8) If the connected graph G is a complete graph K n or a strong regular graph with two m ( nontrivial eigenvalues both with absolute values m n ) then it is easy to check that n 1 the equality holds.
8 Conversely, if the equality holds, according the above proof, we have (G) λ 1 = d k+1, (G) d k m λ which implies that G is pseudosemiregular and λ i = 1 for i =, 3,..., n. Note n 1 that a graph has only one distinct eigenvalue if and only if it has no edges and a graph has two distinct eigenvalues if and only if it is a complete graph. Since G is connected, we reduced to the following two cases: (1) G has two distinct eigenvalues. In this case G = K n. (G) () G has three distinct eigenvalues. In this case, λ 1 = d k+1 and λ i = (G) d k m λ 1 for i =, 3,..., n. Since G is connected, λ 1 > λ i, λ i 0 for i =,..., n. Thus n 1 G must be regular (else G has 0 as an eigenvalue) and nonbipartite ( else G has least four distinct eigenvalues). Hence G is λ 1 regular (λ 1 = m ) and has exact three distinct n eigenvalues. Thus G is strong regular graph with two nontrivial eigenvalues both with (m ( mn ) ) absolute value. n 1 Similarly, (similar to [7, 14, 15]) we may prove Theorem 9 Let G be a connected bipartite graph with n(n ) vertices and m edges. Then (G) E(G) d k+1 (G) d k + (G) (n ) (m d k+1 ) (G) d k. (9) Equality holds if and only if G is the complete bipartite graph or G is the incidence graph of a symmetric (ν, k, λ)design with k = m k(k 1), n = ν and λ = n ν 1. Acknowledgements: This work was supported by NSF ( ) of China and by Scientific Research Fund of Hunan Provinical Education Department(06A037 ).
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