Investigation on spectrum of the adjacency matrix and Laplacian matrix of graph G l

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1 Investigation on spectrum of the adjacency matrix and Lapacian matrix of graph G SHUHUA YIN Computer Science and Information Technoogy Coege Zhejiang Wani University Ningbo 3500 PEOPLE S REPUBLIC OF CHINA Abstract: Let G be the graph obtained from K by adhering the root of isomorphic trees T to every vertex of K and d k j+ be the degree of vertices in the eve j In this paper we study the spectrum of the adjacency matrix A(G ) and the Lapacian matrix L(G ) for a positive integer and give some resuts about the spectrum of the adjacency matrix A(G ) and the Lapacian matrix L(G ) By using these resuts an upper bound for the argest eigenvaue of the adjacency matrix A(G ) is obtained: λ (A(G )) < max max d j + d j+ } d k + d k + d k + + } j k and an upper bound for the argest eigenvaue of the Lapacian matrix L(G ) is aso obtained: µ (L(G )) < max d j + d j + d j+ } d k + d k + d k + max j k d k + + d k + } Key Words: Adjacency matrix Lapacian matrix compete graph spectrum Introduction Let G be a simpe undirected graph with vertex set V = v v v n } which n = V Let A(G) be a (0 )-adjacency matrix of G Since A(G) is a rea symmetric matrix a of its eigenvaues are rea Without oss of generaity that they are ordered in nonincreasing order ie λ (G) λ (G) λ n (G) () and ca them the spectrum of G The argest eigenvaue λ (G) is caed the spectra radius of G If the destine eigenvaues of A(G) are λ (G) > λ (G) > > λ s (G) and their muipicities are then we sha write SpecA(G) = m(λ ) m(λ ) m(λ s ) ( λ (G) λ (G) λ s (G) m(λ ) m(λ ) m(λ s ) ) 36 For exampe the compete graph K n has n vertices and each distinct pair are adjacent Thus the graph K 4 has adjacency matrix A(K 4 ) = and an easy cacuation shows that that spectrum of K 4 is: ( ) 3 SpecA(K 4 ) = 3 About the spectrum and the spectra radius of graphs a great dea of investigation is carried out [][][3] Speciay to the specia graphs for exampe [4] studied the spectra radius of bicycic graphs with n vertices and diameter d [5] studied the spectra radius of trees with fixed diameter In [6] V Nikiforov proved that if G is a graph of order n maximum degree and girth at east 5 then λ (G) min n }

2 where λ (G) is the argest eigenvaue of the adjacency matrix of G In [7] XDZhang proved λ (G) < ( ) n(n ) where G be a simpe connected non-reguar graph of order n and be the maximum degree of G Let d(v i ) denote the degree of v i V i = n and et D(G) = diag(d(v ) d(v ) d(v n )) be the diagona matrix of vertex degrees The Lapacian matrix of G is L(G) = D(G) A(G) Ceary L(G) is a rea symmetric matrix From this fact and Ger sgorin s Theorem it foows that its eigenvaues are nonnegative rea numbers Therefore the eigenvaues of L(G) which are ca the Lapacian eigenvaue of G can be denote by µ (G) µ (G) µ n (G) = 0 We ca µ (G) the Lapacian spectra radius of G If the destine eigenvaues of L(G) are µ (G) > µ (G) > > µ s (G) and their muipicities are then we sha write SpecL(G) = m(µ ) m(µ ) m(µ s ) ( µ (G) µ (G) µ s (G) m(µ ) m(µ ) m(µ s ) For the Lapacian eigenvaues and the Lapacian spectra radius of simpe graphs there are many good resuts In [8] some of the many resuts known for Lapacian matrices are given Fieder [9] proved that G is a connected graph if and ony if the second smaest eigenvaue of L(G) is positive This eigenvaue is ca the agebraic connectivity of G denoted by α(g) In [0] Li and Pan proved the foowing resut: Let G be a simpe connected graph with n vertices and m edges Denote by δ the argest and smaest degrees of vertices in G Then µ (G) + 4m δ(n ) + (δ ) In [] ShuHong and Wenren proved a sharp upper bound as foows: µ (G) d n + + (d n n ) + d i (d i d n ) i= where d d d n is the degree sequence of G ) 363 Preiminaries Let T be an unweighted rooted tree of k eves such that in each eve the vertices have equa degree K be a compete graph on vertices Let G be the graph obtained from K by adhering the root of isomorphic trees T to every vertex of K Simiar to the definition of tree s eve we agree that the compete graph K is at eve and that G has k eves Thus the vertices in the eve k have degree For j = 3 k et n k j+ and d k j+ be the number of vertices and the degree of them in the eve j Observe that n k = is the number of vertices in eve and n the number of vertices in eve k(the number of pendant vertices) Then n k = (d k + )n k n k j = (d k j+ )n k j+ j = 3 k Observe that d k is the degree of vertices of the compete graph K in G d is the degree of the vertices in the eve k n k = The tota number of vertices in the graph G is k n = + j= Exampe Foow (Fig) is an exampe of a such graph G 4 for k = 3 n = 4 n = 8 n 3 = 4 d = d = 4 d 3 = Fig graph G 4 In genera using the abes n n in this order our abeing for the vertices of G is:

3 () First we abe the vertices of K with cockwise direction () For one of vertices of eve j(j = k ) the bigger its abeing is then the vertex of eve j + adjacent to it shoud be abeed first (3) Labe from eve to eve k in turn [] [3] studied the spectrum of the adjacency matrix A(G ) and the eigenvaues of Lapacian matrix L(G ) for case = and = respectivey In this paper we wi study the spectrum of the adjacency matrix A(G ) and the eigenvaues of Lapacian matrix L(G ) for a positive integer We introduce the foowing notations: () 0 is the a zeros matrix the order of 0 wi be cear from the context in which it is used ()I m is the identity matrix of order m m (3) m j = + for j = k (4) e m is the a ones coumn vetor of dimension m For j = k C j is the bock diagona matrix C j = e mj e mj emj with + diagona bocks Thus the order of C j is + For exampe we use these notation with the graph G 4 in Fig m = n n = 3 m = n n 3 = then C = diage 3 e 3 e 3 e 3 e 3 e 3 e 3 e 3 } C = diage e e e } The adjacency matrix A(G 4 ) in Fig become 0 C 0 A(G 4 ) = C T 0 C 0 C T B 4 where B 4 = where L(G 4 ) = U 4 = d 3 I 4 B 4 = and d I 4 C 0 C T d I 8 C 0 C T U In genera our abeing yieds to 0 C C T 0 C C T A(G ) = Ck Ck T B 0 0 where B = 0 L(G ) = d I n C C T d I n C C T dk I nk C k C T k where d k d k U = d k I nk B = d k Appy the Gaussian eimination procedure we obtained the foowing emma: Lemma Let M = α I n C C T α I n C C T αk I nk C k α k I nk + B et and β = α C T k β j = α j β j j = 3 k β j 0 If β j 0 for a j = k then detm = β n βn βn k k (β k + )(β k ) U ()

4 Proof Appy the Gaussian eimination procedure without row interchanges to M to obtain the bock upper trianguar matrix β I n C β I n C β 3I n3 β k I nk C k β k I nk + B Hence Since so detm = β n βn βn k k det(β ki nk + B ) det(λi B ) = (λ + )(λ + ) det(β k I nk + B ) = ( ) det( β k I nk B ) = (β k + )(β k ) Then detm = β n βn βn k k (β k + )(β k ) Thus () is proved # Lemma Let M 0 = α I n C C T α I n C C T αk I nk C k α k I nk B et and β = α C T k β j = α j β j j = 3 k β j 0 If β j 0 for a j = k then detm 0 = β n βn βn k k (β k + )(β k + ) (3) Proof Appy the Gaussian eimination procedure without row interchanges to M 0 to obtain the bock upper trianguar matrix β I n C β I n C β 3I n3 β k I nk C k β k I nk B 365 Hence Since so Then detm 0 = β n βn βn k k det(β ki nk B ) det(λi B ) = (λ + )(λ + ) det(β k I nk B ) = (β k + )(β k + ) detm 0 = β n βn βn k k (β k + )(β k + ) Thus (3) is proved # 3 The spectrum of the adjacency matrix and the Lapacian matrix of G In this section we wi appy Lemma3 Lemma 3 and Lemma 33 to study the spectrum of the adjacency matrix and the Lapacian matrix of G Lemma 3[4] Let H be k k symmetric tridiagona matrix: H = a b b a b b ak b k b k a k and Q j (λ)(j = 0 k) be the characteristic poynomias of the j j eading principa submatrix of matrix H Then Q 0 (λ) = Q (λ) = λ a Q j (λ) = (λ a j )Q j (λ) b j Q j (λ) (j = 3 k) (4) Lemma 3[4] Let H and Q j (λ)(j = 0 k) be matrix as in Lemma 3 then a roots λ (j) i (i = j) of Q j (j = 0 k) are rea and simpe: λ (j) > λ (j) > > λ (j) j and the roots of Q j and Q j respectivey separate each other stricty: λ (j) > λ (j ) > λ (j) > λ (j ) > > λ (j ) j > λ (j) j

5 Lemma 33[5] Let A B be n n Hermitian matrix Assume that B is positive semidefite and that the eigenvaues of A and A+B are arranged in decreasing order as in () Then λ i (A) λ i (A + B) for a i = n 3 The spectrum of the adjacency matrix of G Let φ = k } and Ω = j φ : > + } Observe that n k j = (d k j+ )n k j+ j = 3 k and n k = (d k + )n k Observe aso that if j φ Ω then = + and C j is the identity matrix of order Theorem 3 Let S 0 (λ) = S (λ) = λ S j (λ) = λs j (λ) S j (λ) for j = 3 k and Sk (λ) = (λ + )S k (λ) n k S k (λ) S k + (λ) = (λ + )S k (λ) n k S k (λ) Then (i) If S j (λ) 0 for j = k then det(λi A(G )) = (Sk (λ)) S k + (λ) j Ω S + j (λ) (5) (ii)the spectrum of A(G ) is σ(a(g )) = ( j Ω λ : S j (λ) = 0}) λ : Sk (λ) = 0} λ : (λ) = 0} S + k Proof Suppose S j (λ) 0 for a j = k We appy Lemma to M 0 = λi A(G ) M 0 = λi A(G ) = λi n C C T λi n C C T λink C k Ck T λi nk B We have β = λ = S (λ) 0 β = λ n n β = λ n n S (λ) = λs (λ) n n S 0 (λ) S (λ) = S (λ) S (λ) 0 Simiary for j = 3 4 k k Thus β j = λ β j = λ S j (λ) S j (λ) = λs j (λ) S j (λ) S j (λ) = S j(λ) S j (λ) 0 β k + = S k(λ) S k (λ) + = S k(λ) + S k (λ) S k (λ) = (λ + )S k (λ) n k S k (λ) S k (λ) = S k (λ) S k (λ) β k + = S k(λ) S k (λ) + = S k(λ) ( )S k (λ) S k (λ) = (λ + )S k (λ) n k S k (λ) S k (λ) = S+ k (λ) S k (λ) Therefore from Lemma det(λi A(G )) = S n (λ) Sn (λ) S S n (λ) n k k (λ) S n k k + Sk (λ) (S k (λ)) (λ) S k (λ) S k (λ) = S n n (λ)s n n 3 (λ)s n k n k (λ) j Ω k S + k (λ)(s k (λ)) = (S k (λ)) S + k (λ) S + j (λ) 366

6 Thus (i) is proved Simiar to the proof in [3] we can get (ii) by (i) # Let R k + and R k be the k k symmetric tridiagona matrices R k + = 0 d d 0 d3 d3 0 dk + dk + and Rk = 0 d d 0 d3 d3 0 dk + dk + Observe that R + k = R k + diag0 0 0 } Theorem 3 For j = 3 k et R j be the j j eading principa submatrix R + k Then det(λi R j ) = S j (λ) j = k det(λi R k ) = S k (λ) det(λi R + k ) = S+ k (λ) Proof We appy Lemma 3 in our case a = a = = a k = 0 a k = (or a k = ) and b k = b j = nk n k = d k + nj = d j+ + for j = 3 k For these vaues the recursion formua (4) gives the poynomias S j (λ) j = 0 k S k + (λ) and Sk (λ) This competes the proof # Theorem 33 Let R j j = k R k + and R k as above then (i)σ(a(g )) = ( j Ω σ(r j )) σ(r k + ) σ(r k ) (ii) The mutipicity of each eigenvaue of the matrix R j as an eigenvaue of A(G ) is at east + for j Ω for the eigenvaues of R k + and for the eigenvaues of Rk Proof (i) is an immediate consequence of Theorem 3 and Theorem 3 From Lemma 3 that the eigenvaues of R j j = k R k + and R k are simpy Finay we use (5) and Theorem 3 to obtain(ii) # Theorem 34 Let A(G ) be the adjacency matrix of G Then (a ) σ(r j ) σ(r j ) = φ for j = 3 k (a ) σ(r k ) σ(r k + ) = φ and σ(r k ) σ(rk ) = φ (a 3 ) The argest eigenvaue of R k + is the argest eigenvaue of A(G ) and the argest eigenvaue of Rk is the second argest eigenvaue of A(G ) Proof (a ) and (a ) foow from Lemma 3 By Lemma 33 and R + k = R k + diag0 0 0 } we can get the eigenvaues of R k + are greater or equa to the eigenvaues of Rk Now (a 3) foow from this fact and Lemma3 # Exampe Fig graph G

7 For the graph G 5 in Fig for = 5 k = 4 n = 30 n = 0 n 3 = n 4 = 5 d = d = 4 d 3 = 3 d 4 = 5 R = 0 R = ( ) R 3 = R 4 + = R4 = and Ω = } By Theorem 33 the eigenvaues of A(G 5 ) in Fig are the eigenvaues of R R R 4 + and R4 they are R : 0 R : R4 : : R + 4 The spectra radius of G 5 in Fig is λ (A(G 5 )) = 4696 SpecA(G 5 ) = ( The spectrum of the Lapacian matrix of G Theorem 3 Let P 0 (µ) = P (µ) = µ P j (µ) = (µ d j )P j (µ) P j (µ) for j = 3 k P k + (µ) = (µ (d k + ))P k (µ) n k P k (µ) ) 368 and Pk (µ) = (µ (d k +))P k (µ) n k P k (µ) Then (i) If P j (µ) 0 for a j = k then det(µi L(G )) = (P + k (µ)) P k (µ) j Ω P + j (µ) (6) (ii)the spectrum of L(G ) is σ(l(g )) = ( j Ω µ : P j (µ) = 0}) µ : Pk (µ) = 0} µ : (µ) = 0} P + k Proof Suppose P j (µ) 0 for a j = k We appy Lemma to M = µi L(G ) we denote µ d j = x j (j = k) then M = µi L(G ) = x I n C C T x I n C C T xk I nk C k x k I nk + B We have C T k β = µ d = µ = P (µ) 0 β = (µ d ) n n β = µ d n n P (µ) = (µ d )P (µ) n n P 0 (µ) P (µ) = P (µ) P (µ) 0 Simiary for j = 3 4 k k β j = (µ d j ) β j = µ d j P j (µ) P j (µ) = (µ d j)p j (µ) P j (µ) P j (µ) = P j(µ) P j (µ) 0

8 Thus β k = P k(µ) P k (µ) β k + = = P k(µ) P k (µ) P k (µ) = (µ d k )P k (µ) n k P k (µ) P k (µ) = P + k (µ) P k (µ) P k (µ) P k (µ) + = P k(µ) + ( )P k (µ) P k (µ) = (µ d k + )P k (µ) n k P k (µ) P k (µ) = P k (µ) P k (µ) Therefore from Lemma det(µi L(G )) = P n (µ) P n (µ) P P n (µ) n k k (µ) P n k k Pk (µ) (P + k (µ)) (µ) P k (µ) P k (µ) = P n n (µ)p n n 3 (µ)p n k n k k (µ) P k (µ)(p + k (µ)) = (P + k (µ)) P k (µ) P + j (µ) j Ω Thus (i) is proved Simiar to the proof Theorem 3 we can get (ii) by (i) # Let W k + and W k be the k k symmetric tridiagona matrices W k + = d d d d3 d3 d k dk + dk + d k + and W k = d d d d3 d3 d k dk + dk + d k + Observe that W + k = W k + diag0 0 0 } Theorem 3 For j = 3 k et W j be the j j eading principa submatrix W + k Then det(µi W j ) = P j (µ) j = k det(µi W k ) = P k (µ) det(µi W + k ) = P + k (µ) Proof Simiar to the proof of Theorem 3 in this case a = a j = d j forj = 3 k a k = d k + (or a k = d k + ) and nk b k = = d k + n k b j = nj = d j+ + for j = 3 k For these vaues recursion formua (4) gives the poynomias P j (µ) j = 0 k P k + (µ) and Pk (µ) This competes the proof # Simiar to the proof of Theorem 33 we can get: Theorem 33 Let W j j = k W k + and Wk as above then (i)σ(l(g )) = ( j Ω σ(w j )) σ(w k + ) σ(w k ) (ii) The mutipicity of each eigenvaue of the matrix W j as an eigenvaue of L(G ) is at east + for j Ω for the eigenvaues of Wk and for the eigenvaues of W k + Theorem 34 Let L(G ) be the Lapacian matrix of G Then (b ) σ(w j ) σ(w j ) = φ for j = 3 k (b ) σ(w k ) σ(w k + ) = φ and σ(w k ) σ(wk ) = φ (b 3 ) det W j = for j = k det Wk = 0 and det W k + = (b 4 ) The argest eigenvaue of W k + is the argest eigenvaue of L(G ) (b 5 ) The smaest eigenvaue of W k + is the agebraic connectivity G Proof (b ) and (b ) foow from Lemma 3 Now we appy the Gaussian eimination procedure without row interchanges to reduce W j to the upper trianguar matrix W j d d3 dj dj 369

9 The same procedure appied W k + and W k gives the trianguar matrices W k + d d3 dk + and Wk d d3 dk + 0 respectivey Thus (b 3 ) is proved and 0 is the smaest eigenvaue of W k Since W + k = W k + diag0 0 0 } by Lemma 33 the eigenvaues of W k + are greater or equa to the eigenvaues of Wk Now (b 4) and (b 5 ) foow from this fact and Lemma3 # Exampe 3 For the graph G 5 in Fig ( ) 3 W = W = W 3 = W 4 + = W4 = and Ω = } By Theorem 3 the eigenvaues of L(G 5 ) in Fig are the eigenvaues of W W W 4 + and W4 they are W : W : W4 : : W The spectra radius of G 5 in Fig is µ (L(G 5 )) = 6444 SpecL(G 5 ) = ( An upper bound for the argest eigenvaue of the adjacency matrix A(G ) and the Lapacian matrix L(G ) In this section we wi give an upper bound for the argest eigenvaue of the adjacency matrix and the Lapaican matrix of graph G Lemma 4[3] Let B = (b ij ) be a nonnegative n n matrix and λ is the argest eigenvaue of matrix B Denote the ith row sum of B by s i (B) Then ) min s i(b) λ max s i(b) i n i n the eft equaity hods if and ony if the right equaity aso hods 4 An upper bound for the argest eigenvaue of the adjacency matrix A(G ) Theorem 4 Let G ( is a positive integer) be the graph as above and that G has k eves d k j+ be the degree of vertices in the eve j then λ (A(G )) < max max j k d j + d j+ } + d k + d k + + } Proof Let R k + = 0 d d 0 d3 d3 0 dk + dk + we now appy Lemma 4 to concude λ (R + k ) < max max j k d j + d j+ } + d k + d k + + }

10 From Theorem 334 we easiy have λ (A(G )) < max max j k d j + d j+ } + d k + d k + + } # 4 An upper bound for the argest eigenvaue of the Lapacian matrix L(G ) In this section we give an upper bound for the argest eigenvaue of the Lapacian matrix L(G ) Theorem 4 Let G ( is a positive integer) be the graph as above and that G has k eves d k j+ be the degree of vertices in the eve j then µ (L(G )) < max max j k d j + d j + d j+ } + d k + d k + dk + + d k + } Proof Let W k + = d d d d3 d3 d k dk + dk + d k + we now appy Lemma 4 to concude µ (W + k ) < max max j k d j + d j + d j+ } + d k + d k + dk + + d k + } From Theorem 34 we easiy have µ (L(G )) < max max j k d j + d j + d j+ } + d k + d k + dk + + d k + } # 5 Concusion We studied the spectrum of the adjacency matrix A(G ) and the spectrum of the Lapacian matrix L(G ) for a positive integer with an effective way (I) Let R j j = k R k + and R k as in section 3 We found that: (I )σ(a(g )) = ( j Ω σ(r j )) σ(r k + ) σ(r k ) (I ) The mutipicity of each eigenvaue of the matrix R j as an eigenvaue of A(G ) is at east + 37 for j Ω for the eigenvaues of R k + and for the eigenvaues of Rk (I 3 )The argest eigenvaue of R k + is the argest eigenvaue of A(G ) and the argest eigenvaue of Rk is the second argest eigenvaue of A(G ) It is very convenient with concusions (I )(I )(I 3 ) to cacuate the spectrum of the adjacency matrix A(G ) In section 4 according to the resuts(i )(I )(I 3 ) and Lemma 4 an upper bound for the argest eigenvaue of the adjacency matrix A(G ) is obtained: λ (A(G )) < max max j k d j + d j+ } + d k + d k + + } (II) Let W j j = k W k + and W k as in section 3 We found that: (II ) σ(l(g )) = ( j Ω σ(w j )) σ(w k + ) σ(wk ) (II ) The mutipicity of each eigenvaue of the matrix W j as an eigenvaue of L(G ) is at east + for j Ω for the eigenvaues of Wk and for the eigenvaues of W k + (II 3 ) The argest eigenvaue of W k + is the argest eigenvaue of L(G ) (II 4 ) The smaest eigenvaue of W k + is the agebraic connectivity G In section 4 according to the resuts(ii )(II )(II 3 ) and Lemma 4 an upper bound for the argest eigenvaue of the Lapacian matrix L(G ) is obtained: µ (L(G )) < max max j k d j + d j + d j+ } + d k + d k + dk + + d k + } References: [] Dragan Stevanocic Bounding the angest eigenvaue of trees in terms of the argest vertex degree Linear Agebra and its Appications pp 35 4 [] AiMei Yu Mei Lu and Feng Tian On the spectra radius of graphs Linear Agebra and its Appications pp 4 49 [3] MNEingham XZha The spectra radius of graph on surfaces Journa of Combinatoria Theory Ser B pp [4] ShuGuang Guo On the spectra radius of bicycic graphs with n vertices and diameter d Linear Agebra and its Appications pp 9 3

11 [5] JM Guo JY Shao On the spectra radius of trees with fixed diameter Linear Agebra and its Appications 43() 006 pp 3 47 [6] Vadimir Nikiforov Boundson graph eigenvaues I Linear Agebra and its Appications pp [7] XDZhang Eigenvectors and eigenvaues ofnon-reguar graphs LinearAgebra andits Appications pp [8] RMerris Lapacian matrices of graphs: a survey Linear Agebra and its Appications pp [9] MFieder Agebraic connectivity of graphs Czechosovak Math J pp [0] JSLi YLPan de Cane s inequaity and bounds on the argest Lapacian eigenvaue of a graph Linear Agebra and its Appications pp [] JLShu YHong KWenRen A sharp bound on the argest eigenvaue of the Lapacian matrix of a graph Linear Agebra and its Appications pp 3 9 [] Oscar Rojo Ricardo Soto The spectra of the adjacency matrix and Lapacian matrix for some baanced trees Linear Agebra and its Appications pp 97 7 [3] Oscar RojoThe spectra of some trees and bounds for the agest eigenvaue of any tree Linear Agebra and its Appications pp 99 7 [4] JStoer RBuirsich Introduction to Numerica Anaysis (Second Edition) Springer-Verag 004 [5] RAHorn CRJohnson Matrix Anaysis Cambridge University Press Cambridge 99 37