On the energy of singular graphs

Transcription

1 Electronc Journal of Lnear Algebra Volume 26 Volume 26 (2013 Artcle On the energy of sngular graphs Irene Trantafllou Follow ths and addtonal works at: Recommended Ctaton Trantafllou, Irene (2013, "On the energy of sngular graphs", Electronc Journal of Lnear Algebra, Volume 26 DOI: Ths Artcle s brought to you for free and open access by Wyomng Scholars Repostory It has been accepted for ncluson n Electronc Journal of Lnear Algebra by an authorzed edtor of Wyomng Scholars Repostory For more nformaton, please contact scholcom@uwyoedu

2 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety ON THE ENERGY OF SINGULAR GRAPHS IRENE TRIANTAFILLOU Abstract The nullty, η(g, of a graph G s the algebrac multplcty of the egenvalue zero n the graph s spectrum If η(g > 0, then the graph G s sad to be sngular The energy of a graph, E(G, was frst defned by I Gutman (1985 as the sum of the absolute values of the egenvalues of the graph s adjacency matrx A(G Ths paper consders the energy concept for sngular graphs In partcular, t s proved that the change n energy upon the smple act of deletng a vertex s related to the type of vertces of the sngular graph Certan upper bounds are mproved for the energy of the nduced subgraph, G u, whch s obtaned by deletng vertex u, wth the ad of a parameter known as the null spread of u, η u(g = η(g η(g u Also, some new bounds are gven for the energy of the mnmal confguraton graphs, an mportant class of sngular graphs of nullty one that are related to the graph s core Furthermore, certan graphs that ncrease ther energy when an edge s deleted are consdered, such as the complete multpartte graphs and the hypercubes of even dmensons Key words Nullty, Sngular graphs, Energy of graphs, Null spread, Mnmal confguraton graphs, Complete multpartte graphs, Hypercube AMS subject classfcatons 05C50, 15A18 1 Introducton and prelmnares Let G = (V, E be a fnte, undrected graph wth nonempty vertex set V and edge set E The adjacency matrx, A(G, of a graph G on n vertces s the n n matrx whose entres a j denote the number of edges from vertex u to vertex u j For a smple, undrected graph the adjacency matrx s a symmetrc (0, 1-matrx Thus, A(G has real egenvalues and zeros on the dagonal, meanng that the sum of these egenvalues equals to zero A graph, G, s sngular f the adjacency matrx, A(G, s a sngular matrx; that s, zero s an egenvalue of G The nullty, η(g, of a sngular graph G s the multplcty of zero n the graph s spectrum It s clear that there exst correspondng vectors x, such that Ax = 0 These vectors are defned as the kernel egenvectors of a graph G Let us consder a graph G of nullty one, wth a kernel egenvector x = [x 1,x 2,, x m,0,,0] T, where x 0, = 1,2,,m The subgraph F of G, nduced by the frst m vertces correspondng to the frst m entres, s called the core of G The set of the remanng vertces, correspondng to the zero entres of the kernel egenvector, s called the perphery Receved by the edtors on November 14, 2012 Accepted for publcaton on July 28, 2013 Handlng Edtor: Xngzh Zhan Department of Mathematcs, Natonal Techncal Unversty of Athens, Zografou Campus, Athens, Greece (ern trantafllou@hotmalcom 535

3 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety Irene Trantafllou Defnton 11 [9] Let x be a kernel egenvector of a sngular graph on at least two vertces If x has only non-zero entres, then G s referred to as a core graph Defnton 12 [10] A graph G, G 3, s a mnmal confguraton, wth core (F,x F of nullty η(f, f t s a sngular graph of nullty one, havng F +(η(f 1 [ ] xf vertces, wth F as an nduced subgraph, satsfyng F 2, Fx F = 0, and G 0 [ ] 0 = The vector x F s sad to be the non-zero part of the kernel egenvector of 0 G Example 13 A path on 2k 1 vertces s a mnmal confguraton graph (η(p 2k 1 = 1 that has as a core the null graph N k (η(n k = k Fg 11 The core of P 7, N 4 colored black One of the most mportant theorems, consderng the egenvalues of a graph, s perhaps the nterlacng theorem Theorem 14 (Interlacng Theorem, [8] Let G be a graph wth spectrum λ 1 λ 2 λ n, and let the spectrum of G u 1 be µ 1 µ 2 µ n 1 Then, the spectrum of G u 1 s nterlaced wth the spectrum of G, and λ 1 µ 1 λ 2 µ 2 µ n 1 λ n It s clear from nterlacng that the nullty of a graph can change, at most one, upon deletng (addng a vertex We wll gve next, the followng defnton: Defnton 15 [4] Let G u be the nduced subgraph of graph G obtaned on deletng vertex u The null spread of vertex u s: n u (G = η(g η(g u Observaton 16 By nterlacng: 1 n u (G 1 Observaton 17 Let G be a mnmal confguraton graph wth core F of nullty η(f The η(f 1 vertces of G belong to the perphery of the graph, and ther deleton ncreases the nullty of the graph, meanng that each vertex n the perphery has a null spread of 1 The energy of a graph G was frst set by Gutman n 1978 as the sum of the absolute values of ts egenvalues: E(G = n =1 λ The concept of energy of graphs orgnates from theoretcal chemstry and has been studed rather ntensvely n the last decade Ths paper focuses on the energy of

4 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety On the Energy of Sngular Graphs 537 sngular graphs, a subject frst studed n [13] In Secton 2 of ths paper, we mprove some upper bounds for the energy of the nduced subgraph of G, G u, by dentfyng the vertces of the graph In Secton 3, we study the energy of certan classes of sngular graphs, such as the mnmal confguratons and the r-partte graphs We conclude ths paper wth some results on the energy change after deletng an edge of a complete multpartte graph or a hypercube 2 Energy of subgraphs Let G be a sngular graph and G u an nduced subgraph of G, obtaned from G by deletng vertex u In ths secton, we wll mprove the bound E(G u E(G, by dentfyng the vertces n G Theorem 21 Let G = (V,E be a graph and u V If n u (G = 1, then E(G u E(G The equalty holds f and only f u s an solated vertex Proof If, upon deletng u, the nullty decreases by one, then the set of the nonzero egenvalues n G and G u have the same cardnalty By the nterlacng theorem, E(G u E(G We wll now prove that the equalty holds Let u be an solated vertex Then, snce u s assocated wth zero entres n the adjacency matrx and related to a zero egenvalue n the spectrum of the graph, ts removal has absolutely no effect to the sum of the absolute values of the non-zero egenvalues of G Thus, E(G u = E(G Let E(G = E(G u and n u (G = 1, then n =1 λ = n 1 =1 µ, meanng that f we rearrange only the non-zero egenvalues n non ncreasng order, λ = µ, and λ 2 = µ 2 It s well known that λ 2 = 2m, for a graph G wth m edges, and thus, u s an solated vertex Theorem 22 Let G = (V,E be a graph and u V If n u (G = 1, then E(G u E(G ( λ l + λ m, where λ l and λ m are the smallest non-negatve and the largest non-postve egenvalue, respectvely In the case where G s a connected graph of nullty η(g = n 2, the equalty holds f and only f G s a star graph and u s the center vertex of the graph Proof If, upon deletng u, the nullty ncreases by one, then G u has two less non-zero egenvalues than G By nterlacng, λ 1 µ 1 λ 2 λ l > 0 = = 0 > λ m µ m λ n, and snce E(G = n =1 λ, we have E(G u E(G ( λ l + λ m

5 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety Irene Trantafllou Now, let G be a star graph on n vertces Then G has two non-zero egenvalues, wth multplcty one, namely n 1 and n 1, and n 2 zero egenvalues Upon deletng the center of the graph, the obtaned graph s null and the nullty ncreases by one Thus, E(G u = 0 = E(G ( n 1 + n 1 Let us suppose that G s a graph of nullty η(g = n 2 Then, f the nullty ncreases upon deletng a vertex, the obtaned subgraph has n 1 vertces and a nullty of η(g u = n 1 It s well known that the only graph that s of the same nullty as ts order s the null graph Thus, G s a star graph, wth vertex u as ts center, and the equalty E(G u = E(G ( λ l + λ m holds Observaton 23 Another example of a graph on n vertces that acheves the above equalty s the graph that s a unon of m complete graphs, K 2, and n 2m solated vertces The energy of G = mk 2 (n 2mK 1 s E(G = m( Upon deletng a vertex u of K 2, the nullty ncreases and the energy of G u s E(G u = (m 1( Thus, E(G u = E(G ( λ l + λ m The nterlacng nequaltes mply the followng Theorem 24 Let G = (V,E be a graph and u V If n u (G = 0, then E(G u E(G λ, where λ s ether the smallest non-negatve or the largest non-postve egenvalue of G Example 25 We gve an example of equalty for Theorem 24 The spectrum of graph G, n Fgure 21, s {2,0, 1, 1}, and ts energy s E(G = 4 When we delete the whte vertex u, the obtaned subgraph has {1,0, 1} as ts egenvalues, and the energyofg use(g u = 2 Thus, n u (G = 0and E(G E(G u = 2 = λ l, where λ l s the only non-negatve egenvalue of G Fg 21 The graphs G, G u It s clear, from the above theorems, that the type of vertces are mportant n determnng the energy of a sngular graph Smlar results, however, may be obtaned for any λ, nonzero egenvalue, whch could also nclude non-sngular graphs G Let G = (V,E be a graph and u V Suppose m(g s the multplcty of a non-negatve egenvalue λ for G, m(g u s the multplcty of λ for G u, and m u (G = m(g m(g u s the vertex spread of λ

6 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety On the Energy of Sngular Graphs 539 Then, n the case of m u (G = 0, t can be shown by usng the nterlacng theorem: E(G u max{e(g λ l,e(g λ m }, where λ l (resp λ m s the smallest postve (resp largest negatve egenvalue of G Observaton 26 It s obvous that f H s an nduced subgraph of a graph G, then E(H E(G 3 Energy of sngular graphs In ths secton, we study the energy of sngular graphs We gve some new bounds for the energy of the mnmal confguraton graphs and mprove some known bounds for energy, n the case that the graph s sngular Proposton 31 Let G be a mnmal confguraton wth core of order at least three Then, E(G > 2 5 Proof A mnmal confguraton wth core of order at least three has the path P 4 as an nduced subgraph [11] By Observaton 26, and snce P 4 s not sngular, E(G > 2( Theorem 32 Let G be a mnmal confguraton, wth core F of nullty η(f Then, E(G E(F+(η(F 1( λ l + λ m, where λ l and λ m are the smallest non-negatve and the largest non-postve egenvalue of G, respectvely Proof Let w, = 1,,η(F 1 be the vertces of the perphery P By Theorem 22, snce n w (G = 1: E(G w 1 E(G ( λ l + λ m, where λ l and λ m are the smallest non-negatveand the largestnon-postve egenvalue of G, respectvely Then, by the same theorem: E(G w 1 w 2 E(G w 1 ( µ l 1 + µ m, where µ l 1 and µ m are the smallest non-negatve and the largest non-postve egenvalues of G w 1, respectvely Snce by nterlacng, µ l 1 λ l and µ m λ m, E(G w 1 w 2 E(G w 1 ( µ l 1 + µ m E(G 2( λ l + λ m It s then clear that E(G w 1 w 2 w η(f 1 E(G (η(f 1( λ l + λ m Lemma 33 [2] Let H be an nduced subgraph of a graph G, wth edge set m Then, E(G E(H E(G m E(G+E(H

7 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety Irene Trantafllou Proposton 34 Let G be a mnmal confguraton graph wth core F, perphery P, and nullty η(f If n F s the number of vertces of the core adjacent to some vertex of the perphery and λ k s the smallest absolute value of the graph s egenvalues, then: λ k < 1 2 nf (1+ 8 η(f 1 ( n F +η(f 1+ 2 Proof Let the edge set of core F be m Snce the core F s an nduced subgraph of the mnmal confguraton graph G, by Lemma 33: E(G E(F E(G m When we remove the edges from F, the obtaned graph conssts only of edges between the core F and the perphery P Snce those two sets are ndependent, the G m graph s bpartte Koolen and Moulton [7] proved that for a bpartte graph K F,P : E(K F,P n 8 ( n+ 2 The vertces of G m are the η(f 1 vertces of the perphery and the vertces of the core, n F, adjacent to those of the perphery By Theorem32: E(G E(F > (η(f 1( λ l + λ m, whereλ l andλ m arethesmallest non-negatve and the largest non-postve egenvalues of G, respectvely Thus, or, f λ k = mn( λ l, λ l : λ l + λ m < η(f 1+nF 8(η(F 1 ( η(f 1+n F + 2, λ k < 1 2 nf (1+ 8 η(f 1 ( n F +η(f 1+ 2 Observaton 35 Let us try to construct a mnmal confguraton graph G from a null graph N p It has been shown that the mnmal confguraton graph s a connected graph [9], whch mples that all vertces of the null graph wll be adjacent to some vertex of the perphery Snce the nullty of the null graph s equal to ts order, η(f = n F = p By Proposton 34, for the smallest absolute value of the egenvalues of G, λ k : λ k < (1+ p p 1 ( 2p 1+ 2 For example, f we construct a graph from the null graph N 4, ts smallest, n absolute value, egenvalue s not greater than In Fgure 31, the mnmal confguraton graph has 1 as the smallest absolute value of egenvalues Fg 31 A mnmal confguraton, constructed by the null graph N 4 McClelland s bounds (1971 for the energy of a G(n, m graph, contanng the vertces and edges of the graph, are:

8 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety On the Energy of Sngular Graphs 541 2m+n(n 1 deta 2/n E(G 2mn The upper and lower bound of the above nequalty can be mproved for sngular graphs, as shown n Propostons 36 and 37, respectvely Proposton 36 [5] Let G be a graph on n vertces, and nullty η(g Then, E(G 2(n η(gm Proposton 37 [1] Let G be a graph on n vertces, and nullty η(g Then, E(G n η(g We conclude ths secton wth an upper bound for the energy of r-partte graphs Lemma 38 Let G be a complete r-partte graph, on n vertces and m edges Then, 2(r 1m E(G 2 r Proof Frst, we rearrange the n η(g = r non-zero egenvalues of the graph n non ncreasng order (λ 1 λ 2 λ n η(g, after omttng the η(g zero egenvalues We apply the Cauchy-Schwartz nequalty to (1,1,,1 and (λ 2, λ 3,, λ n η(g, ( n η(g =2 λ 2 n η(g =2 λ 2 n η(g =2 1 2 =(n η(g 1 n η(g =2 λ 2 Snce, λ 1 = n η(g =2 λ : λ 2 1 (n η(g 1 n η(g =2 λ 2 and (n η(gλ 2 1 (n η(g 1 n η(g =1 λ 2 = (n η(g 12m Snce G s a complete r-partte graph, t has only one postve egenvalue λ 1 [12] and E(G = 2λ 1 Thus, E(G 2 2(n η(g 1m n η(g It s clear that the equalty holds f and only f G s a regular r-partte graph 4 Deletng an edge It has been shown that the energy of a graph may ncrease, decrease, or stay the same after deletng an edge [3] In ths secton, we wll study certan graphs that ncrease ther energy when an edge s deleted, such as the complete multpartte graphs and the hypercube of even order Proposton 41 Let K p,q be a complete bpartte graph Then, f we remove an edge e: E(K p,q e = 2 pq 1+2 (p 1(q 1

9 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety Irene Trantafllou Proof Let the matrx of K p,q e be: A = The matrx has four ndependent rows, and so η(k p,q e = p+q 4 Let µ 1 µ 2 µ 3 µ 4 be the remanng non-zero egenvalues Then, snce K p,q e s bpartte, µ 1 = µ 3 and µ 2 = µ 4 The characterstc polynomal can now be wrtten as: x p+q 4 (x µ 1 (x+µ 1 (x µ 2 (x+µ 2 = x p+q 4 (x 4 (µ 2 1 +µ2 2 x2 +µ 2 1 µ2 2 It s well known that, µ 2 = 2m, and so µ2 1 +µ 2 2 = pq 1 Also, µ k = trak The dagonal entres of A 4 are: (q 1 2 p A 4 = (q q 2 (p 1 (p 1 2 q (p p 2 (q 1 (p p 2 (q 1 and tra 4 = (q 2 2q + 1(p 2 + 2p 1 + (p 2 2p + 1(q 2 + 2q 1 Snce, µ 4 1 +µ4 2 = (µ2 1 +µ µ 2 1 µ2 2, after some easy calculaton µ2 1 µ2 2 = (p 1(q 1 The energy of the graph s: E(K p,q e = 2( µ 1 + µ 2 = 2 µ 2 1 +µ µ 1 µ 2 and thus, E(K p,q e = 2 pq 1+2 (p 1(q 1 By Lemma 38, t s clear that: E(K p,q e E(G 2( pq 1+2 (p 1(q 1 pq Proposton 42 Let K t,t,,t be a complete r-partte graph, wth r 3, t 2 If K t,t,,t e s ts subgraph after removng edge e, then E(K t,t,,t e E(K t,t,,t Proof Let K t,t,,t e be the subgraph of a complete r-partte graph, K t,t,,t, after deletng an edge e between the frst two sets of vertces The graph s matrx wll be of the form:

10 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety A = On the Energy of Sngular Graphs It s clear that K t,t,,t e has r+2 ndependent rows Workng as n the proof of Proposton 41, we fnd that t s an egenvalue of the graph, wth multplcty r 3 Let µ 1 µ 2 µ 3 µ 4 µ 5 be the remanng non-zero egenvalues Then, we fnd that: (t 12 +4(t 1 (t 1 (t 12 +4(t 1+(t 1 µ 2 = 2, µ 4 = 2 Also µ 2 + µ 4 µ 3 + µ 5, where µ 3, µ 5 and µ 1 satsfy the polynomal: µ 3 ((r 2t 1µ 2 (m (r 2(r 1 2 t 2 +(r 2tµ (r 1(t 1t = 0, and m are the edges of K t,t,,t e, (m = r(r 1 2 t 2 1 Snce (t (t 1 t+ 1 rt and µ 1 2m rt = (r 1t 2 rt, E(K t,t,,t e = 5 µ +(r 3 t =1 µ 1 +2( µ 2 + µ 4 +(r 3t (r 1t 2 rt +2(t+ 1 rt +(r 3t = 2(r 1t = E(K t,t,,t The hypercube Q n s an undrected regular graph defned recursvely, wth reference to the cartesan product of two graphs, by Q 1 = K 2 and Q n+1 = Q n K 2 The characterstc polynomal of the hypercube Q n s ϕ(q n = n k k=0 (x n+2k(n [6] It s straghtforwardthat the hypercube Q n s sngularf and only [ f n s even The adjacency matrx of the hypercube can be wrtten as: A(Q n = 2 n 1 ] A(Qn 1 I, I 2 n 1 A(Q n 1 where I 2 n 1 denotes the dentty matrx

11 Electronc Journal of Lnear Algebra ISSN A publcaton of the Internatonal Lnear Algebra Socety Irene Trantafllou [ A X Lemma 43 [3] For a parttoned matrx C = Y B are square matrces, we have: j s j(a+ j s j(b j s j(c, where s j ( denote the sngular values of a matrx ], where both A and B Theorem 44 Let Q 2k be a sngular hypercube If Q 2k e s ts subgraph after removng edge e, then: E(Q 2k e E(Q 2k Proof Let e be an edge correspondng to the dentty matrx I 2 2k 1 The adjacency matrx of Q 2k e after deletng edge e s of the form: [ ] A(Q2k 1 J A(Q 2k e = 2 2k 1, J 2 2k 1 A(Q 2k 1 where J 2 2k 1 s formed from the dentty matrx by changng one dagonal entry to zero By Lemma 43, E(Q 2k e 2E(Q 2k 1 The energy of Q 2k s: E(Q 2k = 2k =0( 2k = k 2k =0( = k 2k =0( = 2 k 2k =0( = 4k k 2k =0( 2k 2 2k (2k 2 2k (2k 2 2k (2k 2 k 4 =0 ( 2k =k+1( 2k =0( 2k =0( 2k (2k 2 k (2k 2+ =0 (2k 2 2k + 2k =0 2k( 2k = 4k(2 2k 1 + k (2k 2 4k22k 1 2k2 2k +2 2k2 2k 1 = 2k ( 2k k Snce, In a smlar way, we fnd that E(Q 2k 1 = 2k ( 2k 1 k the proof s complete E(Q 2k = 2k ( 2k k = 2k ( 2k 1 k 2k 2k k = 4k ( 2k 1 k = 2E(Q 2k 1, ( 2k (2k 2 =0 2( 2k Acknowledgment The author would lke to thank an anonymous referee and the edtors for ther useful suggestons whch led to sgnfcant mprovements of ths paper

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