LET G = (V (G), E(G)) be a graph with vertex

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1 O Maximal Icidece Eergy of Graphs with Give Coectivity Qu Liu a,b Abstract Let G be a graph of order. The icidece eergy IE(G) of graph G, IE for short, is defied as the sum of all sigular values of its icidece matrix. I this paper, we determie the maximal icidece eergy IE(G) amog all coected graphs with the coectivity κ ad edge coectivity κ ad K κ (K 1 K κ 1) has the maximal icidece eergy. Idex Terms Sigless Laplacia matrix, Icidece eergy, Coectivity, Edge coectivity I. INTRODUCTION LET G = (V (G), E(G)) be a graph with vertex set V (G) = {v 1, v,..., v } ad edge set E(G) = {e 1, e,..., e m }, the adjacecy matrix A(G) = (a ij ) of G is a (vertex-vertex) symmetric matrix with a ij = 1 if v i ad v j are adjacet, ad a ij = 0 otherwise. Deote the degree of vertex v i by d(v i ), the sigless Laplacia matrix Q(G) of G is defied as Q(G) = D(G) + A(G), where D(G) = diag(d(v 1 )), d(v ),..., d(v )) is the diagoal matrix of degree of G. Sice Q(G) are symmetric matrix, their eigevalues are real umbers. So, we ca assume that q 1 (G) q (G)... q (G) are the sigless Laplacia eigevalues of G. Let I(G) be the (vertex-edge) icidece matrix of the graph G, the (i, j)-etry of I(G) is 0 if v i is ot icidet with e j ad 1 if v i is icidet with e j. Jooyadeh et al. [1] itroduced the icidece eergy IE of G, which is defied as the sum of the sigular values of the icidece matrix of G. Gutma et al. [] showed that IE = IE(G) = qi (G). Some basic properties of IE may be foud i [1 3]. Let G be a coected graph with vertices ad m edges. Let S(G) be the subdivisio graph of G, that is, S(G) is obtaied from G by isertig a ew vertex i each edge. Clearly, S(G) is a bipartite graph with + m vertices ad m edges. We deote σ(g, x) the characteristic polyomial det(xi A(G)) of G. It is well kow [4] that if G is a bipartite graph, the σ(g, x) = det(xi A(G)) = / ( 1) t b(g, t)x t, t=0 Mauscript received December 10, 017; revised September 3, 018. This work was supported by the Natioal Natural Sciece Foudatio of Chia (Nos , ), the Research Foudatio of the Higher Educatio Istitutios of Gasu Provice, Chia (018A-093) ad the Sciece ad Techology Pla of Gasu Provice(18JR3RG06). Q. Liu is with (a) School of Computer Sciece, Fuda Uiversity, Shaghai 00433, Chia ad (b) School of Mathematics ad Statistics, Hexi Uiversity, Gasu, Zhagye, , P.R. Chia. liuqu@fuda.edu.c. (1) where b(g, 0) = 1 ad b(g, t) 0 for all t = 1,,..., /. This expressio for σ(g, x) iduce a quasi-order relatio (i.e. reflexive ad trasitive relatio) o the set of all bipartite graphs with vertices: If G 1 ad G are bipartite graphs with characteristic polyomials i the form G 1 G b(g 1, t) b(g, t) for all t = 0, 1,,..., /. If G 1 G ad there exist t such that b(g 1, t) > b(g, t), the we write G 1 G. Gutma [5] itroduced this quasi-order relatio i order to compare the eergies of a pair of graphs. It is kow [5, 6] that for the bipartite graph S(G), E(S(G)) ca be also expressed as the Coulso itegral formula E(S(G)) = π + 0 Thus for m, we have [7] IE(G) = 1 π + where p i (G) = ( 1) i b i (S(G)). 0 (+m)/ 1 x I[1 + b i (S(G))x i ]dx. () 1 x I[1 + ( 1) i p i (G)x i ]dx, (3) From Eq.() ad Eq.(3) we kow that for two bipartite graphs S(G 1 ) ad S(G ), S(G 1 ) S(G ) IE(G 1 ) IE(G ), S(G 1 ) S(G ) IE(G 1 ) < IE(G ). For two oadjacet vertices v i ad v j, we use G + e to deote the graph obtaied by isertig a ew edge e = v i v j i G. For two vertex disjoit graph G 1 ad G, we deote by G 1 G the graph which cosists of two coected compoets G 1 ad G. The joi of G 1 ad G, deoted by G 1 G, is the graph with vertex set V (G 1 ) V (G ) ad edge set E(G 1 ) E(G ) {u i u j : u i V (G 1 ), u j V (G )}. The coectivity κ(g) of a graph G is the miimum umber of vertices whose removal from G yields a discoected graph or a trivial graph. The edge-coectivity κ is defied aalogously. It is both iterestig ad sigificat to determie the graph with extremal eergies amog a give class of graphs. Numerous results o this subject have bee put forward, for details see [8-13]. Zhou ad Triajstić [9] determied the extremal Kirchhoff idex of graphs with respect to give matchig umber. I [10], Xu characterized the extremal Laplacia-eergy-like with give matchig umber. Rojo [8] obtaied the extremal icidece eergy with respect to (Advace olie publicatio: 7 November 018)

2 the coectivity. I [14], Hu et al. determied the maximal eergy amog all subdivisios of graphs with vertices ad chromatic umber k. Zhag [18] characterize the graphs with the maximum icidece eergies amog all graphs with give chromatic umber ad give pedet vertex umber. Ispired by those works, i this paper, we determie the maximal icidece eergy amog all graphs with vertices ad the coectivity κ ad edge-coectivity κ. ( i + s x) ( s j,i j ( i + s x))]. Proof Sice the sum of all etries o every row of sigless Laplacia matrix of K is ( 1), (4) implies that T Q (x) =, usig Theorem.1, we ca get it immediately. x 1) III. RESULTS II. THE SIGNLESS LAPLACIAN CHARACTERISTIC POLYNOMIAL OF G 1 (G G 3 ) FOR REGULAR GRAPH I this sectio, we determie the sigless Laplacia characteristic polyomials of G = G 1 (G G 3 ) with the help of the coroal of a matrix. The M-coroal T M (x) of a matrix M is defied [15, 16] to be the sum of the etries of the matrix (xi M) 1, that is T M (x) = j T (xi M) 1 j, where j deotes the colum vector of dimesio with all the etries equal oe. It is well kow [15, P ropositio] that, if M is a matrix with each row sum equal to a costat t, the T M (x) = x t. (4) Theorem.1 Let G i (i = 1,, 3) be three graphs o i vertices. Also let T Qi (λ)(i = 1,, 3) be the Q i -coroal of G i. The the sigless Laplacia characteristic polyomial of the matrix Q(G 1 (G G 3 )) is P Q (G 1 (G G 3 )) = P Q (G 1, x 3 )P Q (G, x 1 )P Q (G 3, x 1 ) (1 T Q(G3)(x 1 )T Q(G1)(x 3 ) T Q(G)(x 1 )T Q(G1)(x 3 )). Proof With a proper labelig of vertices, the sigless Laplacia characteristic polyomial of Q(G) = Q(G 1 (G G 3 )) is give by Let M = xi 1 Q(G 1 ) + 3 )I 1, N = xi Q(G ) 1 I, T = xi 3 Q(G 3 ) 1 I 3, the P Q (G) where B = ( = det M j 1 j 1 3 j 1 N 0 3 j T = det(xi 3 Q(G 3 ) 1 I 3 )det(b) = P Q(G3)(x 1 )det(b), ( ) ( ) M j 1 j1 3 j 1 N 0 3 ((x 1 )I 3 Q(G 3 )) ( ) 1 j is the Schur complemet of λi 3 Q(G 3 ) 1 I 3. Thus the result follows. Corollary. Let G = K s (K 1 K ), where K s, K i (i = 1, ) deote the complete graph o s ad i vertices, respectively. The the characteristic polyomial of the sigless Laplacia matrix of G is P Q (G, x) = (x + ) s 1 (x s i + ) i 1 [( + s x) ) First, we defie the relatio (,, ) as follows. Defiitio 3.1. ([13]) We say p is partial larger tha q if p > q, deoted by p q. Similarly, we have p q, p q, p q. Defiitio 3.. ([13]) Let p(x) = i=0 p ix i ad q(x) = i=0 q ix i. If p i q i (resp. p i q i )for each 0 i, the we call p(x) q(x) (resp. p(x) q(x)). If p(x) q(x)(resp. p(x) q(x)), ad there exist a j {0, 1,,..., } such that p i q i (resp. p j q j ), we call p(x) q(x)(resp. p(x) q(x)). By the defiitio above, the followig result is immediate. Lemma 3.3 ([14]) Suppose a i b i 0 for i = 1,,...,. The (x a i ) (x b i ), furthermore, if there exist a j {1,,..., } such that a j > b j, the (x a i ) (x b i ). Theorem 3.4 Let, a ad k be three positive itegers ad a k. The (x +3) k (x k a+) a 1 (x +a+) k a 1. Proof Note that a k ad 1 > a a+k. By Lemma 3.3, (x +3) k (x k a+) a 1 (x +a+) k a 1. The result follows. Theorem 3.5 Let, a i ad b i be positive itegers ad a 1 a, where s = a 1 + a. If b 1 = a 1, b = a +, the (+s x) (b i +s x ) sb j ( t,i j (+s x) (a i +s x ) sa j ( Proof Note that (b i +s x ))),i j (a 1 + s x)(a + s x) = x (a 1 + a + s )x + 4a 1 a + s(a 1 + a ) 4(a 1 + a ) + (s ). (a i +s x ))). (Advace olie publicatio: 7 November 018)

3 ad So (b 1 + s x)(b + s x) = = x (b 1 + b + s )x + 4b 1 b + s(b 1 + b ) 4(b 1 + b ) + (s ). (b 1 +s x)(b +s x)a 1 +s x)(a +s x) Agai, ad Thus = 8s(a 1 a ). (5) ( sa 1 )(a + s x ) + ( sa )(a 1 + s x ) = s(a 1 + a )x 4sa 1 a + s(a 1 + a ) s (a 1 + a ) ( sb 1 )(b + s x ) + ( sb )(b 1 + s x ) = s(b 1 + b )x 4sb 1 b + s(b 1 + b ) s (b 1 + b ). [( sb 1 )(b + s x ) + ( sb )(b 1 + s x )] [( sa 1 )(a + s x ) + ( sa )(a 1 + s x )] Hece by (5) ad (6) ( + s x) + s x) = 8s(a 1 a ). (6) (b i + s x ) (a i + s x ) = ( + s x)[ 8s(a 1 a )] (7) + (sb j (sa j Hece, by (7) ad (8),,i j,i j (b i + s x)) (a i + s x)) = [ 8s(a 1 a )]. (8) [( + s x) sb j (,i j [( + s x) sa j ( t,i j (b i + s x ) (b i + s x )))] (a i + s x ) (a i + s x )))] = ( + s 1 x + a i )[ 8s(a 1 a )]. Thus by Defiitio 3. ( + s x) sb j (,i j ( + s x) sa j (,i j (b i + s x ) (b i + s x ))) (a i + s x ) (a i + s x ))). Hece we have fiished the proof of the Theorem. Lemma 3.6([16]) Let G be a o-complete coected graph of order ad e E(G). The σ(q(g + e), x) = det(xi Q(G + e)) det(xi Q(G)) = σ(q(g), x), where G deote the complemet of a graph G. Lemma 3.7([17]) Let G be a graph with vertices ad m edges. The σ(s(g), x) = x m σ(q(g), x ) = x m det(x I Q(G)). Now we preset our mai result. Theorem 3.8 Let G be a coected graph with vertices ad coectivity κ. The σ(q(g), x) (x + ) κ 1 (x + 3) κ [( + κ x)(κ x)( 4 κ x) κ( κ 4 x) κ( κ 1)(κ x)]. The equality holds if ad oly if G = K κ (K 1 K κ 1 ). Proof Let G 0 be a graph havig the maximal coefficiets of the sigless Laplacia characteristic polyomial amog all coected graphs of order with coectivity κ. The there is a vertex subset X 0 V (G 0 ) ad V 0 = κ such that G 0 V 0 = G 1 G... G t, where G 1, G,..., G t (t ) coected compoets of G 0 V 0. By Lemma 3.6, t =, G 1 ad G ad G[U] are complete, ad each vertex of G 1 ad G is adjacet to each vertex x i V 0. Let i = G i for i = 1,. The G 0 = K κ (K 1 K ) ad 1 + = κ. Assume that 1. By Corollary., P Q (G 0, x) = (x + ) κ 1 [( + κ x) κ j,i j (x κ i + ) i 1 ( i + κ x) ( i + κ x))]. If 1 κ, the by Lemma 3.4, (x +3) κ (x κ a+) a 1 (x +a+) κ a 1. (Advace olie publicatio: 7 November 018)

4 By Theorem 3.5 ( + κ x)(κ x)( 4 κ x) κ( κ 4 x) κ( κ 1)(κ x)] ( + κ x) (a i + κ x ) κa j (,i j (a i + κ x ))), where b 1 = 1, b = = κ 1. Hece, we fid the coefficiets of P Q (G 0, x) with 1 + = κ ad 1 is maximum if ad oly if 1 = 1, = κ 1. It follows that G 0 = K κ (K 1 K κ 1 ), the σ(q(g 0 ), x) (x + ) κ 1 (x + 3) κ [( + κ x)(κ x)( 4 κ x) κ( κ 4 x) κ( κ 1)(κ x)]. This completes the proof of Theorem. Lemma 3.9([14]) Let G 1 ad G be two bipartite graphs with 1 ad vertices, respectively. For ay two positive itegers p 1 ad p satisfyig 1 + p 1 = + p, the x p1 σ(g 1, x) x p σ(g, x) E(G 1 ) E(G ); x p1 σ(g 1, x) x p σ(g, x) E(G 1 ) E(G ). By Lemma 3.7 ad Theorem 3.8 ad Lemma3.9, the followig result is obvious. Theorem 3.10 Let G be a simple graph of order whose coectivity is κ. The IE(G) IE(K κ (K 1 K κ 1 )), with equality holds if ad oly if G = K κ (K 1 K κ 1 ). Theorem 3.11 Let G be a coected graph with vertices ad edge coectivity κ. The σ(q(g), x) (x + ) κ 1 (x + 3) κ [( + κ x)(κ x)( 4 κ x) κ ( κ 4 x) κ ( κ 1)(κ x)]. The equality holds if ad oly if G = K κ (K 1 K κ 1). Proof Let G be a coected graph of order with coectivity κ ad edge-coectivity κ, the κ κ. Note that (x + ) κ κ (x + 3) κ κ ad [( + κ x)(κ x)( 4 κ x) κ( κ 4 x) κ( κ 1)(κ x)] [( + κ x)(κ x)( 4 κ x) κ ( κ 4 x) κ ( κ 1)(κ x)]. By Theorem 3.8, σ(q(g), x) (x +) κ 1 (x +3) κ [(+κ x) (κ x)( 4 κ x) κ( κ 4 x) κ( κ 1)(κ x)] (x +) κ 1 (x +3) κ [(+κ x)(κ x) ( 4 κ x) κ ( κ 4 x) κ ( κ 1)(κ x)]. Ad the equality holds if ad oly if G = K κ (K 1 K κ 1 ) ad κ = κ, that is G = K κ (K 1 K κ 1). The result follows. Similarly Theorem 3.10, we get the followig result. Theorem 3.1 Let G be a simple graph of order whose edge coectivity is κ. The IE(G) IE(K κ (K 1 K κ 1), with equality holds if ad oly if G = K κ (K 1 K κ 1). IV. CONCLUSION I this paper, we use quasi-order relatio to compare the icidece eergy of two graphs with respect to the coectivity ad edge coectivity ad further obtai the maximal icidece eergy amog all coected graphs with give coectivity ad edge coectivity. ACKNOWLEDGMENT The author is very grateful to the aoymous referees for their carefully readig the paper ad for costructive commets ad suggestios which have improved this paper. REFERENCES [1] M. Jooyadeh, D. Kiai, ad M. Mirzakhah, Icidece eergy of a graph, MATCH. Commu. Math. Comput. Chem., vol. 6, o. 3, pp , 009. [] I. Gutma, D. Kiai, ad M. Mirzakhah, O icidece eergy of graphs, MATCH. Commu. Math. Comput. Chem., vol. 6, o. 3, pp , 009. [3] I. Gutma, D. Kiai, M.Mirzakhah ad B.Zhou, O icidece eergy of a graph, Liear Algebra Appl., vol. 431, o. 8, pp , 009. [4] D.M. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs, Academic Press, New York, [5] I. Gutma, Acyclic systems with extremal Huckel π-electro eergy, Theoret. Chim. Acta (Berli), vol. 45, o., pp , [6] I. Gutma, O. E. Polasky, Mathematicl Cocepts i Orgaic Chemistry, Spriger, Berli, [7] N. Biggs, Algebraic Graph Theory, d ed, Cambridge, Cambridge Uiversity Press, [8] O. Rojo, E. Lees, A sharp upper boud o the icidece eergy of graphs i terms of coectivity, Liear Algebra Appl., vol. 438, o. 3, pp , 013. [9] B. Zhou ad N. Triajstić, The Kirchhoff idex ad the matchig umber, It. J. of Quatum Chem., vol. 109, o. 13, pp , 009. [10] K.X. Xu, K.C. Das, Extremal Laplacia-eergy-like ivariat of graphs with give matchig umber, Electro Joural of Liear Alg. Appl., vol. 6, o. 1, pp , 013. [11] G. Zhog, M.X. Yi, ad J. Huag, Numerical method for solvig fractioal covectio diffusio equatios with time-space variable coefficiets, IAENG Iteratioal Joural of Applied Mathematics, vol. 48, o. 1, pp. 6-66, 018. [1] M. Asgari, Numerical solutio for solvig a system of fractioal itegro-differetial equatios, IAENG Iteratioal Joural of Applied Mathematics, vol. 45, o., pp , 015. [13] W. Qiu, W.G. Ya, Coefficiets of Laplacia characteristi polyomials of graphs, Liear Algebra Appl., 436 (01), vol. 38, o. 4, pp , 015. [14] M.L. Hu, W.G. Ya, W. Qiu, Maximal eergy of subdivisio of graphs with a fixed chromatic umber, Bull. Malay. Math. Sci. Soc., vol. 38, o. 4, pp , 015. (Advace olie publicatio: 7 November 018)

5 [15] S.Y. Cui, G.X. Tia, The spectrum ad the sigless Laplacia spectrum of coroae, Liear Algebra Appl., vol. 437, o. 7, pp , 01. [16] C. McLema, E. McNicholas, Spectra of coroae, Liear Algebra Appl., vol. 435, o. 5, pp , 011. [17] S. Shioda, O the characteristic polyomial of the adjacecy matrix of the subdivisio graph of a graph, Discrete Appl. Math., vol., o. 4, pp , [18] J.B. Zhag, H.B. Zhag ad X.D. Liu, Graphs with extremal icidece eergy, Filomat, vol. 9, o. 6, pp , 015. (Advace olie publicatio: 7 November 018)

Research Article The Average Lower Connectivity of Graphs

Applied Mathematics, Article ID 807834, 4 pages http://dx.doi.org/10.1155/2014/807834 Research Article The Average Lower Coectivity of Graphs Ersi Asla Turgutlu Vocatioal Traiig School, Celal Bayar Uiversity,