ARTICLE IN PRESS Discrete Applied Mathematics ( )

Transcription

1 Discrete Applied Mathematics Cotets lists available at ScieceDirect Discrete Applied Mathematics joural homepage: Average distaces ad distace domiatio umbers Fag Tia a, Ju-Mig Xu b, a Departmet of Applied Mathematics, Shaghai Uiversity of Fiace ad Ecoomics, Shaghai, 004, Chia b Departmet of Mathematics, Uiversity of Sciece ad Techology of Chia, Hefei, 006, Chia a r t i c l e i f o a b s t r a c t Article history: Received 8 May 005 Received i revised form 7 February 008 Accepted 9 March 008 Available olie xxxx Keywords: Distace Average distace Diameter Domiatio umber Distace domiatio umber Let be a positive iteger ad G be a simple coected graph with order. The average distace µg of G is defied to be the average value of distaces over all pairs of vertices of G. A subset D of vertices i G is said to be a -domiatig set of G if every vertex of VG D is withi distace from some vertex of D. The miimum cardiality amog all -domiatig sets of G is called the -domiatio umber γ G of G. I this paper tight upper bouds are established for µg, as fuctios of, ad γ G, which geeralizes the earlier results of Daelma [P. Daelma, Average distace ad domiatio umber, Discrete Appl. Math ] for = Elsevier B.V. All rights reserved. 1. Itroductio For termiology ad otatio o graph theory ot give here, the reader is referred to [18]. Let G = V, E be a fiite simple coected graph with vertex set V = VG ad edge set E = EG. The distace d G x, y betwee two vertices x ad y is the legth of a shortest xy-path i G. For S VG, G[S] deotes the subgraph of G iduced by S ad for v VG, d G v, S = mi{d G v, u : u VS}. The eccetricity e G v of v is max{d G v, x : x VG}. The radius radg ad the diameter diamg of G are the smallest ad the largest eccetricities of the vertices i G, respectively. A vertex with e G v = diamg is called a diametral vertex. A vertex v is a cetral vertex if e G v = radg ad the ceter of G is the set of all cetral vertices. The degree of a vertex x VG, deoted by deg G x, is the umber of edges icidet to the vertex x. A vertex of degree oe is called a ed-vertex. Let P deote a path of order ad P xy a path with ed-vertices x ad y. If the legth of a path P xy is equal to diamg, the we call P xy a diametral path i G. The average or mea distace of G is defied to be the average over all pairs of vertices of G, i.e., µg = 1 1 d G x, y. x,y V Lie diameter, Wieer idex [1,17] or other parameters, apart from their ow graph-theoretic iterests, the average distace has umerous applicatios i aalyzig problems i commuicatio etwors, geometry ad physical chemistry. It is the reaso why this cocept has received cosiderable attetio i the literature. There are several excellet surveys of earlier results o average distace of graphs, oe of which is due to Plesi [15]. Thus, may efforts have bee made by several authors to establish the relatioships betwee average distace ad other graph parameters see, for example, [1,, The wor was supported by NNSF of Chia No Correspodig author. address: xujm@ustc.edu.c J.-M. Xu X/$ see frot matter 008 Elsevier B.V. All rights reserved.

2 F. Tia, J.-M. Xu / Discrete Applied Mathematics 6 8,15,16]. For coveiece, let σx = σx, G = y V d G x, y, σg = σx = x V d G x, y, x,y V V be the trasmissio of a vertex x V, ad the trasmissio of the graph G, respectively. I order to avoid large fractios, we will ofte deal with σg rather tha µg. Apart from average distace, σg also occurs i the computatio of other graph-theoretical parameters, such as the forwardig idex of a routig [5,1], ad physical chemistry [9]. A subset I of vertices i G is said to be -idepedet if every vertex i I is at distace at least + 1 from every other vertex of I i G. The -idepedece umber of G, deoted by α G, is defied to be the maximum cardiality amog all -idepedet sets of G. If = 1, α 1 G is αg, the idepedece umber of G. Daelma, Oellerma ad Swart [7] gave the bouds o the average distace with order ad idepedece umber αg. Firby ad Havilad [8] established sharp lower bouds for the average distace of G, i terms of the -idepedece umber α G, ad described the associated extremal graphs, thereby extedig the aforemetioed wor of Daelma et al. for = 1. A subset D of vertices i G is said to be a -domiatig set of G if every vertex of VG D is withi distace from some vertex of D. The miimum cardiality amog all -domiatig sets of G is called the -domiatio umber of G ad is deoted by γ G. For the special case of = 1, γ 1 G is the classic domiatio umber of G. The cocept of -domiatig set was itroduced by Chag ad Nemhauser [,4] ad fids applicatios i may situatios ad structures which give rise to graphs, see the boos by Hayes, Hedetiemi ad Slater [10,11]. Daelma [6] gave the sharp upper bouds o the average distace of a graph of give order ad domiatio umber γg, ad determied the extremal graphs. I this paper, by geeralizig Daelma s techique, we establish the sharp upper bouds o the average distace of G, i terms of -domiatio umber γ G, ad describe the extremal graphs, extedig the results of Daelma for = 1 i [6]. The proofs of our mai results are i Sectio ad some lemmas are give i Sectio.. Lemmas Lemma.1. Let G be a otrivial coected graph, ad be a positive iteger. The γ G = mi γ T, where the miimum is tae over all spaig trees T of G. Proof. Let G be a otrivial coected graph ad T be a spaig tree of G. The ay -domiatig set of T is also a - domiatig set of G. Therefore, γ G γ T. Thus we have that γ G mi γ T, where the miimum is tae over all spaig trees T of G. Now we show the reverse iequality. If G is a tree, the the theorem holds trivially. So we may assume that G is a coected graph cotaiig cycles. Let D be a miimum -domiatig set of G ad C be a cycle i G. If we ca prove that D is also a - domiatig set of G e for some cycle edge e EC, the γ G e D = γ G. By iteratig the above operatio fiitely, we get γ T γ G for some spaig tree T of G. Thus, we have that mi γ T γ G, where the miimum is tae over all spaig trees T of G. If VC VD, the obviously the vertices i VG D are also all withi distace to G[D] e for ay edge e EC. If VC VD, the we select two adjacet vertices x ad y i VC such that d G x, D + d G y, D = max{d G u, D + d G v, D : uv EC}. Now we will show that D is also a -domiatig set of G {xy}. First for ay two adjacet vertices u ad v i G, we have d G u, D d G v, D 1. The if w is a vertex i VC such that d G w, D = max{d G v, D : v VC}, we have that w = x or w = y. Without loss of geerality, suppose that d G x, D = max{d G v, D : v VC}. Let z be aother eighbor of x differet from y i VC. So we immediately have that d G z, D d G y, D. Thus, we get the distace betwee a vertex i VG D ad D is ot iflueced by deletig the edge {xy}. That is to say, d G xy v, D = d G v, D for all vertices v i VG. Hece, D is also a -domiatig set of G e for some cycle edge e. From Lemma.1, we get that every coected graph G cotais a spaig tree T with the same -domiatio umber. That is to say, every extremal graph G with give order, -domiatio umber ad maximum average distace is a tree. So we have to cosider oly trees below. Let S deote a -geeralized star which is a tree cotaiig oe vertex whose eccetricity is at most. Lemma.. Let H be a graph. The γ H e > γ H for each edge e EH if ad oly if H is the uio of several vertex disjoit -geeralized stars S. Proof. Let H be a graph such that γ H e > γ H for each edge e EH, ad D be a miimum -domiatig set of H. If γ H = 1, by Lemma.1 ad the property γ H e > γ H for each edge e EH, the H must be a tree ad we ca easily see that H must be a -geeralized star S. If γ H, the for ay two vertices x ad y i D, we have d H x, y + 1. Otherwise, if d H x, y, the there must exist a edge e o the shortest path betwee x ad y i H such that γ H e = γ H. We partitio the graph H ito balls of radius, deoted H 1, H,..., H γ, whose ceters are the vertices i D.

3 F. Tia, J.-M. Xu / Discrete Applied Mathematics The the balls are all disjoit. Furthermore, there are o edges joiig ay two balls. Otherwise, if there exist a edge e joiig two balls, say H i ad H j, the γ H e = γ H, a cotradictio to the hypothesis o H. Ad sice H i is a iduced subgraph of H with γ H i = 1, by the aforemetioed, H i is a -geeralized star S. The coverse is easily verified. Corollary.. If G is a tree with γ G, the there exists a edge e i a diametral path i G such that γ G e = γ G. Proof. Let D be a miimum -domiatig set of G. We partitio the graph G ito balls of radius, deoted G 1, G,..., G γ, whose ceters are the vertices i D. Sice G is coected, there must exist a edge e joiig two such balls. The e must be i a diametral path i G, ad its deletio does ot chage γ G. Lemma.4 Meir ad Moo [14]. γ G +1 for ay coected graph G of order with + 1. Defiitio.5. Suppose that +1 < γ, ad let s ad t be, respectively, the quotiet ad the remider of the +1 divisio of + 1γ by, amely + 1γ = s + t, where s 0 ad 0 t 1. I particular, for = 1 we get t = 0, ad cosequetly s = γ. The we defie the followig umbers: A = s + 1 t, B = s + 1, C = s + 1, D = s + 1 t. As fuctios of γ, we have A = γ + t, B = γ t, C = γ + + 1t, D = γ 1t. Note that, for = 1, A, B, C ad D tae the followig values, which appear i the results by Daelma [6]: A = 6γ, B = 6γ = γ, C = 5 6γ, D = 4 6γ = γ. Defiitio.6. For give positive itegers ad γ, a class of graphs H,γ is defied as follows. i If γ, +1 H,γ cosists of a sigle path P +1γ 1 = v 1, v,..., v +1γ 1 ad idepedet vertices w 1, w,..., w +1 +1γ that are joied with v +1γ 1 see Fig. 1. ii If γ =, H +1,γ is a sigle path P = v 1, v,..., v. iii If +1 < γ, H +1,γ is obtaied from a sigle path P +1γ + = v 1, v,..., v +1γ + by attachig exactly oe P to the vertex v i for A + + i + 1γ + ; ad attachig exactly oe P t to the vertex v A+t+1 see Fig.. The reaso for the differet shapes of the extremal graphs for γ ad γ +1 > is the fact that the path P +1 is the uique graph of order with the maximum trasmissio of a vertex, which has the -domiatio umber γ = Lemma.7. Let G be a tree with order ad -domiatio umber γ [ + 1γ 1] + 1 γ, γ σv, G + 1 ; 1, γ =. + 1 The equality holds if ad oly if G = H,γ ad v = v The, for each vertex v VG, +1.

4 4 F. Tia, J.-M. Xu / Discrete Applied Mathematics Fig. 1. The extremal graph H,γ with 1 γ +1. Fig.. The extremal graph H,γ with +1 < γ Proof. It is well ow that i a tree, each vertex havig maximum trasmissio is a ed-vertex, i.e., a vertex with degree oe see [19] by Zelia. Thus, we oly prove this result for a diametral vertex. Let P be a diametral path, ad let D be a miimum -domiatig set of G. Sice every vertex of D ca -domiate at most + 1 vertices of P, we have diamg + 1γ 1, ad thus we have σv, G γ + [ + 1γ 1] + 1γ + 1 = [ + 1γ 1] + 1 γ. The uiqueess of the extremal graph is obvious. For γ =, the result follows immediately. +1 Lemma.8. Let G be a tree of order ad -domiatio umber γ > diamg + 1γ + 1., the +1 Proof. The proof proceeds by iductio o. For +, by < γ +1, the value of γ +1 is small. Thus, it is easy to verify that the statemet holds for all graphs with maximum diameter ad -domiatio umber at least γ. For γ =, we see that the path P also satisfies this statemet. So, we cosider this statemet as γ 1. Suppose that the statemets hold for all trees of order less tha. Let G be a tree with maximum diameter amog all trees of order ad -domiatio umber at least γ. Let x ad y be two vertices i G such that d G x, y = diamg, ad let P xy be a diametral path, P xx = x, x 1,..., x be a subpath of P xy. First we have degx i = for i = 1,...,. Otherwise, if degx j for some 1 j, the x j must be adjacet to a vertex x j ot o P xy. Let G = G x j 1 x j + x j 1 x j. Thus diamg = diamg + 1 ad γ G γ G, a cotradictio to the choice of G. Let P = c 1, c,..., c be a subpath of P xy {x, x 1,..., x } such that c 1 is adjacet to the vertex x o P xy. Case 1. degc i = for i = 1,,...,. The G {x, x 1,..., x, c 1,..., c } is coected, has -domiatio umber γ G 1 ad has diameter at least diamg + 1. Sice γ G 1 > +1 +1, by the iductio hypothesis, we have diamg diam G {x, x 1,..., x, c 1,..., c } γ = + 1γ + 1. Case. degc i for some i = 1,,...,. Let c i be the earest vertex to x o P xy such that degc i. Let d deote oe vertex farthest from c i ot o P xy ad c i be the vertex adjacet to c i o P ci d, the we have i < d G c i, d. I fact, if d G c i, d i, the d G x, d. Let G = G c i c i+1 + c i+1 c i, the diamg = diamg + 1 ad γ G = γ G, a cotradictio to the choice of G

5 But if d G c i, d >, let G = G c i 1 c i + c i 1 c i, F. Tia, J.-M. Xu / Discrete Applied Mathematics 5 the diamg = diamg + 1, γ G = γ G, a cotradictio to the choice of G. Thus γ G {x, x 1,..., x, c 1,..., c i 1 } = γ G 1, ad γ G 1 > +i. The diameter of G +1 {x, x 1,..., x, c 1,..., c i 1 is the path P dy, thus, by i < d G c i, d, diamg diamg {x, x 1,..., x, c 1, c,..., c i 1 } i 1. By the iductio hypothesis, we have diamg diam G {x, x 1,..., x, c 1, c,..., c i 1 } + i 1 = + i + 1γ i 1 = + 1γ + 1. Lemma.9. Let G be a tree of order ad -domiatio umber γ > if 1mod, the σv, G 1 i= For each vertex v VG, + 1γ + + i A + + i 1 A + A At + t + t ;.1 here, let 1 A + + i = 0 if = 1. if 0mod, the σv, G + 1γ + + i A + + i i= γ γ + + At + t + t.. The equality holds if ad oly if G = H,γ ad v = v 1. Proof. The proof proceeds by iductio o. For +, by < γ +1, it ca verify that +1 v = v 1 H,γ is the vertex satisfyig σv, H,γ gettig the maximum. Thus, the statemet holds by some calculatios. Let + 4. Suppose that the statemet holds for all trees of order less tha. Now let G be a tree ad v VG such that σv, G is maximum amog all trees of order ad -domiatio umber at least γ. Sice each vertex havig maximum trasmissio i a tree is a ed-vertex, we ca assume that v is a diametral vertex. Let u be a eccetric vertex of v with d G u, v = diamg ad P uv be a diametral path i G. The u must be a ed-vertex of P uv ad the eighbor u 1 is uique. By γ >, we +1 get + 1γ Hece, diamg + 1γ + 1. Sice u must be withi distace from some vertex of G, ad we aim to get a upper boud for σv, G, the we ca assume the existece of a subpath P uu = u, u 1, u,..., u of P uv. By the choice of u, we have deg G u i = for all i = 1,,...,. Otherwise, if u i is adjacet to aother vertex u ot o i P uv, the G = G u i 1 u i + u i 1 u satisfies γ i G γ G ad σv, G > σv, G. Hece G {u, u 1,..., u } is coected ad has -domiatio umber at least γ G 1. By the iductio hypothesis ad by Lemma.8, we have that, for 1mod, σv, G σv, G {u, u 1,..., u } + + 1γ γ γ + 1 σv, H +1,γ γ γ γ + 1 = 1 i=0 ad for 0mod, 1 + 1γ + + i A + + i 1 A + A At + t + t,. σv, G σv, G {u, u 1,..., u } + + 1γ γ γ + 1 σv, H +1,γ γ γ γ + 1 = i=0 + 1γ + + i A + + i γ γ + + At + t + t..4

6 6 F. Tia, J.-M. Xu / Discrete Applied Mathematics Fig.. The extremal graph G,γ with 1 γ It remais to prove the uiqueess of the extremal graph. If the equality holds i.1 or., the it also holds i. or.4. By the iductio hypothesis, we have that G {u, u 1,..., u } = H +1,γ 1, ad v = v 1. Notice that the vertices u, u 1,..., u are exactly at distace + 1γ + 1, + 1γ +,, + 1γ + 1 from v 1, which implies that G = H,γ ad v = v 1. From Lemmas.7 ad.9, we get the followig corollary. Corollary.10 Daelma, Lemma i [6]. Let G be a tree of order ad domiatio umber γ. The, for each vertex v VG, γ 1 σv, G γ, if γ ; γ γ + 6γ +, if γ >. The equality holds if ad oly if G = H,γ ad v = v Mai results Now we prove the followig sharp upper bouds o the average distace of a graph with give order ad -domiatio umber γ. The shape of the extremal graphs also differs depedig o γ, γ +1 = or γ +1 >. We will treat +1 the three cases separately. Defiitio.1. For positive itegers ad γ, a class of graphs G,γ is defied as follows. i If γ, the +1 G,γ is obtaied from a sigle path P +1γ with ed-vertices v 1 ad v, ad two idepedet sets of vertices W 1 ad W of order +1γ + ad +1γ +, by joiig each vertex of W i to v i, where i = 1, see Fig.. ii If γ =, the G,γ is a sigle path P = v 1, v,..., v. +1 Theorem.. Let G be a coected graph of order ad -domiatio umber γ +1. The we have γ + 1γ γ 7, 6 1 if γ + 1 ad γ is eve; γ + 1γ γ 7 + 1γ, µg 6 1 if γ + 1 ad γ is odd; + 1, if γ =. + 1 The equality holds if ad oly if G = G,γ.

7 F. Tia, J.-M. Xu / Discrete Applied Mathematics 7 Fig. 4. The structure of G i Claim. Proof. The proof proceeds by iductio o. It is easy to chec that the statemet holds for 4 +. Assume that the statemet holds for all values smaller tha. We will prove the statemet for a fixed 4+ by iductio o γ 1. Clearly, it holds for γ = 1, so we may assume γ. If γ = or γ +1 =, the the result follows immediately from the fact that the graph +1 G, or G, is isomorphic to the graph P. So let γ <. +1 Let G be a coected graph of order ad -domiatio umber γ that has maximum trasmissio. The G is a tree by Lemma.1. Sice γ, by Corollary., we ca choose a edge xy i a diametral path P, whose deletio does ot chage γ G. Let G x ad G y deote the compoets of G xy that cotais x ad y, respectively. Sice 4 +, we also ca choose the edge xy such that G x ad G y cotai at least + 1 vertices, respectively. Claim 1. There exists oe vertex at distace from the vertices x ad y i G x ad G y, respectively. Proof. Without loss of geerality, we oly prove the statemet for G x. Suppose that every vertex of G x is at distace less tha to x. The {x} is a miimum -domiatig set for G x. Tae the farthest vertex x of x i G x o the path P, ad let P xx deote the path betwee x ad x o P i G x. Sice G x + 1, there exists a vertex x 1 o P xx such that degx 1. Suppose that x is the eighbor of x 1 which is earer to x o P xx, ad x is a eighbor of x 1 ot o P xx. Let G = G x 1 x + x x. Thus, we have G = x G x x 1 x + x x ad γ G = γ G. Sice σg > x σg x, σx, G > σx, x G x, ad σg = a,b VG x + a,b VG y + a VGx b VGy d Ga, b = σg x + σg y + VG x VG y + VG y σx, G x + VG x σy, G y, we have σg > σg, a cotradictio to the maximality of σg. P yy By Claim 1, there exist paths of legth which belogs to P i G x ad G y, deoted by P xx = y, y 1,..., y 1, y, respectively. = x, x 1,..., x 1, x ad Claim. deg x =, deg y =, degx i = ad degy i = for i = 1,..., 1. Proof. We first prove that deg x =. Suppose that deg x ad let x deote a eighbor of x ot o P. Let G = G xy+x y. By the same proof i Claim 1, we will get γ G = γ G ad σg > σg, a cotradictio to the maximality of σg. Similarly, we ca prove deg y =. Thus, x ad y are diametral vertices i G x ad G y, respectively. By γ G xy = γ G, we have x ad y must be i a miimum -domiatig set of G. Now, we prove degx i = for i = 1,..., 1. Let x i be the earest vertex to x o P xx such that degx i ad y j be the earest vertex to y o P yy such that degy j. Without loss of geerality, we assume that i j. Let x be a vertex farthest from i x i ot o P xx. Sice x is a diametral vertex i G x, we have d G x i, x i i. Let M deote the vertices i all coected compoets of G x i which cotai o vertices o P xx, ad let N deote the set of vertices adjacet to x i i M see Fig. 4. If VG x > VG y + M, the let G = G x i N + y i N. By d G x i, x i, γ i G = γ G ad σg σg = d G a, b d G a, b a M b VG a M M b VG M = M VG x M i i + 1 M VG y i i + 1 = M i + 1 VG x VG y M > 0, a cotradictio to the choice of G.

8 8 F. Tia, J.-M. Xu / Discrete Applied Mathematics If VG x VG y + M, the let G = G x i N + x i+1 N. By d G x i, x i, γ i G = γ G ad σg σg = d G a, b d G a, b a M b VG a M M b VG M = M VG y + i + 1 VG x M i 1 = M VG y VG x + M + i + > 0 a cotradictio to the choice of G. Hece, o matter VG x > VG y + M or VG x VG y + M, we get both the cotradictios with σg. Thus deg x i = ad deg y i = for i = 1,..., 1. By Claim, ow suppose that G is the graph obtaied from G by idetifyig the + vertices x, x 1,..., x 1, x ad y, y 1,..., y 1, y with a ew vertex z ad deletig loops. The, G has + 1 vertices ad γ G = γ G 1 satisfyig γ G VG. +1 Let X = VG x {x, x 1,..., x }, Y = VG y {y, y 1,..., y }, Z = {x,..., x 1, x, y, y 1,..., y }, p = VG x, q = γ G x. By the iductio hypothesis, we have, σg = + + d G a, b + σ G[Z] a,b X a X Y a,b Y a X,b Y + d G a, x i + d G a, x + d G a, y + d G a, y i = d G a, b + d G a, b + a,b X a,b Y a X,b Y d G a, b σ G[Z] + + 1d G a, x + d G a, z a X + + 1d G a, y + d G a, z a Y = d G a, b + d G a, b + d G a, b a,b X a,b Y + σ G[Z] + a X,b Y d G a, z a X Y a X d G a, y + 1 a Y d G a, x + 1 Let p +1. = σg X Y σx, G x + σy, G y X + Y = σg X Y σx, G x + σy, G y σ 1 G +1,γ [ p + 1 p σv 1, H p,q + σv 1, H p,γ q ]..1 Fp, q = p + 1 p σv 1, H p,q + σ v 1, H p,γ q. Case 1. q p +1 or γ q p. Without loss of geerality, we oly prove q +1 p +1, the γ q γ p +1

9 If q = p +1, the pp 1 Fp, q = p + 1 p F. Tia, J.-M. Xu / Discrete Applied Mathematics γ q 1 p + 1 γ q. Sice F p, q 1 > 0, the Fp, q p F + 1q, q. Sice G is a graph with -domiatio umber γ ad maximum trasmissio, the σg σg,γ. By.1 ad. ad γ <, we get a cotradictio as follows, 0 σ 1 G +1,γ F + 1q, q σ G,γ = γ < 0. If q > p, the σv +1 1, H p,q < σ v 1, H p, σ v 1, H p p,γ. Thus, Fp, q < F By γ q γ p p p, p ad we ca get the same cotradictio as above. +1 Case. q p ad γ +1 q p. +1 We have, [ ] + 1γ Fp, q = p + + 1q γ that is, + 1γ Fp, q F + 1q γ 4 if γ = is eve γ 4 if γ is odd. γ γ + + 1, γ γ, q γ ; γ ;. p, we have +1 σv 1, H p,γ q < Sice G is a graph with -domiatio umber γ ad maximum trasmissio, the σg σg,γ. We ca calculate σ G +1,γ 1 ad σg,γ by the shape of G,γ defied i Defiitio.1. By.1 ad., we obtai that, 0 σ 1 G +1,γ γ + F + 1q +, q σg σ 1 G +1,γ γ + F + 1q +, q σg,γ = 0. This yields σg = σg,γ. Now we oly eed to prove the uiqueess of the extremal graph G,γ. Sice equality σg = σg,γ implies the equality i., we have σg = σ G +1,γ 1, σx, Gx = σ v 1, H p,q ad σy, Gy = σ v 1, H p,γ q. By the iductio hypothesis, G = G +1,γ 1. By the uiqueess of Lemma.9, we get G x = H p,q ad G y = H p,γ q. It is easy to see that G,γ is the oly class of graphs satisfyig all of these properties ad the theorem holds. Corollary. Daelma, Theorem 1 i [6]. Let G be a coected graph of order ad domiatio umber γ. The we have + 1 γ γ + + γ 7, if γ is eve; µg + 1 The equality holds if ad oly if G = G,γ. 6 1 γ γ + + γ 7 9γ 1, if γ is odd

10 10 F. Tia, J.-M. Xu / Discrete Applied Mathematics Fig. 5. The extremal graph D,γ with +1 < γ +1. Defiitio.4. For positive itegers ad γ with +1 < γ +1, let s ad t be the quotiet ad the remider of the divisio of + 1γ by ad let A be defied as i Defiitio.5. Let D,γ be the graph obtaied from a sigle path P +1γ = v 1, v,..., v +1γ, by attachig exactly oe P to the vertex v i for 1 i s ; by attachig exactly oe P to the vertex v +1γ +1 j for 1 j s ; ad by attachig exactly oe path P t to the vertex v m+a+t see Fig. 5. Theorem.5. Let G be a graph of order with -domiatio umber γ > +1. Let s ad t be the quotiet ad the remider of the divisio of + 1γ by, amely + 1γ = s + t, where s 0 ad 0 t 1, ad assume A, B, C, D as i Defiitio.5. If γ t is eve, the µg + 1 If γ t B 6 1 [ + 1γ t C + 1 A + t 1 + t D ] t t 1 is odd, the µg + 1 B [ + 1γ t C t D + + D + 1t + 1] t t The equality holds if ad oly if G = D,γ...5 A + t..6 1 Proof. The proof proceeds by iductio o. Sice the bouds i.5 ad.6 are strictly decreasig i γ, it suffices to prove the statemets for all graphs with -domiatio umber greater tha or equal to a give umber γ. For +, by < γ +1, we ca see, by some calculatios, that the statemet holds. So we ca assume Let G be a coected graph of order ad -domiatio umber γ G γ with maximum trasmissio. By Lemma.1, G is a tree. Let P h P deote the graph obtaied by attachig a path P to each vertex of P h. We will first show that G cotais at least oe iduced subgraph isomorphic to P h P for some positive iteger h. The the graph obtaied by shriig P h P to P h 1 P will have -domiatio umber less tha γ G, to which the iductio hypothesis ca be applied. Let a ad b be two vertices of G such that d G a, b = diamg, ad let P ab deote a diametral path i G. Let P aa = a, a 1,..., a ad P bb = b, b 1,..., b be two subpaths of P ab. Note that they do ot overlap. Otherwise, we get γ G = 1, a cotradictio to γ > ad + 4. By the choice of the vertices a ad b, dega = degb = 1. Furthermore, +1 dega i = ad degb i =, for i = 1,,...,. Otherwise, if a i is adjacet to aother vertex a i ot o P ab, the G = G a i 1 a i + a i 1 a satisfies γ i G γ G ad σg > σg, a cotradictio to the assumptio of G. Let {c 1,..., c } ad {d 1,..., d } be two subpaths of P ab P aa P bb such that c 1 is adjacet to a, ad d 1 is adjacet to b. Note that they may overlap. Case 1. degc 1 = or degd 1 =. Without loss of geerality, we assume that degc 1 =. If degc i = for i = 1,,...,, let G = G {a, a 1,..., a, c 1,..., c }. The G is coected ad has -domiatio umber γ G 1 γ 1. Let V = VG. By Lemma.9 ad the iductio hypothesis, σg = + d G x, y x,y V x V x,y VG V y VG V + = σg + x V + 1d G a, x

11 F. Tia, J.-M. Xu / Discrete Applied Mathematics 11 Fig. 6. The structure of G as degc i for i {,..., } ad degd j for j {,..., }. = σg σa, G σd +1,γ σ a, H,γ Sice the assumptio o G, we have σg σ D,γ. By Lemma.9 ad.7, we get, 0 σ D,γ σ D +1,γ 1 + 1σ v1, H,γ t γ γ γ if γ t is eve; = t γ γ t γ if γ t is odd. Let F, γ be the latter expressio i.8. For costats ad γ G , we get df,γ dγ > 0. Whe γ G = ++1, the + 1γ +1 = + 1. Thus, we have t = 0 if = 1, ad t = 1 if 1. Hece, we have F, γ F, t 1 if γ t is eve; = t t 1 if γ.9 t is odd. Iequality.9 is correct oly i two cases. The first oe is whe γ t is eve, = + 1γ 1, ad t = 1. The secod oe is whe γ t is odd, = + 1γ 1, = 1 ad t = 0 or = ad t = 1. Thus, equality holds i.7 oly at the above two cases. The σa, G = σ a, H +1γ 1,γ ad σg = σd+1γ 1,γ. By Lemma.9, we have G = H +1γ 1,γ. Notice that H +1γ 1,γ = D +1γ 1,γ. Therefore, the result follows i this case. Suppose ow that degc i for some i {,..., }. Thus,. If degd j = for j = 1,...,, the the result follows with the same argumet as above. So we assume that degd j for some j {,..., } below. Let c i be the earest vertex to a o P ab such that degc i, ad d j be the earest vertex to b o P ab such that degd j. Let c ad d deote the vertices, ot o P ab, farthest from c i ad d j, respectively. With the same method employed i the proof of Lemma.7, we have i < d G c i, c ad j < d G d j, d. I fact, we ca also prove that d G c i, c = i+1. If d G c i, c i +, let c be the eighbor of c i o P ci c ad G = G c i c i 1 + c c i 1,the γ G = γ G ad σg > σg, a cotradictio to the assumptio of G. Similarly, let d be the eighbor of d j o P dj d, we ca prove d Gd j, d = j + 1. Furthermore, let M M deote all vertices i the coected compoets of G c i G d j which cotais o vertices i P ab. We ca prove G[M] = P c c ad G[M ] = P d d. Suppose that v VM P c c exists such that v is adjacet to some vertex i {c i } VP c c. If G = G c i 1 c i + c i 1 v + vc i, the we have γ G = γ G ad σg > σg, a cotradictio to the assumptio o G see Fig. 6. If c i = d j, the γ G = ad {a, b, c i } is a miimum -domiatig set for G. Sice G has the maximum trasmissio σg, we have i = j =. Hece, = 4 +. By + 1γ = + 1 = s + t ad, we have t = 1. The we have G = D 4+,. If c i d j, we ca calculate that σg < σd,γ by the defiitio of σg = u,v V V d Gu, v, see Figs. 5 ad 6. Case. Cosider ow the case degc 1 > ad degd 1 >. We first deal with the cases whe t = 0 ad o assumptio o γ t, or t 0 ad γ t is odd. Let w c be a eighbor of c 1 ot o P ab. The w c must be a ed-vertex of P with o vertices o P ab. I fact, suppose that c 1 is adjacet to a ed-vertex.8

12 1 F. Tia, J.-M. Xu / Discrete Applied Mathematics of some P l = w 1, c w,..., c wl, where l <. Let c G = G a c 1 + a w l, the γ c G γ G ad σg > σg, a cotradictio to the choice of G. Deote the path P by w 1, c w,..., c w. c Hece G cotais a iduced subgraph H 1 = G[{a, a 1,..., a, c 1, w 1,..., c w c }] with the followig properties H 1 is isomorphic to P h1 P for some h 1 ; P 1 : N G VH 1 = {u} for some vertex u VP h1 with degu = H 1 where N G VH deote the set of all vertices i H which are adjacet to some vertex i G H. Amog all iduced subgraphs H 1 of G with properties P 1, choose oe of maximum order. The the vertex u has two eighbors i H 1, oe ed-vertex u 1 of P = u 1, u,..., u ad oe vertex u with degree at least i H 1. Let Z deote the set of the remaiig eighbors of u i G H 1. We defie a ew graph G = G {u, u 1,..., u } + u Z, i.e., we delete the vertices {u, u 1,..., u } ad joi the eighbors of u i VG VH 1 to u. With X = VH 1 {u, u 1,..., u }, ad Y = VG VH 1, we have σg = + + d G x, y + x,y X x,y X x,y Y x,y Y x X,y Y x X,y Y x VG {u,u 1,...,u } d G u, x + d G u i, x = + + d G x, y + + 1h h 1 + x VG {u,u 1,...,u } d G u, x = σg + + 1h h σu, H 1 + σu, G X It is easy to chec that γ G = γ G 1 ad γ G > VG. By the iductio hypothesis ad +1 we have γ H 1 = h 1, γ G X = γ h VG X + 1, σg σ D +1,γ h h σ u, H +1h1,h σ u, H +1h1 ++1,γ h Let Fh 1 deote the latter expressio i.10. By.1 ad. i Lemma.9, we get σ u, H +1h1,h 1 ad σ u, H +1h1 ++1,γ h σ D +1,γ 1 is also obtaied from.5 ad.6 because of the iductio hypothesis. By replacig them ito Fh 1, we get the derivative of Fh 1 o h 1 as dfh 1 dh 1 = γ 0. That is, for costats ad γ, ad h γ t, Fh 1 is a decreasig fuctio o h 1 ad attais its maximum at h 1 = +1γ t. Thus, h 1 = B +1 if γ t is eve, ad h 1 = B if γ t is odd. By.1,.,.5,.6 ad.10, we see that the right-had side of.10 equals the value of σ D,γ, that is, + 1γ t σg F = σ D +1,γ 1 + 1

13 F. Tia, J.-M. Xu / Discrete Applied Mathematics 1 B B + + 1σ u, H B, B σ u, H B + ++1,γ +1 B +1 if γ t is eve; B B σ u, H B++1 B, σ u, H B ,γ +1 B + 1 if γ t is odd. = σ D,γ. Thus, as the cases whe t = 0 ad o assumptio o γ t, or t 0 ad γ t is odd, the iequality of Theorem.5 is proved. If the equality holds i Theorem.5, we have G = D +1,γ 1 ad H 1 = H +1h1,h 1 for h 1 = +1γ t. By the iductio hypothesis ad the uiqueess of Lemma.9, D,γ is the oly class of graphs satisfyig all of the above properties. Now we prove Theorem.5 whe γ t is eve ad t 0. By the same argumet as above we ca see that d 1 must be adjacet to a ed-vertex w 1 of a path d P = w 1, d w,..., d w d which has o vertices o P ab. Hece G cotais aother iduced subgraph H = G [{ b, b 1,..., b, d 1, w 1,..., d d}] w with the followig properties P. H is isomorphic to P h P for some h ; P : N G VH = {v} for some vertex v VP h with degv =. H I additio to the graph H 1, amog all subgraphs H of G with the property P, choose oe of maximum order. The the vertex v has two eighbors i H, oe ed-vertex v 1 of P = v 1,..., v ad oe vertex v with degree at least i H. Let Z deote the remaiig eighbors of v i G H. Let G = G {u, u 1,..., u } {v, v 1,..., v } + u Z + v Z, that is, we delete the vertices u, u 1,..., u ad v, v 1,..., v, joiig the eighbors of u ad v i VG VH 1 VH to u ad v, respectively. Let Let G = G H 1 H + {u, u 1,..., u } + {v, v 1,..., v }. X = VH 1 V{u, u 1,..., u }, Y = VH V{v, v 1,..., v }, W = VG VH 1 VH. By σg = x,y V V d Gx, y, we obtai, σg = = x,y X + x,y Y x X Y W x,y W x X,y Y + x X,y W + x Y,y W d G x, y d G u, x + d G u i, x + d G v, x + d G v i, x [ + x,y X + x X,y W x,y Y x {u,u 1,...,u }, y {v,v 1,...,v } + d G x, y + x,y W d G x, y x Y,y W d G x, y x X,y Y d G x, y + d G x, y d G u, x + d G u i, x + d G v, x + d G v i, x x W x {u,u 1,...,u } y {v,v 1,...,v } d G x, y + σg[w] σg[w] + + 1d G u, x + + 1d G v, x + 1 x X ]

14 14 F. Tia, J.-M. Xu / Discrete Applied Mathematics Fig. 7. The structure of G[W] as the equality holds i d G u, x + + 1d G v, x + 1 x Y = σg h 1 1h h 1 + h + 1 h 1 + h + σg + + 1d G u, x d G u, x + d + 1 x X d G v, x + d + + 1d G v, x + 1 σg[w] x Y = σg + σg σ u, H h1 +1,h σ v, H h +1,h h 1 1h h 1 + h d h 1 + h + 1 h 1 + h σg[w].11 where d = d G u, v. It is easy to chec that γ G = γ G ad γ G > VG +1 the iductio hypothesis ad.11, we have σg σ D +1,γ + σ D h1 +h +1,γ h 1 +h σ u, H h1 +1,h σ v, H h +1,h h 1 1h h 1 + h d ; γ G = γ G h 1 h + ad γ G > VG h 1 + h + 1 h 1 + h σg[w]..1 Let Fh 1, h deote the latter expressio i.1. The equality holds i.1 if ad oly if G = D +1,γ ad G = D h1 +h +1,γ h 1 +h. Thus, by the structure of D +1,γ or D h1 +h +1,γ h 1 +h, we get that the shape of G[W] see Fig. 7, ad d = h 1 + h t. By σg[w] = x,y VG[W] VG[W] d Gx, y ad Fig. 7, we have d 1dd tt + 1t 1 σg[w] = + + t t t + + td + t 1d + t + tt 1d By By.1,.1 ad d = h 1 + h t, we get that Fh 1, h h 1 < 0 ad Fh 1, h h < 0. That is, for costats ad γ, h γ t ad Fh 1, h attais its maximum at h 1 = +1γ t B h 1 = h = + 1 Now we ca get ad h 1 + h = B + 1. σ u, H h1 +1,h 1 = σ v, H h +1,h = σ v, H B, B +1 h 1 + 1γ t, ad h = +1γ t. Thus, by.1 ad. i Lemma.9; σ D +1,γ ad σ D h1 +h +1,γ h 1 +h = σ D B +1,γ B +1 +

15 by the iductio hypothesis; σg[w] by.1, ad d = h 1 + h t. F. Tia, J.-M. Xu / Discrete Applied Mathematics 15 Hece, we ca calculate that, for γ t eve ad t 0, + 1γ t σg F, + 1γ t = σ D +1,γ + σ D B +1,γ B +1 + σg[w] B B B t B σ v 1, H B, B σ v 1, H B +1, B B = σd,γ. The uiqueess of the graph ca be easily verified by the the iductio hypothesis. Corollary.6 Daelma, Theorem i [6]. Let G be a graph of order with domiatio umber γ. The + 1 γ γ 5 6γ 4, if γ is eve; µg γ 1γ 5 6γ + 6 γ 1, if γ is odd. 1 The equality holds if ad oly if G = D,γ. Acowledgemets The authors would lie to express their gratitude to the aoymous referees for their id suggestios ad useful commets o the origial mauscript, which resulted i this revised versio. Refereces [1] R.A. Beezer, J.E. Riegsecer, B.A. Smith, Usig miimum degree to boud average distace, Discrete Math [] D. Biestoc, E. Gyori, Average distace i graphs with removed elemets, J. Graph Theory [] G.J. Chag, -domiatio ad graph coverig problems, Ph.D. Thesis, School of OR ad IE, Corell Uiversity, Ithaca, NY, 198. [4] G.J. Chag, G.L. Nemhauser, The -domiatio ad -stability problems o su-free chordal graphs, SIAM J. Algebr. Discrete Methods [5] F.R.K. Chug, E.G. Coffma, M.I. Reima, B. Simo, The forwardig idex of commuicatio etwors, IEEE Tras. Iform. Theory [6] P. Daelma, Average distace ad domiatio umber, Discrete Appl. Math [7] P. Daelma, O.R. Oellerma, H.C. Swart, Average distace ad idepedece umber, Discrete Appl. Math [8] P. Firby, J. Havilad, Idepedece ad average distace i graphs, Discrete Appl. Math [9] I. Gutma, O.E. Polasy, Mathematical Cocepts i Orgaic Chemistry, Spriger, Berli, [10] T.W. Hayes, S.T. Hedetiemi, P.J. Slater, Fudametals of Domiatio i Graphs, Marcel Deer, New Yor, [11] T.W. Hayes, S.T. Hedetiemi, P.J. Slater, Domiatio i Graphs Advaced Topics, Marcel Deer, New Yor, [1] M.C. Heydema, J.C. Meyer, D. Sotteau, O forwardig idices of etwors, Discrete Appl. Math [1] H. Hosoya, Topological idex. A ewly proposed quatity characterizig the topological ature of structural isomers of saturated hydrocarbos, Bull. Chem. Soc. Jp [14] A. Meir, J.W. Moo, Relatios betwee pacig ad coverig umbers of a tree, Pacific J. Math [15] J. Plesi, O the sum of all distaces i a graph or digraph, J. Graph Theory [16] I. Tomescu, O the sum of all distaces i chromatic blocs, J. Graph Theory [17] H. Wieer, Structural determiatio of paraffi boilig poits, J. Am. Chem. Soc [18] J.-M. Xu, Theory ad Applicatio of Graphs, Kluwer Academic Publishers, Dordrecht, Bosto, Lodo, 00. [19] B. Zelia, Medias ad peripherias of trees, Arch. Math.Bro