Research Article The Average Lower Connectivity of Graphs

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1 Applied Mathematics, Article ID , 4 pages Research Article The Average Lower Coectivity of Graphs Ersi Asla Turgutlu Vocatioal Traiig School, Celal Bayar Uiversity, Turgutlu, Maisa 45400, Turkey Correspodece should be addressed to Ersi Asla; ersi.asla@cbu.edu.tr Received 14 August 2013; Revised 8 Jauary 2014; Accepted 9 Jauary 2014; Published 12 February 2014 Academic Editor: Bo-Qig Dog Copyright 2014 Ersi Asla. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. For a vertex V of a graph G, thelower coectivity, deotedbys V (G), is the smallest umber of vertices that cotais V ad those vertices whose deletio from G produces a discoected or a trivial graph. The average lower coectivity deoted by κ av (G) is the value ( V V(G) s V (G))/ V(G). It is show that this parameter ca be used to measure the vulerability of etworks. This paper cotais results o bouds for the average lower coectivity ad obtais the average lower coectivity of some graphs. 1. Itroductio I a commuicatio etwork, the vulerability parameters measure the resistace of the etwork to disruptio of operatio after the failure of certai statios or commuicatio liks. The best kow ad most useful measures of how well a graph is coected is the coectivity, defied to be the miimum umber of vertices i a set whose deletio results i a discoected or trivial graph. As the coectivity is the worst-case measure, it does ot always reflect what happes throughout the graph. Recet iterest i the vulerability ad reliability of etworks (commuicatio, computer, ad trasportatio) has give rise to a host of other measures, some of which are more global i ature; see, for example, [1, 2]. Let G be a fiite simple graph with vertex set V(G) ad edge set E(G). IthegraphG, deotes the umber of vertices. The miimum degree of a graph G is deoted by δ(g). A subset S V(G) of vertices is a domiatig set if every vertex i V(G) S is adjacet to at least oe vertex of S. The domiatio umber γ(g) is the miimum cardiality of a domiatig set. A subset S of V(G) is called a idepedet set of G if o two vertices of S are adjacet to G. A idepedet set S is maximum if G has o idepedet set S with S > S. The idepedece umber of G, α(g),is the umber of vertices i a maximum idepedet set of G. Heig [3] itroduced the cocept of average idepedece ad average domiatio. For a vertex V of a graph G, thelower idepedece umber, deoted by i V (G), isthe miimum cardiality of a maximal idepedet set of G that cotais V, ad the lower domiatio umber, deoted by γ V (G), is the miimum cardiality of a domiatig set of G that cotaisv.the average lower idepedece umber of G, deoted by i av (G),isthevalue( V V(G) i V (G))/ V(G) ad the average lower domiatio umber of G, deoted by γ av (G),is the value ( V V(G) γ V (G))/ V(G). Siceγ V (G) i V (G) holds for every vertex V, wehaveγ av (G) i av (G) for ay graph G. Also, it is clear that i(g) = mi{i V (G) V V(G)} ad γ(g) = mi{γ V (G) V V(G)} so γ(g) γ av (G) ad i(g) i av (G). The (u, V)-coectivity of G, deoted by κ G (u, V), is defied to be the maximum value of k for which u ad V are k-coected. It is a well-kow fact that the coectivity κ(g) equals mi{κ G (u, V)u, V V(G)}. I 2002, Beieke et al. [4]itroducedaparametertogivea more refied measure of the global amout of coectivity. If the order of G is, the the average coectivity of G, deoted by κ(g),isdefiedtobeκ(g) = ( u,v κ G (u, V))/ ( 2 ). The expressio u,v κ G (u, V) is sometimes referred to as the total coectivity of G.Clearly,for ay graph G, κ(g) κ(g). Therearealotofresearchesothecoectivityofagraph [5]. May works provide sufficiet coditios for a graph to be maximally coected [6 8]. The average coectivity has bee extesively studied [4, 9]. 2. The Average Lower Coectivity of a Graph We itroduce a ew vulerability parameter, the average lower coectivity. For a vertex V of a graph G, thelower

2 2 Applied Mathematics a c d O the other had, the average lower coectivity of G 1 ad G 2 is differet: b Figure 1:3-cycle G: with oe additioal vertex ad edge. κ av (G 1 ) = 1, 8, (2) κ av (G 2 ) = 1, 6. Thus, the average lower coectivity is a better parameter tha the coectivity ad average coectivity to distiguish these two graphs. The average parameters have bee foud to be more useful i some circumstaces tha the correspodig measures based o worst-case situatios. G 1 (a) G 2 (b) Theorem 2. Let G be a coected graph. The, κ av (G) <κ(g) +2. (3) Proof. For every vertex of G, s V (G) κ(g)+2.foratleastoe vertex V, s V (G) = κ(g).hece,theiequalityisstrict.the, κ av (G) <κ(g) +2. (4) Figure 2: The graphs G 1 ad G 2. Theorem 3. Let G be a coected graph. The, κ av (G) κ av (G+e). (5) coectivity, deoted by s V (G), isthesmallestumberof vertices that cotais V ad those vertices whose deletio from G produces a discoected or a trivial graph. We observe that (i) 1 s V (G) 1; (ii) s V (G) = 1 if ad oly if V is a cut vertex; (iii) s V (G) = 1 if ad oly if G V is complete. The average lower coectivity deoted by κ av (G) is the value ( V V(G) s V (G))/,where will deote the umber of vertices i graph G ad V V(G) s V (G) will deote the sum over all vertices of G.ForaygraphG, κ(g) = mi{s V (G) V V(G)} so κ av (G) κ(g).wealsoobservethat (i) κ av (G) 1; (ii) κ av (G) = 1 if ad oly if G is complete. Example 1. Let the graph G be 3-cycle with oe additioal vertex ad edge, as show i Figure 1. Itiseasytoseethat s a (G) = 2, s b (G) = 2, s c (G) = 1, ads d (G) = 3 ad we have κ av (G) = ( )/4 = 2. Let G 1 ad G 2 be graphs. Now oe ca ask the followig questio: is the average lower coectivity a suitable parameter? I other words, does the average lower coectivity distiguish betwee G 1 ad G 2? Proof. It is easy to see that s V (G) s V (G + e). Therefore, V V(G) s V (G) V V(G) s V (G+e). (6) So, we have κ av (G) κ av (G+e). (7) Theorem 4. Let G be a k-coected ad k-regular graph. The, κ av (G) =k. (8) Proof. The cardiality of s V (G)-sets is always the same for every vertex of ay graph G ad equals k.the,wehave κ av (G) = V V(G) s V (G) = k =k. (9) This meas that the proof is completed. It is obvious that we ca give the followig equality for the average lower coectivity of the cycle C. (i) The average lower coectivity of the cycle C is 2. Theorem 5. Let G be a coected graph. The, κ av (G) δ(g) +2. (10) Proof. For every vertex of G, s V (G) δ(g) + 2.Thus, For example, cosider the graphs i Figure 2. Itcabeeasilyseethatthecoectivityadaverage coectivity of these graphs are equal: κ (G 1 ) =κ(g 2 ) = κ(g 1 ) = κ(g 2 ) =1. (1) κ av (G) = V V(G) s V (G) κ av (G) δ(g) +2. (δ (G) +2), (11)

3 Applied Mathematics 3 3. Average Lower Coectivity of Several Specific Graphs I this sectio, we determie the average lower coectivity of several special graphs. Theorem 6. Let T be a tree with order. IfT has k vertices with degree 1, the κ av (T) = +k. (12) Proof. Assume that T has k vertices with degree 1 ad k vertices with degree at least 2. Let vertices set of T be V(T) = V(G 1 ) V(G 2 ) where i V(G 1 ) the set cotais k vertices with degree 1; i V(G 2 ) the set cotais kvertices with degree at least 2. If V V(G 1 ),thes V (G 1 )=2.Wehavetorepeat this process for k vertices with degree 1. If V V(G 2 ),the s V (G 2 )=1.Wehavetorepeatthisprocessfor kvertices with degree at least 2. Thus, we have κ av (T) = V V(T) s V (T) V (T) = k 2+( k) 1 Corollary 7. The average lower coectivity of (a) the path P is ( + 2)/; (b) the star K 1, 1 is (2 1)/; (c) the comet C t,r is (2r+t+1)/(t+r). = +k. (13) Theorem 8. Let K r,s be a complete bipartite graph. The κ av (K r,s )= { s 2 +sr+r, if s < r; { r+s { r, if r = s. (14) Proof. Let the partite sets of K r,s be R ad S with R = r ad S = s.wedistiguishtwocases. Case 1. If r=s,thebytheorem 4 we have κ av (K r,s )=r. Case 2 (r < s). Forx R, a miimum discoectig set of G that cotais x must be S {x},sos x (G) = s + 1. Othe other had, for y S, a miimum discoectig set of G that cotais y must be R,sos y (G) = r. Elemetary computatio yields the result. Defiitio 9. The wheel graph with 1spokes, W,isthe graph that cosists of a ( 1)-cycle ad oe additioal vertex, say u, that is adjacet to all the vertices of the cycle. I Figure 3,wedisplayW 7. Theorem 10. Let W be a wheel graph. The, κ av (W ) =3. (15) u Figure 3: The wheel graph W 7. u Figure 4: The gear graph G 6 Proof. The wheel graph W has vertices. The cardiality of s V (G)-setsisalwaysthesameforeveryvertexofayW ad equals 3. The, we have κ av (W )= V V(W ) s V (W ) This meas that the proof is completed. = 3 =3. (16) Defiitio 11. Thegeargraphisawheelgraphwithavertex added betwee each pair adjacet to graph vertices of the outer cycle. The gear graph G r has 2r+1 vertices ad 3r edges. I Figure 4 we display G 6. Theorem 12. Let G r be a gear graph. The, κ av (G r )= 5r + 3 2r + 1. (17) Proof. Let vertices set of G r be V(G r )=(H 1 ) V(H 2 ) V(H 3 ) where i V(H 1 ) the set cotais 1 vertex with degree, V(H 2 ) the set cotais vertices with degree 2, ad V(H 3 ) the set cotais vertices with degree 3.If V V(H 3 ),thes V (G) = 3. If V V(H 3 ),thes V (G 3 )=2. Therefore, κ av (G r )= V V(G r ) s V (G r ) 2r + 1 = (r+1) 3+2 r 2r + 1 = 5r + 3 2r + 1. Now we give the defiitio of Cartesia product. (18) Defiitio 13. The Cartesia product G 1 G 2 of graphs G 1 ad G 2 has V(G 1 ) V(G 2 ) as its vertex set ad (u 1,u 2 ) is adjacet to (V 1, V 2 )ifeitheru 1 = V 1 ad u 2 is adjacet to V 2 or u 2 = V 2 ad u 1 is adjacet to V 1. Coectivity of graph products has already bee studied by differet authors. I [10] itisprovedthatκ(g 1 G 2 ) κ(g 1 )+κ(g 2 ).

4 4 Applied Mathematics Theorem 14. Let G 1 ad G 2 be two coected graphs; the κ av (G 1 G 2 ) κ(g 1 G 2 ). (19) Proof. We kow κ av (G) κ(g). Therefore, we get κ av (G 1 G 2 ) κ(g 1 G 2 ). (20) Propositio 15. For positive iteger m 3, (i) κ av (K 2 P m )=2; (ii) κ av (K 2 C m )=3. [7] J. Fàbrega ad M. A. Fiol, Maximally coected digraphs, Graph Theory,vol.13,o.6,pp ,1989. [8] T.Soeoka,H.Nakada,M.Imase,adC.Peyrat, Sufficietcoditios for maximally coected dese graphs, Discrete Mathematics,vol.63,o.1,pp.53 66,1987. [9] P. Dakelma ad O. R. Oellerma, Bouds o the average coectivity of a graph, Discrete Applied Mathematics,vol.129, o. 2-3, pp , [10] W.-S. Chiue ad B.-S. Shieh, O coectivity of the Cartesia productoftwographs, Applied Mathematics ad Computatio, vol. 102, o. 2-3, pp , Propositio 16. Let r 3ad t 3be positive itegers. The 23, ifr=3, t=3; { 9 (i) κ av (P r P t )= 3 2 t, ifr=3, t 4; { 3 8 { r t, ifr 4, t 4. (ii) κ av (P r C t )= { 3, if r = 3, 4, t 3; { 4 4 { r, ifr 5, t 4. (iii) κ av (C r C t )=4. (21) Coflict of Iterests The author declares that there is ocoflict of iterests regardig the publicatio of this paper. Ackowledgmet The author wishes to thak Bo-Qig Dog for his helpful suggestios ad correctios. Refereces [1] K. S. Bagga, L. W. Beieke, R. E. Pippert, ad M. J. Lipma, A classificatio scheme for vulerability ad reliability parameters of graphs, Mathematical ad Computer Modellig, vol. 17, o. 11, pp , [2] O. R. Oellerma, Coectivity ad edge-coectivity i graphs: a survey, Cogressus Numeratium, vol.116,pp , [3] M. A. Heig, Trees with equal average domiatio ad idepedet domiatio umbers, Ars Combiatoria, vol. 71, pp , [4] L.W.Beieke,O.R.Oellerma,adR.E.Pippert, Theaverage coectivity of a graph, Discrete Mathematics,vol.252,o.1 3, pp.31 45,2002. [5] A. Hellwig ad L. Volkma, Maximally edge-coected ad vertex-coected graphs ad digraphs: a survey, Discrete Mathematics,vol.308,o.15,pp ,2008. [6] Y.-C. Che, Super coectivity of k-regular itercoectio etworks, Applied Mathematics ad Computatio, vol.217,o. 21, pp , 2011.

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