EXTREMAL PROPERTIES OF ZAGREB COINDICES AND DEGREE DISTANCE OF GRAPHS

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1 Miskolc Mathematical Notes HU e-issn Vol. 11 (010), No., pp ETREMAL PROPERTIES OF ZAGREB COINDICES AND DEGREE DISTANCE OF GRAPHS S. HOSSEIN-ZADEH, A. HAMZEH AND A. R. ASHRAFI Received 7 December, 009 Abstract. The degree distace, Zagreb coidices ad reverse degree distace of a coected graph have bee studied i mathematical chemistry. I this paper some ew extremal values of these topological ivariats over some special classes of graphs are determied. 000 Mathematics Subject Classificatio: 05C35, 05C1, 05A0, 05C05 Keywords: Zagreb coidex, Zagreb idex, degree distace, reverse degree distace 1. INTRODUCTION All graphs i this paper are fiite ad simple. A graph ivariat is a fuctio o a graph that does ot deped o the labelig of its vertices. Such quatities are also called topological idices. Two of these graph ivariats are kow uder various ames, the most commoly used oes are the first ad secod Zagreb idices. Due to their chemical relevace they have bee subject of umerous papers i the chemical literature [9,1,13,16,0], while the first Zagreb idex attracted a sigificat attetio of mathematicias. The problem of determiig extremal values ad the correspodig extremal graphs of some graph ivariats is the topic of several papers [, 4, 10, 13 15, 17, 0]. The aim of this paper is to ivestigate similar problems for a recetly itroduced geeralizatio of Zagreb idices. Let G be a coected graph with vertex ad edge sets V.G/ ad E.G/, respectively. For every vertex u V.G/, the edge coectig u ad v is deoted by uv ad deg G.u/ deotes the degree of u i G. The distace d G.u;v/ is defied as the legth of a shortest path coectig u ad v ad the diameter of G, diam G.G/, is the maximum possible distace betwee ay two vertices i the graph. We will omit the subscript G whe the graph is clear from the cotext. This research was i part supported by a grat from IPM (No ). c 010 Miskolc Uiversity Press

2 130 S. HOSSEIN-ZADEH, A. HAMZEH AND A. R. ASHRAFI ad The first ad secod Zagreb idices were origially defied as M 1.G/ D deg.u/ M.G/ D uv.g/ uve.g/ deg.u/ deg.v/; respectively. The first Zagreb idex ca be also expressed as a sum over edges of G, M 1.G/ = P uve.g/ Œdeg.u/ C deg.v/ : We refer the reader to [16] for the proof of this fact. The readers iterested i more iformatio o Zagreb idices ca be referred to [4, 9, 10, 1, 14, 16, 17] ad to the refereces therei. The Zagreb idices ca be viewed as the cotributios of pairs of adjacet vertices to certai degree-weighted geeralizatios of Wieer polyomials [7]. It tured out that computig such polyomials for certai composite graphs depeds o such cotributios from pairs of o-adjacet vertices. The first ad secod Zagreb coidices were first itroduced by Došlić. They are defied as follows: NM 1.G/ D Œdeg.u/ C deg.v/ ; NM.G/ D uv6e.g/ uv6e.g/ Œdeg.u/deg.v/ : Dobryi ad Kochetova [6] ad Gutma [8] itroduced a ew graph ivariat, degree distace, defied as follows: the degree distace of a vertex x, deoted by D 0.x/, is defied as D 0.x/ D D.x/deg.x/, where D.x/ D P yv.g/ d.x;y/ ad the degree distace of G, deoted by D 0.G/, is D 0.G/ D D 0.x/ D D.x/ deg.x/ D 1 xv.g/ x;yv.g/ xv.g/ d.x;y/œdeg.x/ C deg.y/ : If G is vertex graph the the reverse Wieer matrix is a matrix RW.G/ D ŒRW ij such that RW ij D diam.g/ d.v i ;v j /, if i j ad 0 otherwise. The reverse degree distace of G is defied as r D 0.G/ D P id1 deg.v i/ P j D1 RW ij [1]. The girth of G is the legth of a shortest cycle cotaied i G. A Moore graph is a graph of diameter k with girth k C 1. Those graphs have the miimum umber of vertices possible for a regular graph with give diameter ad maximum degree. For a real umber k, let P k.g/ deote the sum of the k th powers of the degrees of G. We deote by.;m/ the maximum value of P.G/ whe G is a graph (ot ecessarily coected) with vertices ad m edges. Also, let P.;m/ D max G} ;m P.G/ where } ;m deotes the family of coected graphs o vertices ad m edges.

3 ETREMAL PROPERTIES IN GRAPHS 131 For the sake of completeess we state here some results from the refereces [1, 3, 10, 11, 14, 17, 1] which will be useful throughout the paper. Lemma A (Ashrafi, Došlić ad Hamzeh [1]). Suppose G is a coected graph with exactly vertices ad m edges. The we have: (a) NM 1.G/ D m. 1/ M 1.G/, (b) NM.G/ D m 1 M.G/ M 1.G/. The subdivisio graph S.G/ of a graph G is obtaied by isertig a ew vertex of degree two o each edge of G. If G has vertices ad m edges, the S.G/ has Cm vertices ad m edges. Lemma B (Ilić ad Stevaović [10]). Suppose G is a graph with exactly vertices ad m edges. The (a) It holds that M 1.G/ 4m. The equality is attaied if ad oly if G is regular. (b) It holds that M.G/ 4m3. The equality is attaied if ad oly if G is regular. (c) Let be the maximum vertex degree i G. The M 1.G/ ad M.G/ m M 1.G/ m M 1.G/ m. Equality is attaied simultaeously i both iequalities if ad oly if G is regular. (d) M 1.S.G// D M 1.G/ C 4m ad M.S.G// D M 1.G/. Lemma C (Liu ad Liu [14]). Let G be a coected graph with vertices ad m edges. The m ı. 1/ (a) If, the M 1.G/ m. C ı C 1/. (b) M 1.G/.Cı/ m with equality if ad oly if G is regular. ı Lemma D (Su ad Wei [17]). If G is a coected bicyclic graph with exactly vertices, m edges ad without pedat vertices, the M 1.G/ M.G/ m ; with equality if ad oly if G D K ;3. Lemma E (Zhou ad Triajstić [1]). Let G be a coected graph with vertices, m edges ad diameter l. The (a) r D 0.G/ D m. 1/l D 0.G/. (b) If the.l 1/M 1.G/ r D 0.G/ m.l /. 1/ + M 1.G/ with either equality if ad oly if l. (c) Let G be a coected triagle-free ad quadragle-free graph with 3 vertices, miimal degree ı ad maximal degree. The Œl 1Cı.l / M 1.G/ m.l / ı r D 0.G/. C /M 1.G/ + mœ.l 3/. 1/ with right equality if ad oly if G is a Moore graph of diameter, or a regular graph of diameter 3 (ad girth 5, 6 or 7) ad with left equality if ad oly if G D S, or G is a Moore graph of diameter, or a regular graph of diameter 3 (ad girth 5, 6 or 7).

4 13 S. HOSSEIN-ZADEH, A. HAMZEH AND A. R. ASHRAFI (d) M 1.G/. 1/ with equality if ad oly if G is the star or a Moore graph of diameter. (e) Let G be a coected graph with miimal degree ı ad maximal degree. The l. 1/m W.G/ r D 0.G/ l. 1/m ıw.g/ with either equality if ad oly if G is a regular graph. (f) Let G be a tree with vertices ad diameter diam.g/. The r D 0.G/ D. 1/ diam.g/ 4W.G/ C. 1/. The joi G C H of graphs G ad H with disjoit vertex sets V.G/ ad V.H / ad edge sets E.G/ ad E.H / is the graph uio G [ H together with all the edges joiig V.G/ ad V.H /. Lemma F (Bucicovschi ad Cioabă [3]). If 1 1 m ad G is a coected graph with vertices ad m edges, the D 0.G/ 4. 1/m P.;m/ D 4. /m C ;m C 1/: Equality happes if ad oly if G is a joi of K 1 ad a graph G 0 o 1 vertices ad m C 1 edges with P.G0 / D. 1;m C 1/. Let } 1 ad } deote the classes of coected uicyclic ad bicyclic graphs, respectively. Note that ay graph i } 1 cotais a uique cycle ad it has edges ad every graph i } cotais two liearly idepedet cycles, cycles without commo edges, havig C 1 edges. Lemma G (Tomescu [11]). Suppose G is a coected graph with vertices, m edges. The we have: (a) For every 3 we have mi G} 1 D 0.G/ D ad the uique extremal graph is K 1; 1 C e. (b) For every 4 we have mi G} D 0.G/ D 3 C 18. The extremal graph is uique ad may be obtaied from K 1; 1 by addig two edges havig a commo extremity. Throughout this paper our otatio is stadard. For terms ad cocepts ot defied here we refer the reader to ay of several stadard moographs such as, e.g., [5] or [18].. RESULTS I this sectio some ew extremal values of Zagreb coidices, degree distace ad reverse degree distace over some special classes of graphs are determied. We begi by Zagreb coidices of graphs..1. Zagreb coidices The aim of this subsectio is to obtai ew lower ad upper bouds for the first ad secod Zagreb coidices of graphs. Propositio 1. Let G be a simple graph with vertices ad m edges. 1.G/ 4m C m. 1/ with equality if ad oly if G is regular. The

5 Proof. Apply Lemma A(a) ad Lemma B(a). ETREMAL PROPERTIES IN GRAPHS 133 Propositio. Let G be a simple graph with vertices, m edges. The m M 1.G/. C 1/.G/ m m 1.1 /. The right had (left had) side of this iequality is satisfied if ad oly if G is regular. Proof. The right had side is a cosequece of Lemma A(b) ad Lemma B(b). To prove the left had side of iequality, we otice that by Lemma A ad Lemma B(c), we have:.g/ m D m M.G/ 1 m m M 1.G/ m M 1.G/ 1 m m M 1.G/ D m M 1.G/. C 1/; which completes our argumet. By Lemma A ad Lemma B(c) the equality holds if ad oly if G is regular. Propositio 3. Let G be a simple graph with vertices, m edges. (a) 1.S.G// 4m. / + 4m 1.1 /; the equality holds if ad oly if G is regular. (b).s.g// 8m 10m m ; the equality holds if ad oly if G is regular. (c).s.g// = m + 18m 10m + 5 N M 1.S.G//. Proof. The parts (a) ad (b) are obtaied from Lemma A ad Lemma B(a) ad the followig iequalities: N M 1.S.G// D 4m. C m 1/ M 1.S.G// ad, N M.S.G// D 8m D 4m. C m 1/ M 1.G/ 4m D 4m. C m / M 1.G/ 4m C 4m 8m 4m D 4m. / C 4m 1.1 /; M.S.G// D 8m m 5 M 1.G/ 8m m 10m : 1 M 1.S.G//

6 134 S. HOSSEIN-ZADEH, A. HAMZEH AND A. R. ASHRAFI The equalities i parts (a) ad (b) are also obtaied from Lemma A ad Lemma B(a). For part (c), we apply Lemma A, Lemma B(d) ad the followig equality:.s.g// D 8m m C 5. 4m 4m C 8m C 1.S.G/// This completes our proof. D m C 18m 10m C 5 N M 1.S.G//: Propositio 4. Suppose G is a graph with vertices ad m edges. The 1.G/ m. 1/ m. + ı + 1/: Proof. Apply Lemma A(a) ad Lemma C(a). Propositio 5. Suppose G is a coected graph with vertices, m edges..cı/ 1.G/ m. 1/ m. The equality holds if ad oly if G is regular. ı Proof. Apply Lemma A(a) ad Lemma C(b). Propositio 6. Let G be a coected bicyclic graph with vertices ad m edges M N 1.G/ with equal- without pedat vertices, the ity if ad oly if G is isomorphic to K ;3. Proof. By Lemmas A ad D.G/ m D m M.G/ M 1.G/ m m M N 1.G/ = m. 1/ M C N.G/ m m. 1/ M 1.G/ mc M 1.G/ m m. 1/ M.G/ m ad, as desired. Propositio 7. Let G be a graph of order cotaiig m edges, the.m C Nm /. 1/. 1/.G/C. NG/.m C Nm /. 1 The equality i right had side is satisfied if ad oly Š 1.mod 4/ ad G is 1 - regular. The equality i left had side is satisfied if ad oly G is isomorphic to complete graph K. Proof. Apply [19, Theorems. ad 3.1]... Degree Distace of Graphs /. 1/. The degree distace of a graph is a degree aalogue of the Wieer idex of the graph. So, it is useful to fid the lower ad upper bouds of this topological idex o some classes of graphs as trees, triagle- ad quadragle-free graphs ad so o. Propositio 8. Let G be a coected graph with vertices, m edges ad diameter diam.g/. The 4. 1/m M 1.G/ D 0.G/.diam.G/ 1/M 1.G/ C. 1/mdiam.G/ with either equality if ad oly if diam.g/. Proof. Apply Lemma E(1) ad Lemma E().

7 ETREMAL PROPERTIES IN GRAPHS 135 Propositio 9. Let G be a coected triagle- ad quadragle-free graph with 3 vertices, miimal degree ı, maximal degree, m edges ad diameter l. The. C /M 1.G/ C mœ3. 1/ C D 0.G/ Œl 1 C ı.l / M 1.G/ C.l /mı C. 1/ml with right equality if ad oly if G is a Moore graph of diameter, or a regular graph of diameter 3 (ad girth 5, 6 or 7) ad with left equality if ad oly if G D S, or G is a Moore graph of diameter, or a regular graph of diameter 3 (ad girth 5, 6 or 7). Proof. Apply Lemma E(1) ad Lemma E(3). Propositio 10. Let G be a coected graph with vertices, maximal degree ad m edges. The. C /. 1/ C mœ3. 1/ C D 0.G/ with equality if ad oly if G is the star or a Moore graph of diameter. Proof. Apply Lemma E(1) ad Lemma E(4). Propositio 11. Let G be a coected graph with miimal degree ı ad maximal degree. The ıw.g/ D 0.G/ W.G/ with either equality if ad oly if G is a regular graph. Proof. Apply Lemma E(1) ad Lemma E(5). Propositio 1. Let G be a tree with vertices ad diameter diam.g/. The D 0.G/ D. 1/Œdiam.G/.m C / C 4W.G/: Proof. Apply Lemma E(1) ad Lemma E(6)..3. Reverse Degree Distace of Graphs The reverse degree distace of graphs is a ew topological idex proposed by Zhou ad Triajstić [1]. I the ext two propositios, some extremal properties of this topological idex are ivestigated. Propositio 13. If 1 1 m ad G is a coected graph with vertices ad m edges, the r D 0.G/ 4. 1/m C P.;m/ C. 1/mdiam.G/: Equality happes if ad oly if G is a joi of K 1 ad a graph G 0 o 1 vertices ad m C1 edges with P.G0 / D. 1;m C 1/: Proof. Apply Lemmas E(1) ad F. Propositio 14. Suppose G is a coected graph with vertices ad m edges. The for every 3 we have r D 0.G/ 3 C 3 C 6 C m. 1/diam.G/ ad the uique extremal graph is K 1; 1 C e: Proof. Apply Lemmas E(1) ad G(a).

8 136 S. HOSSEIN-ZADEH, A. HAMZEH AND A. R. ASHRAFI Propositio 15. Suppose G is a coected graph with vertices ad m edges. The for every 4 we have r D 0.G/ 3 C 18 C m. 1/diam.G/ ad the extremal graph is uique ad may be obtaied from K 1; 1 by addig two edges havig a commo extremity. Proof. Apply Lemmas E(1) ad G(b). ACKNOWLEDGEMENT The authors are idebted to the referee for various valuable commets leadig to improvemets of the paper. REFERENCES [1] A. R. Ashrafi, T. Došlić, ad A. Hamzeh, The zagreb coidices of graph operatios, Discrete Appl. Math., vol. 158, o. 15, pp , 010. [] A. R. Ashrafi, T. Došlić, ad A. Hamzeh, Extremal graphs with respect to the zagreb coidices, MATCH Commu. Math. Comput. Chem., vol. 65, o. 1, pp. 85 9, 011. [3] O. Bucicovschi ad S. M. Cioabă, The miimum degree distace of graphs of give order ad size, Discrete Appl. Math., vol. 156, o. 18, pp , 008. [4] K. C. Das ad I. Gutma, Some properties of the secod zagreb idex, MATCH Commu. Math. Comput. Chem., vol. 5, pp , 004. [5] M. V. Diudea, I. Gutma, ad L. Jätschi, Molecular Topology, ser. Hutigto. New York: Nova Sciece, 001. [6] A. A. Dobryi ad A. A. Kochetova, Degree distace of agraph: A degree aalogue of the wieer idex, J. Chem. If. Comput. Sci., vol. 34, o. 5, pp , [7] T. Došlić, Vertex-weighted wieer polyomials for composite graphs, Ars Math. Cotemp., vol. 1, o. 1, pp , 008. [8] I. Gutma, Selected properties of the schultz molecular topological idex, J. Chem. If. Comput. Sci., vol. 34, o. 5, pp , [9] I. Gutma ad K. C. Das, The first zagreb idex 30 years after, MATCH Commu. Math. Comput. Chem., vol. 50, pp. 83 9, 004. [10] A. Ilić ad D. Stevaović, O comparig zagreb idices, MATCH Commu. Math. Comput. Chem., vol. 6, o. 3, pp , 009. [11] A. Ioa Tomescu, Uicyclic ad bicyclic graphs havig miimum degree distace, Discrete Appl. Math., vol. 156, o. 1, pp , 008. [1] M. H. Khalifeh, H. Yousefi-Azari, ad A. R. Ashrafi, The first ad secod zagreb idices of some graph operatios, Discrete Appl. Math., vol. 157, o. 4, pp , 009. [13] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, ad S. G. Wager, Some ew results o distacebased graph ivariats, Europea J. Combi., vol. 30, o. 5, pp , 009. [14] M. Liu ad B. Liu, New sharp upper bouds for the first zagreb idex, MATCH Commu. Math. Comput. Chem., vol. 6, o. 3, pp , 009. [15] M. J. Nadjafi-Arai, G. H. Fath-Tabar, ad A. R. Ashrafi, Extremal graphs with respect to the vertex pi idex, Appl. Math. Lett., vol., o. 1, pp , 009. [16] S. Nikolić, G. Kovačević, A. Miličević, ad N. Triajstić, The zagreb idices 30 years after, Croat. Chem. Acta, vol. 76, o., pp , 003. [17] L. Su ad S. Wei, Comparig the zagreb idices for coected bicyclic graphs, MATCH Commu. Math. Comput. Chem., vol. 6, o. 3, pp , 009. [18] D. B. West, Itroductio to graph theory. Upper Saddle River: Pretice Hall, 1996.

9 ETREMAL PROPERTIES IN GRAPHS 137 [19] L. Zhag ad B. Wu, The ordhaus-goddum-type iequalities for some chemical idices, MATCH Commu. Math. Comput. Chem., vol. 54, o. 1, pp , 005. [0] B. Zhou ad I. Gutma, Relatios betwee wieer, hyper-wieer ad zagreb idices, Chem. Phys. Lett., vol. 394, o. 1-3, pp , 004. [1] B. Zhou ad N. Triajstić, O reverse degree distace, J. Math. Chem., vol. 47, o. 1, pp , 010. Author s address S. Hossei-Zadeh, A. Hamzeh ad A. R. Ashrafi Departmet of Mathematics, Faculty of Sciece, Uiversity of Kasha, Kasha , Ira School of Mathematics, Istitute for Research i Fudametal Scieces (IPM), P. O. Box: , Tehra, Ira address: ashrafi@kashau.ac.ir