Sum-connectivity indices of trees and unicyclic graphs of fixed maximum degree

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1 1 Sum-coectivity idices of trees ad uicyclic graphs of fixed maximum degree Zhibi Du a, Bo Zhou a *, Nead Triajstić b a Departmet of Mathematics, South Chia Normal Uiversity, uagzhou , Chia zhoubo@scu.edu.c b The Rugjer Bošković Istitute, P. O. Box 180, HR-1000 Zagreb, Croatia tria@irb.hr Abstract We obtai the maximum sum-coectivity idices of graphs i the set of trees ad i the set of uicyclic graphs respectively with give umber of vertices ad maximum degree, ad determie the correspodig extremal graphs. Additioally, we deduce the -vertex uicyclic graphs with the first two maximum sum-coectivity idices for 4. Keywords: sum-coectivity idex; product-coectivity idex; Radić coectivity idex; maximum degree; trees; uicyclic graphs *Correspodig author.

2 1. Itroductio The well-kow Radić coectivity idex, proposed by Radić [1] i 1975, is the most used molecular descriptor i quatitative structure property relatioship (QSPR) ad quatitative structure activity relatioship [e.g., 1,4,5,6,11,14]. Mathematical properties of this idex have also bee studied [e.g.,,7,9,10]. Let be a simple (molecular) graph with vertex set V ( ) ad edge set E ( ). For v V( ), () v deotes the set of its (first) eighbors i ad the degree of v is d ( ) ( ). v d v v The Radić coectivity idex R R( ) of a graph is defied as [1] R R d d 1/ ( ) ( u v). uve ( ) defied as [1] A related coectivity idex of a graph, called the sum-coectivity idex, is 1/ ( ) ( du dv). uve ( ) Correspodigly, we call R ( ) the product-coectivity idex of. We determied i [15] the uique tree with fixed umbers of vertices ad pedat vertices (vertices of degree oe) with the miimum value of the sum-coectivity idex, ad trees with the miimum, secod miimum ad third miimum, ad the maximum, secod maximum ad third maximum values of the sum-coectivity idex, ad we discussed its properties for a class of trees represetig acyclic hydrocarbos. The product- ad sum-coectivity idices are highly itercorrelated quatities; for example, the value of the correlatio coefficiet is for 134 trees represetig the lower alkaes from [3]. But, i geeral, the sum-coectivity idex has a wider rage [15]. Properties o the product-coectivity idex for trees ad uicyclic graphs may be foud i [7,8,13]. Particularly, the product-coectivity idex for trees with give maximum degree was ivestigated i [8]. For 1, let T (, ) be the set of trees with vertices ad maximum degree, ad U (, ) the set of uicyclic graphs with vertices ad maximum degree. Let P

3 3 ad C respectively be the path ad the cycle o 3 vertices. I particular, T (,) { P } ad U (, ) { C }. I this paper, we obtai the maximum sum-coectivity idices of graphs i T (, ) ad U (, ), respectively, ad determie the correspodig extremal graphs. Recall that the -vertex trees with the first two maximum sum-coectivity idices for 4 have bee kow [15]. From the result related to maximum degree for uicyclic graphs, we ow deduce the -vertex uicyclic graphs with the first two maximum sum-coectivity idices for 4.. Prelimiaries For a edge subset E 1 of the graph (the complemet of, respectively), E1 ( E1, respectively) deotes the graph resultig from by deletig (addig, respectively) the edges i E 1. Lemma 1. [15 For a coected graph Q with at least two vertices ad vertex u V( Q), let 1 be the graph obtaied from Q by attachig two paths P a ad P b to u, ad the graph obtaied from Q by attachig a path Pa bto u, where ab 1. The ( 1) ( ). Lemma. For a coected graph M with VM ( ) 3, a vertex u of degree two, let H be the graph obtaied from M by attachig a path P a to u. Deote u 1 ad u by the two eighbors of u i M, ad u by the pedat vertex of the path attached to u i H. If 1 H mi d ( u ), d ( u ) 4, the for H H { uu} { uu }, ( H) ( H). H Proof. Assume that dh( u1) dh( u), ad thus, d ( ) 4. H u If dh ( u, u) 1, the ( H) ( H) d ( u ) d ( u ) d ( u ) 3 d ( u ) H 1 H H 1 H

4 4 If dh ( u, u), the as 1 1 x x3 is decreasig for x 0, we have , d ( u ) d ( u ) H H ad thus ( H) ( H) dh( u1) dh( u) dh( u1) 3 dh( u) d ( u ) d ( u ) 3 d ( u ) d ( u ) H 1 H 1 H H dh( u) dh( u) The result follows. Attachig a path of legth r to a vertex u of a graph meas that addig a edge betwee u ad a termial vertex of a path o r vertices. Particularly, if r 1, the a pedat vertex is attached. For 1, let T, be the tree obtaied by attachig 1 pedat vertices ad 1 paths of legth two to a vertex. For 1, let U, be the uicyclic graph obtaied by attachig 1 pedat vertices ad 1 paths of legth two to a vertex of a triagle. 3. Maximum Sum-Coectivity Idex i T (, ) First we obtai the maximum sum-coectivity idex of trees i T (, ) ad determie the correspodig extremal graphs. Theorem 1. Let T (, ), where 1. The

5 if ( 1 ) if 3 with equality if ad oly if T, for 1, ad is a tree obtaied by attachig 1 paths of legth at least two to a vertex for. Proof. The case is trivial. Suppose that 3. Let be a tree i T (, ) with maximum sum-coectivity idex. Let v be a vertex of degree i. If there exists some vertex of degree more tha two i differet from v, the by Lemma 1, we may get a tree i T (, ) with larger sum-coectivity idex, a cotradictio. Thus, v is the uique vertex of degree more tha two i. Let k be the umber of eighbors of v with degree two. The k mi{ 1, }. If 1, i.e., i.e.,, the 0 k 1. It is easily see that 1, the 1 k. If 1, , ad the

6 6 k k k 1 ( ) ( 1 k) ( 1 ) 3 k ( 1) ( 1 ) if ( 1 ) if if ( 1 ) if 3 with equality if ad oly if k 1, i.e., each of the 1 eighbors of vertex v of degree two is adjacet to a pedat vertex, i.e., T, for 1, ad k, i.e., is a tree obtaied by attachig paths of legth at least two to a vertex for 1. I Fig. 1, all the extremal graphs i Theorem 1 with 7 are give. Fig. 1 comes here Fig. 1. The 7-vertex trees with maximum sum-coectivity idices for,3,4,5,6. 4. Maximum Sum-Coectivity Idex i U (, ) Now we obtai the maximum sum-coectivity idex of graphs i U (, ) ad determie the correspodig extremal graphs. As a cosequece, we deduce the -vertex uicyclic graphs with the first ad secod maximum sum-coectivity idices for 4. Theorem. Let U (, ), where 1. The

7 if ( ) 1 1 ( ) if 3 with equality if ad oly if U, for 1, ad is a uicyclic graph obtaied by attachig paths of legth at least two to a vertex of a cycle for 1. Proof. The case is trivial. Suppose that 3, is a graph i U (, ) with maximum sum-coectivity idex, ad C is the uique cycle of. Let v be a vertex of degree i. First we cosider 3. If there is some vertex outside C with degree three, the by Lemma 1, we may get a graph i U (,3) with larger sum-coectivity idex, a cotradictio. If there are at least two vertices o C with degree three, the by Lemma, we may get a graph i U (,3) with larger sum-coectivity idex, also a cotradictio. Thus, v V ( C) ad v is the uique vertex i with degree three. The either 1 ( ) ( ) whe 5 v is adjacet to a vertex of degree oe ad two vertices of degree two for 4, or ( ) ( 4) whe v is adjacet to three vertices of degree two for Obviously, 1 ( ) 1 ( 4) 3 1 for 5. Hece, is the graph obtaied by attachig a pedat vertex to a triagle for 4, i.e., U, 4,3 ad a graph obtaied by attachig a path of legth at least two to a cycle for 5. Now suppose that 4. We will show that v lies o C. Suppose that v is ot o C. Let w be the vertex o C such that d ( v, w) mi d ( v, x): x V( C). If there is some vertex outside C with degree more tha two differet from v, or if there is some vertex o C with degree more tha two differet from w, the by Lemmas 1 ad, we may get a graph i U (, ) with larger sum-coectivity idex, a cotradictio. Thus, v ad w are the oly vertices of degree more tha two i, ad d ( v), d ( w) 3 or 4. Let Q be the

8 8 path coectig v ad w. Suppose that d ( w) 4. Deote w 1 by a eighbor of w o, C w by the eighbor of w o Q, ad w by the pedat vertex of the path attached to w. Cosider 1 { ww1} { ww1 } U(, ). If d ( w, w) 1, the ( 1 ) ( ) d ( w ) d ( w ) d ( w) 3 d( w) If d w, w, the ( 1 ) ( ) d ( w ) d ( w ) d( w) 3 d( w) I either case, ( 1 ) ( ), a cotradictio. Thus, d ( w) 3. Suppose that v1, v,, v 1 are the eighbors of v outside Q. Let di d ( vi ) for i 1,, 1. Note that d1, d 1 or. Cosider { vv3,, vv 1} { wv3,, wv 1} U(, ). Note that d ( ) w ad d ( ) 3. v The ( ) ( ) d 3 d d 3 d

9 9 Sice d ( ) 3, v the by Lemma 1, we may get a graph i U (, ) such that ( ) ( ) ( ), a cotradictio. Hece, we have show that v lies o C. If there is some vertex outside C with degree more tha two, the by Lemma 1, we may get a graph i U (, ) with larger sum-coectivity idex, a cotradictio. Thus, is a graph obtaied from C by attachig paths to v, ad attachig at most oe path to some vertex o C differet from v. If there is some vertex o C with degree three, the by Lemma, we may get a graph i U (, ) with larger sum-coectivity idex, a cotradictio. Thus, is a graph obtaied from C by attachig paths to v. Let k be the umber of eighbors of v with degree two. The k mi{ 1, }. If 1, i.e., 1, the 0 k. If 1, i.e., easily see that, the 0 k 1. It is , ad the k k k 1 ( k ) k ( ) ( 1) ( ) if ( ) ( ) if if ( ) if 3

10 10 with equality if ad oly if k 1 for 1, i.e., U,, ad k 1 for, i.e., is a uicyclic graph obtaied by attachig paths of legth at least two to a vertex of a cycle. I Fig., all the extremal graphs i Theorem with 7 are give. Fig. comes here Fig.. The 7-vertex uicyclic graphs with maximum sum-coectivity idices for,3,4,5,6 (for 3, there are three such graphs). Theorem 3. Amog the uicyclic graphs o 4 vertices, C is the uique graph with maximum sum-coectivity idex, which is equal to, for 4, U 4,3 is the uique graph with the secod maximum sum-coectivity idex, which is equal to 1, while for 5 5, the graphs obtaied by attachig a path of legth at least two to a cycle are the uique graphs with the secod maximum sum-coectivity idex, which is equal to ( 4). 3 5 Proof. The case 4 may be checked directly. Suppose that 5 ad is a uicyclic graph o vertices. Let be the maximum degree of, where 1. Let x x 1 f ( x) ( x ) 3 x If for x. 1, the by Theorem, ( ) f ( ) ( 1) f ( ). 3 1

11 11 If 1, the by Theorem, ( ) f( ). Note that for, 1 x x x 3/ is decreasig o, 1 3/ 1 3/ 3. 8 x ad the x x 1 1 f x 1 x x 3/ , Now we have ( ) f ( ) f (3) f () for ( ) f ( ) f (3) f () for Thus, ad the f( x ) is decreasig for x. 3 1, ad 1 3. The obviously, C is the uique -vertex uicyclic graph with maximum sum-coectivity idex f (), while the -vertex uicyclic graphs with maximum degree three ad sum-coectivity idex f (3) are the -vertex uique graphs with the secod maximum sum-coectivity idex, ad by Theorem, such graphs are the graphs obtaied by attachig a path of legth at least two to a cycle. Ackowledgemet This work was supported by the Natioal Natural Sciece Foudatio of Chia (No ), the uagdog Provicial Natural Sciece Foudatio of Chia (No. S ) ad the Miistry of Sciece, Educatio ad Sports of Croatia (rat No ). Refereces [1] R. arcía-domeech, J. álvez, J.V. de Juliá-Ortiz, L. Pogliai, Some ew treds i chemical graph theory, Chem. Rev. 108 (008) [] I. utma, B. Furtula (Eds.), Recet Results i the Theory of Radić Idex, Uiv. Kragujevac, Kragujevac, 008. [3] O. Ivaciuc, T. Ivaciuc, D. Cabrol-Bass, A.T. Balaba, Evaluatio i quatitative structure-property relatioship models of structural descriptors derived from iformatio-theory operators, J. Chem. If. Comput. Sci. 40 (000) [4] L.B. Kier, L.H. Hall, W.J. Murray, M. Radić, Molecular-coectivity I: Relatioship to ospecific local aesthesia, J. Pharm. Sci. 64 (1975) [5] L.B. Kier, L.H. Hall, Molecular Coectivity i Chemistry ad Drug Research, Academic Press, New York, 1976.

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