WEIGHTED SZEGED INDEX OF GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvib.org /JOURNALS / BULLETIN Vo. 8(2018), 11-19 DOI: 10.7251/BIMVI1801011P Formr BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) WEIGHTED SZEGED INDEX OF GRAPHS K. Pattabiraman and P. Kandan Abstract. Th wightd Szgd indx of a connctd graph G is dfind ( ) as Sz w (G) d G () d G (v) n G () n G v (), whr n G () is th v E(G) nmbr of vrtics of G whos distanc to th vrtx is ss than th distanc to th vrtx v in G. In this papr, w hav obtaind th wightd Szgd indx Sz w (G) of th spic graph S(G 1, G 2, y, z) and ink graph L(G 1, G 2, y, z). 1. Introdction Lt G G(V, E), b th graph, whr V V (G) and E E(G) dnots th vrtx st and dg st of th graph G, rspctivy. A th graphs considrd in this papr ar simp. Graph thory has sccssfy providd chmists with a varity of sf toos [3, 5, 6, 7], among which ar th topoogica indics. A mocar graph is a simp graph sch that its vrtics corrspond to th atoms and th dgs to th bonds. In thortica chmistry, assigning a nmrica va to th mocar strctr that wi cosy corrat with th physica qantitis and activitis. Mocar strctr dscriptors (aso cad topoogica indics) ar sd for moding physicochmica, pharmacoogic, toxicoogic, bioogica and othr proprtis of chmica componds. Thr xist svra typs of sch indics, spciay thos basd on dgr and distancs. Th dgr of a vrtx x V (G) is dnotd by d G (x). A vrtx x V (G) is said to b qidistant from th dg v of G if d G (, x) d G (v, x), whr d G (, x) dnots th distanc btwn and x in G; othrwis, x is a nonqidistant vrtx anaogosy, d G (v, x). For an dg v E(G), th nmbr of vrtics of G whos distanc to th vrtx is ss than th distanc to th vrtx v in G is dnotd by n G () ( or n (, G);) anaogosy,n G v () ( or n v (, G)) is th nmbr of vrtics of G whos distanc to th vrtx v in G is 2010 Mathmatics Sbjct Cassification. 05C12, 05C76. Ky words and phrass. wightd Szgd indx, spic, ink. 11

12 K. PATTABIRAMAN AND P. KANDAN ss than th distanc to th vrtx ; th vrtics qidistant from both th nds of th dg v ar not contd. Th Szgd indx of a connctd graph G is dfind as Sz(G) n G () n G v (). v E(G) Simiary, th wightd Szgd indx of a connctd graph G which is introdcd by Iić and Miosavjvić [9], dfind as ( ) Sz w (G) d G () d G (v) n G () n G v (). v E(G) For mor rcnt rsts on th wightd Szgd indx and th wightd PI indx rfr[10, 11, 13]. A varity of topoogica indics of spic graphs and ink graphs hav bn comptd arady in [1, 2, 4, 8, 14]. Wightd Szgd indx of gnraizd hirarchica prodct of two graphs is ontaind in [12]. In this papr, w aim at contining work aong th sam ins, for finding th xact va of th wightd Szgd indx of spic and ink graphs. 2. Wightd Szgd indx of spic of graphs Lt G 1 (V 1, E 1 ) and G 2 (V 2, E 2 ) b two graphs with disjoint vrtx sts V 1 and V 2. Lt y V 1 and z V 2. A spic S S(G 1, G 2, y, z) of G 1 and G 2 by vrtics y and z, is dfind by idntifying th vrtics y and z in th nion of G 1 and G 2 (s Fig. 1), introdcd by Došić [4]. In this sction, w compt th Wightd Szgd indx of spic of two givn graphs. G 1 G 2 Fig.1 Spicgraph S G, G,y,z) ( 1 2 W dfin th st N G () {x V (G) d G (x, ) < d G (x, v)}. Th proof of th foowing mma is foows from th strctr of spic of two connctd graphs. Lmma 2.1. Lt G i b th graphs with n i vrtics and E i dgs, i 1, 2, thn for th spic graph S w hav th foowing. For v E 1. (i) If y N G1 () and y, thn n S () n G 1 () 1 n 2, d S () d G1 () n S v () n G1 v (), d S (v) d G1 (v) (ii) If y N G 1 () and, thn n S () n G1 () 1 n 2, d S () d G1 () d G2 (z) n S v () n G 1 v (), d S (v) d G1 (v)

WEIGHTED SZEGED INDEX OF GRAPHS 13 (iii) If d G1 (y, ) d G1 (y, v), thn n S () n G 1 (), d S () d G1 () n S v () n G 1 v (), d S (v) d G1 (v) Anaogos rations hod if v E 2. Thorm 2.1. Lt G i b th graphs with n i vrtics, i 1, 2 thn th wightd Szgd indx of th Spic graphs is Sz w (S(G 1, G 2, y, z)) Sz w (G 1 ) (n 2 1) () d G 1 (v))(n G 1 (z))(n G 1 ())(n G 1 (n 2 1) Sz w (G 2 ) (n 1 1) (n G 2 (n 1 1) z y ()d G 1 (v)d G 2 () d G 2 (v))(n G 2 () d G 2 (v) d G 1 (y))(n G 2. (z))(n G 1 (z))(n G 2 ()) Proof. For a spic graph S S(G 1, G 2, y, z), t th dg st can b partitiond as E 1 E(G 1 ) and E 2 E(G 2 ), by th dfinition of Sz w Sz w (S) (d S () d S (v))n S ()n S v () y v E(S) By Lmma 2.1, w hav Sz w (S) () d G1 (v))n G1 ()n G1 v () () d G2 (v))n G2 ()n G 2 v () () d G1 () 1 n 2 )(n G1 () d G1 d G1 (y,) d G1 (y,v) z (v) d G2 () d G1 (z))(n G1 () 1 n 2 )(n G1 ())(n G1 () d G 2 (v))(n G 2 () 1 n 1 )(n G 2 () d G 2 (v) d G 1 (y))(n G 2 () 1 n 1 )(n G 2 d G2 (z,) d G2 (z,v) () d G 2 (v))(n G 2 ())(n G 2

14 K. PATTABIRAMAN AND P. KANDAN y () d G 1 (v))(n G 1 ())(n G 1 () d G 1 (v))(n G 1 ())(n G 1 d G1 (y,) d G1 (y,v) (n 2 1) (n 2 1) z y (z))(n G1 () d G2 () d G2 d G2 (z,) d G2 (z,v) (n 1 1) (n 1 1) z (y))(n G2 () d G 1 (v))(n G 1 ())(n G 1 () d G1 ())(n G1 () d G1 (v) d G2 ())(n G2 ())(n G2 (z))(n G1 () d G 2 (v))(n G 2 ())(n G 2 () d G2 ())(n G2 () d G 2 (v) d G 1 (y))(n G 2 By th dfinition of Sz w, for G 1 and G 2 w hav Sz w (S) Sz w (G 1 ) (n 2 1) () d G 1 (v))(n G 1 (n 2 1) y (z))(n G 1 ())(n G 1 () d G 1 (v) d G 2 (z))(n G 1

WEIGHTED SZEGED INDEX OF GRAPHS 15 Sz w (G 2 ) (n 1 1) (n 1 1) z (y))(n G 2 ())(n G 2 () d G 2 (v))(n G 2 () d G 2 (v) d G 1 (y))(n G 2. Using th Thorm 2.1, w hav th foowing xamps. Examp 2.2. For th cycs, w hav, Sz w (S(C n, C m, y, z)) n(n 1)(n 2) m(m 1)(m 2) 2nm(m n 2), if n is vn m is vn, n 2 (n 1) (m 1)(m 1)(2n m 3) 2n(m 1)(n 1), if n is vn m is odd, (n 1) 2 (n 1) (m 1) 2 (m 1) 2(n 1)(m 1)(m n 2), if n is odd m is odd. Examp 2.3. For th cyc and path, w hav,sz w (S(C n, P m, y, z)) n 3 n2 2 (n 1)(2m2 m 2) (m 1)(2n 2 n 2) 2(m 1)(m2 m 3) 3, if n is vn, (n 1) 2 (2n1) 2 2m(m 1)(m1) 3 2(n 1)(n 3)(m 1) (n 1)(2m 2 6m 3), if n is odd. A broom T is a tr which is nion of th path and th star, ps on dg joining th cntr of th star to an ndpoint of th path. Cary, T S(K 1,n, P m, y, z). Examp 2.4. For broom T, Sz w (T ) n 2 (n2)2n(n3)(m 1)n(2m 2 6m 3) 2(m 1)(m2 m 3) 3. 3. Wightd Szgd indx of ink of graphs Th ink L L(G 1, G 2, y, z) of G 1 and G 2 by vrtics y and z is dfind as th graph, obtaind by joining y and z by an dg in th nion of G 1 and G 2 (s Fig. 2). In this sction, w obtain th wightd Szgd indx of ink of th givn two graphs.. G 1 G 2 Fig.2 Link graph L G, G,y,z) ( 1 2

16 K. PATTABIRAMAN AND P. KANDAN Th proof of th foowing mma is foows from th strctr of ink of two connctd graphs. Lmma 3.1. Lt G i b th graphs with n i vrtics and E i dgs, i 1, 2, thn for th ink graph L w hav th foowing. For v E 1. (i) If y N G 1 () and y, thn n L () n G 1 () n 2, d L () d G1 () n L v () n G1 v (), d L (v) d G1 (v) (ii) If y N G 1 () and, thn n L () n G1 () n 2, d L () d G1 () 1 n L v () n G 1 v (), d L (v) d G1 (v) (iii) If d G1 (y, ) d G1 (y, v), thn n L () n G 1 (), d L () d G1 () n L v () n G1 v (), d L (v) d G1 (v) Anaogos rations hod if v E 2. Thorm 3.1. Lt G i b th graphs with n i vrtics, i 1, 2 thn th wightd Szgd indx of th ink of G 1 and G 2 graphs is Sz w (L(G 1, G 2, y, z)) Sz w (G 1 ) n 2 () d G1 n 2 n 1 z y ()d G1 () d G 2 (v))(n G 2 n 1 (n G1 ()n G1 v ()n 2 n G1 Sz w (G 2 ) () d G 2 (v))(n G 2 (n G 2 ()n G 2 v () n 1 n G 2 (y) d G 2 (z) 2)n 1 n 2. Proof. For a ink L L(G 1, G 2, y, z), of G 1 and G 2 graphs, t th dg st can b partitiond as E 1 E(G 1 ), E 2 E(G 2 ) and th ink dg yz, thn by th dfinition of Sz w Sz w (L) (d L () d L (v))n L ()n L v () By Lmma 3.1, w hav v E(L) Sz w (L) () d G1 (v))n G1 ()n G1 v () () d G2 (v))n G2 ()n G 2 v () (d L (y) d L (z))n 1 n 2 () d G 1 (v))(n G 1 () n 2 )(n G 1 y (y) 1 d G 1 (v))(n G 1 () n 2 )(n G 1

d G1 (y,) d G1 (y,v) z WEIGHTED SZEGED INDEX OF GRAPHS 17 () d G2 (z) 1 d G2 d G2 (z,) d G2 (z,v) () d G 1 (v))(n G 1 ())(n G 1 () n 1 )(n G2 () n 1 )(n G2 () d G 2 (v))(n G 2 ())(n G 2 y (y) 1 d G2 (z) 1)n 1 n 2 () d G 1 (v))(n G 1 ())(n G 1 () d G 1 (v))(n G 1 ())(n G 1 d G1 (y,) d G1 (y,v) (n 2 ) (n 2 ) z y (n G1 ()n G1 v () d G 1 (v))(n G 1 ())(n G 1 () d G1 () d G1 () d G2 () d G2 d G2 (z,) d G2 (z,v) (n 1 ) (n 1 ) z () n 2 n G1 ())(n G2 ())(n G2 () d G 2 (v))(n G 2 ())(n G 2 () d G2 () d G 2 (v))(n G 2 (n G 2 ()n G 2 v () n 1 n G 2 (y) 1 d G 2 (z) 1)n 1 n 2.

18 K. PATTABIRAMAN AND P. KANDAN By th dfinitions of Sz w, for G 1 and G 2 w hav Sz w (L) Sz w (G 1 ) (n 2 ) (n 2 ) y () d G1 () d G 1 (v))(n G 1 (n G 1 ()n G 1 v () n 2 n G 1 Sz w (G 2 ) (n 1 ) (n 1 ) (n G2 z () d G 2 (v))(n G 2 () d G 2 (v))(n G 2 ()n G2 v () n 1 n G2 (y) d G 2 (z) 2)n 1 n 2. A dob broom T 1 is a tr consisting of two stars, whos cntrs ar joind by a path. Cary, T 1 L(K1,n, K 1,m, y, z), ths by sing Thorm 3.1, w hav th foowing xamp. Examp 3.2. For a dob broom T 1, Sz w (T 1 ) (n m 1)(2(n 1)(m 1) n m) n 2 (n 1) m 2 (m 1). Rfrncs [1] A. R. Ashrafi, A. Hamzh and S. Hossin-zadh. Cacation of som topoogica indics of spics and ink of graphs, J. App. Math. and Inf. 29(1-2)(2011), 327-335. [2] M. Azar. A not on vrtx-dg Winr indics of graphs, Iranian Jorna of Mathmatica Chmistry 7(1)(2016), 11-17. [3] A. T. Baaban (Ed.). Chmica Appications of Graph Thory, Acadmic Prss, London (1976). [4] T. Došić. Spics, inks and thir dgr-wightd Winr poynomias, Graph Thory Nots Nw York, 48(2005), 47-55. [5] A. Graovac, I. Gtman and D. Vkičvić (Eds.). Mathmatica Mthods and Moding for Stdnts of Chmistry and Bioogy, Hm Copis Ltd., Zagrb, (2009). [6] I. Gtman. Introdction to Chmica Graph Thory, Facty of Scinc, Kragjvac, (2003) (in Srbian). [7] I. Gtman (Ed.). Mathmatica Mthods in Chmistry, Prijpoj Msm, Prijpoj, (2006). [8] M. A. Hossinzadh, A. Iranmansh and T. Došić. On Th Narmi-Katayama Indx of Spic and Link of graphs, Ectronic Nots in Discrt Mathmatics 45(2014), 141-146. [9] A. Iić and N. Miosavjvić. Th Wightd vrtx PI indx, Mathmatica and Comptr Moding 57(3-4)(2013), 623-631.

WEIGHTED SZEGED INDEX OF GRAPHS 19 [10] K. Pattabiraman and P. Kandan. Wightd PI indx of corona prodct of graphs, Discrt Math. Agorithm App. 6(4)(2014), 1450055. [11] K. Pattabiraman and P. Kandan. Wightd Szgd indics of som graph oprations, Transactions on Combinatorics 5(1)(2016), 25-35. [12] K. Pattabiraman, S. Nagarajan and M. Chandraskharan. Wightd Szgd indx of gnraizd hirarchica prodct of graphs, Gn. Math. Nots 23(2)(2014), 85-95. [13] K. Pattabiraman and P. Kandan. On Wightd PI indx of graphs, Ectronic Nots in Discrt Mathmatics 53(2016), 225-238. [14] R.Sharafdini and I. Gtman. Spic graphs and thir topoogica indics, Kragjvac J. Sci. 35(2013), 89-98. [15] Z. Yarahmadi and G.H.Fath-Tabar. Th Winr, Szgd,PI, vrtx PI, th First and Scond Zagrb indics of N-branchd phnyactyns Dndrimrs, MATCH Commnications in Mathmatica and in Comptr Chmistry 65(2011), 201-208. [16] Z. Yarahmadi. Eccntric Connctivity and Agmntd Eccntric Connctivity Indics of N- Branchd Phnyactyns Nanostar Dndrimrs, Iranian Jorna of Mathmatica Chmistry 1(2)(2010), 105-110. Rcibd by ditors 13.11.2016; Accptd 01.06.2017: Avaiab onin 12.06.2017. Dpartmnt of Mathmatics, Annamaai Univrsity, Annamaainagar, India E-mai addrss: pramank@gmai.com Dpartmnt of Mathmatics, Annamaai Univrsity, Annamaainagar, India E-mai addrss: kandan2k@gmai.com