Balaban Index of Cubic Graphs

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1 MATCH Communications in Mathematical and in Compute Chemisty MATCH Commun. Math. Comput. Chem. 73 (015) ISSN Balaban Index of Cubic Gaphs Matin Kno a, Riste Škekovski b,c,d, Aleksanda Tepeh d,e a Slovak Univesity of Technology in Batislava, Faculty of Civil Engineeing, Depatment of Mathematics, Radlinského 11, , Batislava, Slovakia, kno@math.sk b Faculty of Infomation Studies, 8000 Novo Mesto, Slovenia, c FAMNIT, Univesity of Pimoska, 6000 Kope, Slovenia, d Faculty of Mathematics and Physics, Univesity of Ljubljana, 1000 Ljubljana, skekovski@gmail.com e Faculty of Electical Engineeing and Compute Science, Univesity of Maibo, Smetanova ulica 17, 000 Maibo, Slovenia, aleksanda.tepeh@gmail.com Abstact (Received Septembe 7, 014) Balaban index is defined as J(G) = m m n+ 1 w(u) w(v), whee the sum is taken ove all edges of a connected gaph G, n and m ae the cadinalities of the vetex and the edge set of G, espectively, and w(u) (esp. w(v)) denotes the sum of distances fom u (esp. v) to all the othe vetices of G. In this pape, we give an uppe bound fo the Balaban index of -egula gaphs on n vetices. Then we concentate on cubic gaphs. We give a bette uppe bound fo fulleene gaphs and we show that the Balaban index tends to zeo as the numbe of vetices inceases. This means that Balaban index does not distinguish well the fulleene gaphs when they ae sufficiently lage. We conclude the pape with a conjectue on the lowe bound of Balaban index fo cubic gaphs. 1 Intoduction In this pape we conside simple and connected gaphs. Fo a gaph G, by V(G) and E(G) we denote the vetex and edge sets of a gaph G, espectively. We set n = V(G) and m = E(G). Fo vetices, u,v V(G), by dist G (u,v) we denote the distance fom u to v in G. Balaban index J(G) of G is defined as m 1 J(G) =, m n+ w(u) w(v) e=uv

2 -50- whee the sum is taken ove all edges e = uv of G and fo x V(G), we have w(x) = y V(G) dist G(x,y). Balaban index is a topological index intoduced by Alexandu T. Balaban nea to 30 yeas ago [7, 8]. This topological index was used successfully in QSAR/QSPR modeling [1, 31]. Seveal ecent uses can be found in [18, 6, 8]. In [10] two diffeent appoaches wee pesented fo the calculation of Balaban index by taking into account the chemical natue of elements. In [11], Balaban index is compaed with Wiene index egading the alkanes, and it was obtained that Balaban index educes the degeneacy of the late index and povides much highe disciminating ability. Theefoe Balaban index is also called shapened Wiene index. See [1] fo anothe efeence that involves these two indicies and infinite polymes. Howeve, mathematical popeties of Balaban index ae still not studied extensively. Thee ae few theoetical esults known. Zhou and Tinajstić [3] give tight lowe and uppe bounds fo geneal gaphs, and Sun [30] and Deng [19] give the bounds in the case of tees. Among all tees with n vetices, the sta S n and the path P n have the maximal and the minimal Balaban index, espectively. Also tees with the second minimal and maximal Balaban index, espectively, wee chaacteized, see [19]. Seveal authos have studied bounds on the Balaban index ove given classes of gaphs. In Balaban, Ionescu-Pallas, Balaban [16], the behavio of J fo vaious infinite families of gaphs is discussed. In many of these cases, J tends to a constant finite value. J has the asymptotic value π fo an infinitely long n-alkane (a path). In Ghobani [5], Balaban index of vetex tansitive gaphs is studied. Fo the study of this index ove fulleene gaphs see [17,4,7]. The lagest Balaban index among all n-vetex unicyclic gaphs and n-vetex bicyclic gaphswasconsideed indengandchang[0]anddongandyou[]. TheBalabanindex of a class of dendimes is computed in Ashafi, Shabani, Diudea [3,4]. Ou eseach was motivated by the following poblem fom Dong and Guo []: Poblem 1. Among n-vetex gaphs, find those with the minimum Balaban index. Using the AutoGaphiX softwae, Aouchiche, Capoossi, Hansen [] showed that, among all connected gaphs on n vetices the path on n vetices is not a gaph fo which the lowe bound is attained, as eoneously stated in [3]. Among gaphs on n vetices, Balaban index attains its maximum fo the complete

3 -51- gaph K n, whee J(K n ) = ( n ) ( n ( n ) n+ ) 1 n 1 = n 3 n (n 3n+4), which is slightly moe than n. Its minimum value among n-vetex gaphs is not known. Howeve, if we conside n-vetex tees, then the Balaban index attains its minimum fo the path on n vetices P n, see [19,30], and lim n J(P n ) = π, see [16]. One may expect that, if G is ann-vetex -egula gaph, then J(P n ) J(G) J(K n ). In the next section we show that this is not the case. We pove that if G is an n-vetex -egula gaph (i.e. a gaph whee each vetex has neighbos), then J(G) tends to 0 as n tends to. In othe wods, zeo is also an accumulation point fo Balaban index. Balaban index of egula gaphs In this section we concentate on -egula gaphs with 3. In the following esult we give an uppe bound fo J(G) fo such gaphs. Theoem. Let G be an -egula gaph on n vetices with 3. Then J(G) ( 1) ( ) ( )n+ log 1 Poof. Let u V(G) and let n i be the numbe of vetices at distance i fom u. Thus, w(u) = i i n i and. n i = n. Since the gaph is -egula, we have n i ( 1) i 1. Let s and c be such that n = 1+ + ( 1)+ + ( 1) s 1 +c and 0 c < ( 1) s. Then we can bound w(u) fom below in the following way: s+1 w(u) = i n i 1 + ( 1)+ +s ( 1) s 1 +(s+1) c. i=0 In othe wods, a lowe bound on w(u) is attained if the beadth-seach tee, ooted at u, is an almost complete tee with all leaves at distance s and maybe s+1 fom u, and evey non-leaf vetex is of degee. So, we have 1+( 1)+( 1) + +( 1) s 1 = n 1 c, i

4 -5- and hence which gives ( 1) s 1 Fom (1) and fom c < ( 1) s we get ( )n+ = n 1 c, ( ) ( )n+ c( ) s = log 1. (1) = ( 1) s + which means that log 1 ( ( )n+ c( ) s = log 1 ( ( )n+ ). Consequently, Thus, ( ( )n+ w(u) s ( 1) s 1 = log 1 ( ) ( )n+ log 1 = J(G) ( ( )n+ log 1 m m n+ m 1 w(u) < ( 1) s +( 1) s ( ) = ( 1) s+1, ) < s + 1. Since log 1 ( ( )n+ ) s by (1), we have ) ) ( )n+ ( )n+. ( 1) n n n+ n ( 1) log 1( ( )n+ ) 1 1 ( 1) 1 log 1 ( ( )n+ ) ( )n+ ( 1) ( 1) n < ( ) ( 1) < ( ). (( )n+) log ( )n+ 1 ( ) log ( )n+ 1 Now we may state the following inteesting consequence. Coollay 3. Fo -egula gaphs G on n vetices, whee 3, it holds lim J(G) = 0. n Poof. Let G be an -egula gaph on n vetices. By Theoem, we have J(G) ( 1) ( ) ( )n+ log 1 < c 1 ln(n)+c, whee c 1 and c ae constants depending on, but not on n. Hence, lim J(G) lim n n c 1 ln(n)+c = 0.

5 -53- In othe wods, Balaban index of egula gaphs which ae eally big in the numbe of vetices, is close to 0. The numbe of such gaphs is enomously lage, and we conclude that the Balaban index does not distinguish them well. 3 Fulleene gaphs Hee we conside chemical stuctues called fulleenes. Fulleenes [9] ae polyhedal molecules made of cabon atoms aanged in pentagonal and hexagonal faces, and thei coesponding gaphs, fulleene gaphs, ae 3-connected, cubic plana gaphs with only pentagonal and hexagonal faces. By Coollay 3, if G is the class of fulleenes, then lim n {J(G); G G and V(G) = n} = 0. We emak that the uppe bound given in Theoem is vey ough. Fo instance, if G is the well-known Buckminste fulleene, then ou bound with = 3 gives J(G) = 4.5, while J(G) = 90 1 = 05 = 0.91 (by a mistake, in [5] log 6/ this value is doubled). In what follows, fo fulleenes we give a bette uppe bound fo the Balaban index. Notice that the bound in the next theoem tends to 0 fo n much faste than 18/ log (n+)/3. Theoem 4. Let G be a fulleene gaph on n 60 vetices. Then J(G) 5 n. Poof. We ague similaly as in the pevious poof. Let G be a fulleene on n vetices and let u V(G). Let n i be the numbe of vetices at distance i fom u. Then n 0 = 1 and n 1 = 3. Moeove in [1, Lemma 6], it is shown that n i+1 n i + 3 fo i 1. This immediately gives the bound n i 3i fo i 1. We obtain a lowe bound of w(u) by assuming each n i = 3i fo i 1, as in this way we have fewe vetices at highe distance. So w(u) = i n i s 3s+(s+1)c, i fo some c and s, whee 0 c < 3s+3 and s+c = n. Hence, 3( s+(s+1)) n, and fom hee s +3s+ n 3.

6 -54- Since n 60, we have s 5, and hence s 3s+, which gives s n/3. Since s is intege, we obtain Consequently, s n/3. w(u) n/3 +c( n/3 +1) 3 n/3 j=1 j Thus, n/3 ) 3 ( n/3 ) ( +3 + n/3 > 3 6 J(G) > n 3 n 3. 3 m m n+ m 1 w(u) n 3 n n+ 3 n 1 n 7 3 n < 5 n. 3 n 3 4 Cubic gaphs with small value of Balaban index Thee ae cubic gaphs fo which the Balaban index tends to 0 even faste than fo the fulleene gaphs. While fo the fulleene gaphs Balaban index is bounded by 5n 1/, these cubic gaphs have the Balaban index 3n 1 and even less. Let n be divisible by 4, and let H n be a gaph obtained fom the cycle of length 3 n 4, in which evey thid vetex is doubled, see Figue 1 fo H 16. Obseve that H n is obtained fom n/4 copies of K 4 e (i.e., K 4 without one edge) which ae joined by n/4 exta edges to fom a connected cubic gaph. Obviously, H n has n vetices. We have the following statement. Figue 1: The gaph H 16. Poposition 5. Fo positive n divisible by 4, it holds J(H n ) 3 n.

7 -55- Poof. Fo simplicity, let l = 3n/4. Analogously as in the pevious poofs, fist we give a lowe bound fo w(u), u V(H n ). In ode to do so, we find the total distance cw(u) fom u to the vetices of the oiginal cycle. If l is even, then cw(u) = l ( l ) ( l = +1 ) l ) +( = l 4. Similaly, if l is odd, then ( cw(u) = l 1 ) = ( l+1 ) = l 1. 4 As thee is at least one vetex in H n not on the oiginal cycle and diffeent fom u, and as the distance of this vetex to u is at least one, fo both the above cases we get w(u) cw(u)+1 > l 4 = 9n 64. Hence, J(G) 3 n 3 n n+ 3n 1 9n 64 < 3 n. The last esult means that if n appoaches to, then J(H n ) appoaches to 0 quite fast. Howeve, among cubic gaphs on n vetices H n does not have the smallest value of Balaban index. We intoduce a class of gaphs L n with J(L n ) < J(H n ). Since it seems to be difficult to find a good uppe bound fo J(L n ), below we pesent a table of values obtained by a compute pogam. Let n be even and n 10. If 4 n, then L n is obtained fom (n 10)/4 copies of K 4 e joined to a path by edges connecting the vetices of degee, to which at the ends we attach two pendant blocks, each on 5 vetices, see Figue fo L 18. On the othe hand if 4 n, then L n is obtained fom (n 1)/4 copies of K 4 e, joined into a path by edges connecting the vetices of degee, to which ends we attach two pendant blocks, one on 5 vetices and the othe on 7 vetices, see Figue 3 fo L 0. Figue : The gaph L 18.

8 -56- Figue 3: The gaph L 0. n /n J(H n ) J(L n ) Table 1: Balaban index fo H n and L n fo small numbes of vetices. In Table 1 we have J(L n ) fo small values of n, as well as J(H n ) (if n is divisible by 4) and the bound 3 n. We conclude the pape with a conjectue about L n. Conjectue 6. Among n-vetex cubic gaphs, L n has the smallest Balaban index. 5 Concluding emaks Beside the above conjectue one can study some moe poblems elated to the esults of this pape. As a divesity of the Balaban index J, futhe indices wee developed omitting the faction facto m/(m n+) befoe the sum, and it would be inteesting to exploe simila bounds as in this pape fo indices intoduced in [13]. Shotly speaking, the main esult of ou pape is showing that 0 (beside π and othe values mentioned in [16]) is also an accumulation point fo Balaban index. Fom this point of view, it would be inteesting poblem to find some moe new accumulation points fo this index. In this pape, we wee esticted to simple cubic but geneal gaphs. Note that fom the point of view of chemisty one inteesting class of cubic multigaphs ae annulenes, and also some othe paticula cubic gaphs ae. So as futhe wok one could extend ou study to multigaphs as well as estict to some paticula egula gaphs. See [5,6,9,14,15] fo some (chemical) esults of such classes of gaphs. Acknowledgements: We would like to thank the anonymous efeee fo poviding us with constuctive comments, and fo pointing out the possibilities of futhe investigations. The fist autho acknowledges patial suppot by Slovak eseach gants VEGA 1/0781/11, VEGA 1/0065/13, APVV and APVV All authos wee patially suppoted by Slovenian eseach agency ARRS, pogam no. P , poject no. L1 49, and Ceative Coe FISNM

9 -57- Refeences [1] V. Andova, T. Došlić, M. Knc, B. Luža, R. Škekovski, On the diamete and some elated invaiants of fulleene gaphs, MATCH Commun. Math. Comput. Chem. 68 (01) [] M. Aouchiche, G. Capoossi, P. Hansen, Refutations, esults and conjectues about the Balaban index, Int. J. Chem. Model. 5 (013) [3] A. R. Ashafi, H. Shabani, M. V. Diudea, Balaban index of an infinite class of dendimes, Coat. Chem. Acta 83 (010) [4] A. R. Ashafi, H. Shabani, M. V. Diudea, Balaban index of dendimes, MATCH Commun. Math. Comput. Chem. 69 (013) [5] A. T. Balaban, Chemical gaphs. VIII. Valence isomeism and geneal cubic gaphs, Rev. Roum. Chim. 15 (1970) [6] A. T. Balaban, Chemical gaphs. XIV. Valence isomes of [1]-annulene, Rev. Roum. Chim. 17 (197) [7] A. T. Balaban, Highly disciminating distance based numeical descipto, Chem. Phys. Lett. 89 (198) [8] A. T. Balaban, Topological indices based on topological distances in molecula gaphs, Pue Appl. Chem. 55 (1983) [9] A. T. Balaban, Chemical gaphs. 44. Isome enumeation fo substituted satuated polyhedal valence isomes of annulenes, Rev. Roum. Chim. 31 (1986) [10] A. T. Balaban, Chemical gaphs. Pat 48. Topological index J fo heteoatom containing molecules taking into account peiodicities of element popeties, MATCH Commun. Math. Comput. Chem. 1 (1986) [11] A. T. Balaban, A compaison between vaious topological indices, paticulaly between the index J and Wiene index W, in: D. H. Rouvay, R. B. King (Eds.), Topology in Chemisty Discete Mathematics of Molecules, Howood, Chicheste, 00, pp [1] T. S. Balaban, A. T. Balaban, D. Bonchev, A topological appoach to pedicting popeties of infinite polymes. Pat VI. Rational fomulas fo the nomalized Wiene index and a compaison with index J, J. Mol. Stuct. (Theochem) 535 (001) [13] A. T. Balaban, A. Beteinghe, T. Constantinescu, P. A. Filip, O. Ivanciuc, Fou new topological indices based on the molecula path code, J. Chem. Inf. Model. 47 (007) [14] A. T. Balaban, R. O. Davies, F. Haay, A. Hill, R. Westwick, Cubic identity gaphs and plana gaphs deived fom tees, Austal. J. Math. 11 (1970) [15] A. T. Balaban, C. Deleanu, Chemical gaphs. 47. Valence isomes of [14]annulene, Rev. Roum. Chim. 3 (1987)

10 -58- [16] A. T. Balaban, N. Ionescu Pallas, T. S. Balaban, Asymptotic values of topological indices J and J (aveage distance sum connectivities) fo infinite cyclic and acyclic gaphs, MATCH Commun. Math. Comput. Chem. 17 (1985) [17] A. T. Balaban, X. Liu, D. J. Klein, D. Babić, T. G. Schmalz, W. A. Seitz, M. Randić, Gaph invaiants fo fulleenes, J. Chem. Inf. Comput. Sci. 35 (1995) [18] A. T. Balaban, P. V. Khadika, C. T. Supuan, A. Thaku, M. Thaku, Study on supamolecula complexing ability vis à vis estimation of pk(a) of substituted sulfonamides: Dominating ole of Balaban index (J), Bioog. Med. Chem. Lett. 15 (005) [19] H. Deng, On the Balaban index of tees, MATCH Commun. Math. Comput. Chem. 66 (011) [0] B. Deng, A. Chang, Maximal Balaban index of gaphs, MATCH Commun. Math. Comput. Chem. 70 (013) [1] J. Devilles, A. T. Balaban (Eds.), Topological Indices and Related Desciptos in QSAR and QSPR, Godon & Beach, Amstedam, [] H. Dong, X. Guo, Chaacte of gaphs with extemal Balaban index, MATCH Commun. Math. Comput. Chem. 63 (010) [3] X. Dong, L. You, The maximum Balaban index (sum Balaban index) of unicyclic gaphs, J. Math. Res. Appl. 34 (014) [4] P. W. Fowle, Resistance distances in fulleene gaphs, Coat. Chem. Acta 75 (00) [5] M. Ghobani, Remaks on the Balaban index, Sedica J. Comput. 7 (013) [6] G. Gassy, B. Calas, A. Yasi, R. Lahana, J. Woo, S. Iye, M. Kaczoek, F. Floćh, R. Buelow, Compute assisted ational design of immunosuppessive compounds, Natue Biotechnol. 16 (1998) [7] A. Ianmanesh, Y. Alizadeh, Balaban and Randić indices of IPR C 80 fulleene isomes, zigzag nanotubes and gaphene, Int. J. Nanosci. Nanotechnol. 7 (011) [8] P. V. Khadika, C. T. Supuan, A. Thaku, M. Thaku, QSAR study on benzene sulphonamide cabonic anhydase inhibitos: topological appoach using Balaban index, Bioog. Med. Chem. 1 (004) [9] H. W. Koto, J. R. Heath, S. C. O Bien, R. F. Cul, R. E. Smalley, C60: Buckminstefulleene, Natue 318 (1985) [30] L. Sun, Bounds on the Balaban index of tees, MATCH Commun. Math. Comput. Chem. 63 (010) [31] R. Todeschini, V. Consonni, Handbook of Molecula Desciptos, Wiley VCH, Weinheim, 000. [3] B. Zhou, N. Tinajstić, Bounds on the Balaban index, Coat. Chem. Acta. 81 (008)

ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi

Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated