Scientific paper Maximal Unicyclic Graphs With Respect to New Atom-bond Connectivity Index Kinkar Ch. Das,1 Kexiang Xu2 and Ante Graovac3 1 Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea. 2 College of Science, Nanjing University of Aeronautics & Astronautics, Nanjing, Jiangsu, PR China 3 Faculty of Science, University of Split, Nikole Tesle 12, HR-21000 Split, Croatia * Corresponding author: E-mail: kinkardas2003@googlemail.com Received: 23-04-2012 Abstract The concept of atom-bond connectivity (ABC) index was introduced in the chemical graph theory in 1998. The atombond connectivity (ABC) index of a graph G defined as ^ where E(G) is the edge set and di is the degree of vertex vi of G. Very recently Graovac et al.1 define a new version of the ABC index as where ni denotes the number of vertices of G whose distances to vertex vi are smaller than those to the other vertex vj of the edge e = vi v., and n. is defined analogously. In this paper we determine the maximal unicyclic graphs with respect to new atom-bond connectivity index (ABC2). Keywords: Unicyclic graph, Atom-bond connectivity (ABC) index, New atom-bond connectivity (ABC2) index, Upper bound 1. Introduction Mathematical chemistry is a branch of theoretical chemistry using mathematical methods to discuss and predict molecular properties without necessarily referring to quantum mechanics.2-4 Chemical graph theory is a branch of mathematical chemistry which applies graph theory in mathematical modeling of chemical phenomena.5 This theory has an important effect on the development of the chemical sciences. Topological indices are numbers associated with chemical structures derived from their hydrogen-depleted graphs as a tool for compact and effective description of structural formulas which are used to study and predict the structure-property correlations of organic compounds. Molecular descriptors play significant role in chemistry, pharmacology, etc. Among them, topological indices have a prominent place.6 One of the best known and widely used is the connectivity index, %, introduced in 1975 by Milan Randic.7 Estrada et al. proposed a new index, known as the atom-bond connectivity index (ABC).8 This index is defined as Where E(G) is the edge set and di is the degree of vertex vi of G. The ABC index has proven to be a valuable predictive index in the study of the heat of formation in alkanes.8-9 The maithematical properties of this index was reported. 10-19 Let G = (V,E) be a simple connected graph with vertex set V(G) = {v1, v2, ..., vn} and edge set E(G), where \V(G)\ = n and \E(G)\ = m. Let dt be the degree of vertex vt for i = 1, 2, ..., n. A vertex of a graph is said to be pendent if its neighborhood contains exactly one vertex. An edge of a graph is said to be pendent if one of its vertices is a pendent vertex. We denote by Cn and Pn, the cycle and the path on n vertices, respectively, throughout this paper. For other undefined notations and terminology from graph theory, the readers are suggested to refer.20 Let G be a connected graph and e = v^j be an edge of G. The number of vertices of G whose distance to the vertex vi is smaller than the distance to the vertex vj is denoted by ni = n(e\G). Analogously, nj = n(e\G) is the number of vertices of G whose distance to the vertex vj is smaller than to v. Graovac et al.1 define a new version of the ABC index as ABC Example 1.1. Dendrimers are nanostructures that can be precisely designed and manufactured for a wide variety of applications, such as drug delivery, gene delivery and diagnostics etc. Let D[n] be a dendrimer where n is the step of growth in it. Note that, in D[n], there are 4(1 + 2 + ... + 2n1) + 1 = 4(2n - 1) + 1 vertices and 4(2n - 1) edges. By the definition of (ABC2)-index, we find that 1 1 2>i-l . - 1 ' 3(2" - 1) + 1 + 2n_1 + • 1 • + 2"-' (2i1_ - 1)[3(2" - 1) + 1 + 2n-1 + ■•+ 2n_i] | 3(2" - 1) + 2" - 2 J (2n-i - 1)[3(2" - 1} + 1 + 21,-1 + •■ + 2n"'] | 2"+2 - 5 (2"-i - ï)[3(2n - 1)+ 1 + 2""1 + •■•+ 2""'] For the edges linking the vertices on ith layer and the ones on (i + 1) th layer each of which occurs 2'+2 times. When n = 1, D[n] is just star S5. So we have ABC2(D[1]) = ABC2(S5) = 2^3. Set Therefore, for n > 2, we have for i = 1,2.....n- 1. -l)|3(2n-l) + l| + ZÏL-112i+2Ai — A I___L Vp-l 7Ï/I, For example, we have Example 1.2. Any connected graph with maximum degree not exceeding 4 is called molecular graph. Any (molecular) graph is called conjugated if it has a perfect matching. Conjugated unicyclic graphs have some important applications in chemistry, especially mathematical chemistry. Conjugated hydrocarbon molecules considered in the Huckel molecule orbital theory are usually represented by the carbon-atom skeleton graphs with perfect matching, of which all vertices have degrees not more than 4. For more details of chemical applications of conjugated molecular graphs, see [21, 22, 23]. Let U(k) be the set of conjugated unicyclic graphs of order 2k. Two conjugated unicyclic molecular graphs, denoted by U1(k) and U2(k), are shown in Fig. 1. Figure 1. Two graphs Uj(k) and U2(k). Now we will calculate the ABC2 index of U1(k) and U2(k) as follows: , , 12k — 2 , . 1 \2k — 4 ABCM m = 0c- 1) — + Ck - 2) - + 2 —. ABC2QJ2m = kJW^I +1) when k is even and ABC2W2m - k I If ï +Jet-«1) when k is odd. The goal of this paper is to determine the maximal unicyclic graphs with respect to new atom-bond connectivity index (ABC2). 2. Some Lemmas In this section, we shall list some results that will be needed in the next section. Lemma 2.1. Let n > 8 be a positive integer. Then and and 1 n-3 ^ ln-4 3{n-4) " ij |n-3 H+ 1 „-3 I n-4 Proof: We have n — 3 2 1 -r = 1+---r Jn— S J n-5 n- 5 Using the above result, we get (V3 + l)n-4V3-3- V3(n - 3) > {V3 + l)n - 4V3 - 3 - V3(n - 3) r - 3 n -5 n2-(8 + 2V3)n + lS + 8V3 n-5; (1) n-5 Let us consider a function ««-JiEs+^-Jf-Ji! Liz.— 3(*-4) x> 8. Then rm= ((V3 + l)x - 4i/3 - 3) vT^S - - 3)3 (i/3 + l)x - 4^3 - 3 ~ - 3) JP!) 2v/3(x-5)(;c--3)3(*-4)3 (x-4-V3) -4 > 0 asx> 8 and by (1). Thus fx) is an increasing function for x > 8. Since f(15) < 0 and f(16) > 0,we get the required result. Lemma 2.2. Let n>l be a positive integer. Then fi+M+M^+M+M'"111*9 (2) and li+ pEE< ji+ [J •J 4 ^n-S ij+(jl-5) — \J 3 lJn-4 ij3(il 3(il-4) for n * 9. (3) Moreover, the equality holds in (3) if and only if n = 8. Proof: For n = 7, -+V2 ~ 2.28 < 2.3 « 2 - + - ■\|4 3 and (3) holds. For n = 8 the equality holds in (3). For n = 9, and (2) holds. We have (i+ -2352 <2353 - J+J!+M,or 71=io- and Now it remains to prove this lemma for n > 14. For this, let us consider a function f(x) = , * > 14. (X-5)\/X-4 Then (* - 5)V* - 4 (2* - 4}(3jc - 13) -(x~4)(x- 5) 2(x - 3)(x - 4)(x - 5){r - 8) > 0 as x > 14 and (2x-4)(3x-13) > (x-4)(x-5). Thus we have f (x) is an increasing function for x > 14 and hence f (x) > 0.699 for x > 14. Now, = yJn^M V3(" - 4) v;4()i - 5) Vl2Cit-4)(/i-5) (a - S)^/^! J5 |12(n - 4){n - 5) (V4(n - 5} + V3fn- 4)) as 4(«-5) > 3(n-4) > 0.0504 > J^ - J^ as n > 14 and /0) > 0.699. Using the above result In-6 In—s for n>6, we get the required result (3) for n > 14 Lemma 2.3. Let x, n be positive integers with (n-3)/2 12. Now we have to prove the following two claims. Claim 1. Let x,n be positive integer numbers with (n-3)/2 < x < n - 6, n > 12. Also let r be a positive integer number such that x + r = n - 3. Then and (i) ,J(x 4- r)xr > n (ii) (x + r)j(x + r)xr > 3 (xr + x + r) + 1. (5) Proof of Claim 1 (i). Since (n-3)/2 < x < n - 6 and x + r = n - 3, we have x > r > 3. Again since both x and r are integers with x + r = n - 3, then the minimum value of xr is 3(n - 6). Now we have to show that V3(n- 3)(n- 6) > n, that is, 2n2 -27n + 54 > 0, that is, n > 12, which, evidently, is always obeyed. Proof of Claim 1 (ii). Now, by Arithmetic-Geometric Mean Inequality <3(x + r)(2 + f) = 30 + r) J 4 + 2Vxr +j Now we have to show that •—xr >■—hii+1 as x + r = — 3, 9 4 that is, that is, > n + 1 as xr > 3(n - 6), that is, 4n2 - 57n + 114 > 0, that is, n > 12, which, evidently, is always obeyed. Claim 2. Let x, n be positive integer numbers with (n-3)/2 < x < n - 6, n > 12. Then 1 1 Jx(x + l)3 y/(n-3-xXn-2-x)2 \/n - 3(2x - n + 3) VO + l)3(n -2-x)3 (6) Proof of Claim 2. Let r be a positive integer number such that x + r = n - 3. Since (n-3)/2 < x < n - 6 and x + r = n - 3, we have x > r > 3. Now, (74* + l)3 + VRr+lF) - r)V(ii - 3)xr > + 1) + r(r + l))(a: - r)J(x + r)xr by Claim 1 (i) and (ii) = (x4 -r+) + 3(x3 - r3) + S^2 - r2) + (x - r) as x + r = n - 3 = x(x + l)3 - r(r + l)3. From the above inequality, we can easily get + l)3 - Vr(r + l):i < (x - r)v'(n - 3)*r, that is, From the above, we get the required result (6). Let us consider a function ffx") — I " + l"~3~x -l- I "~3" (x+l)(n-2-x) 1 — < x < n - 6, n > 12. i Then we have i /'(*) = and by Claim 2. Thus / (x) is strictly increasing function for (n-3)/2 < x < n - 6, n > 12. Hence we get the required result (4) for n > 12. Moreover, the equality holds in (4) if and only if x = n - 6. Next we assume that n < 11. Since (n-3)/2 < x < n -6, we have n > 9. Thus (n,x) = (9,3) or (10, 4) or (11, 5). For (n,x) = (9,3), the equality holds in (4). For (n,x) = (10,4), the equality holds in (4). For (n,x) = (11,4), and (4) holds. Moreover, for (n,x) = (11, 5), the equality holds in (4). This completes the proof. Theorem 2.4. Let x, n be positive integer numbers with (n-3)/2 < x < n - 4. (i) If n = 7, then > 0. /Z+ [EEE+ I -3 ¿2 E+z ijx+l ijn-z-x ^Oi+lXn-2-X) 1J3 3 with equality holding in (7) if and only if x = 2. (ii) If n = 8, then f]l+ pE+ I < ii+ j£+ /1 aJx+1 ^n-2-x (x+l)(tl-2-x) ^4 12 with equality holding in (8) if and only if x = 3. (iii) If n = 9, then (8) I x 111-3—x I n-3 aJ n-2-x -^(x+lKn-2-x) 16, then I x ln-3-x I ri—3 ■\Jx+1 y n-2-x y](x+l)(n-; < V2 + — (11) (x+l)(n-2-x) -\J n-3 with equality holding in (11) if and only if x = n - 4. Proof: (i) Since n = 7, we have either x = 2 or x = 3. We have J+V^2.28<2.3*2J4 Using the above result, we get the required result (7). Moreover, the equality holds in (7) if and only if x = 2. (ii) Since n = 8, we have either x = 3 or x = 4. We have Using the above result, we get the required result (8). Moreover, the equality holds in (8) if and only if x = 3. (iii) Since n = 9, we have either x = 3 or x = 4 or x = 5. We have -+ V2 = 2.327 < 2.343 Using the above result, we get the required result (9). Moreover, the equality holds in (9) if and only if x = 3. (iv) If x = n - 5, then the equality holds in (10). Otherwise, x ± n - 5. Since 10 < n < 15, by Lemmas 2.1, 2.2 and 2.3, we get I x n—3—x n—3 VxTT + Jn - 2 - x + J(x+ l)(n - 2 - x) 2 In — 5 < |3 + Jn-4 + j3(n-4) ri - 3 r- Iri - 4 >Vz+ I--. n — 3 (v) If x = n - 4, then the equality holds in (11). Otherwise, x ± n - 4. Since n > 16, by Lemmas 2.1, 2.2 and 2.3, we get This completes the proof. 3. Upper Bound on the ABC2 Index of Unicyclic Graphs Figure 2. S(0, 2, 5, 4, 2, 1, 4, 3). Now we turn to determine the maximal new atombond connectivity index (ABC2) among all connected unicyclic graphs of order n. Let S(m1, m2, ..., mk) be a uni-cyclic graph of order n with girth k and n - k pendent vertices, where mi is the number of pendent vertices adjacent to i-th vertex of the cycle. We consider that the vertices in the cycle are numbered clockwise (see Fig. 2). Clearly, Xki=1 mi = n - k and S(0, 0, ..., 0) = Cn The cycle of a unicyclic graph G is denoted by C(G). Denote by C'4 is a uni-cyclic graph of order 5 obtained from cycle C4 with one pendent edge attached to any one vertex of cycle C4. Denote by C'3, is a unicyclic graph of order 5 obtained from cycle C3 with one end of path P3 attached to any one vertex of cycle C3. Let G be a connected graph of order n. For 2 < l < n < n, vvvj e E(G), ± + S—i + ifl-i) + as ni > / ni rij rtirij n/ rij \ tii/ I \ n*/ 1 <1(1-1) as m>l. Thus we have < I .ha _ ii (12) J tti ny rijiij 1 y with equality holding if and only if ni = h. = l. Moreover, for pendent edge vivj e E(G), 11 1 2 _ jn-2 (13) Lemma 3.1. Suppose that vv. is a cut-edge of con' J nected unicyclic graph G of order h (>3), but v.v. is not a pendent edge. Let vt denote the vertex obtained from identifying vi and vj in Gvivj, and G1 = Gvivj + vv (See, Fig. 3). Then ABC2(G) < ABC2(G1). Proof: Denote by Figure 3. Two graphs g and G1. Then we have ABC2(G)= ^ ABC2 (G,vriQ. vrvs£E(C) From given condition we get ABC2(G, vrvs) = ABC2 (G\ v^, v^ * vvj. Moreover, for vv. e E(G), h. > 2 and h. > 2 as vv. is i j i j i j not a pendent edge in G. Thus we have ABC2{G,vivj)= I + i.-Lfi 1 Moreover, vv. is a pendent edge in G1 and hence 1 J ABC%(G\vivj)= + — = & 1 ]J ^Jii rij n/ii; ijn-l Now we have ABC2(Gl) - ABC2(G) = ^ ABC2(G\vrVs) VrftEBW) ^ ABC2(G,vrvs) = ^BCaCGSvii?,)- j4BC2(G,v£Vy) Theorem 3.2. Let G be a connected unicyclic graph of order h (>3) with girth k. Then ABCZ(G) < ABC^Sim^ m2l..., mk)) (14) with equality holding if and only if G £ S(m1, m2, ..., mk). Proof: If G is isomorphic to S(m1, m2, ..., mk), then the equality holds in (14). Otherwise, G £ S(m1, m2, ..., mk). Then there exists a non-pendent edge v.v. in G such that vv<£ E(C(G)). We consider the transformation dei. fined in Lemma 3.1. Then by Lemma 3.1, we have ABC2 (G) < ABC2 (G1), that is, we have increased the value of (ABC2)-index. If G1 is S(m1, m2, ..., mk), then we are done. Otherwise, we continue the same transformation for sufficient number of times, we arrive at S(m1, m2, ..., mk). This completes the proof. Now we give an upper bound on the ABC2 - index of unicyclic graph G in terms of order H. Theorem 3.3. Let G be a connected unicyclic graph of order h (> 3). (i) If H = 4, then ABC2(G) < 2V2 (15) with equality holding in (15) if and only if G £ C4 (ii) If h = 5, then ABC2(G) < V3 (16) with equality holding in (16) if and only if G £ S(1, 1, 0). (iii) If h = 6, then ABC2(G)<3Jj + V2 + Jl (17) with equality holding in (17) if and only if G £ S(2, 1, 0). (iv) If H = 7, then ABC2(C)<{n-3)j£ + 2jj+§ (18) with equality holding in (18) if and only if G £ S(2, 2, 0). (v) If « = 8, then ABC2(G)<(n-3)J^+JI + Jt + JI with equality holding in (19) if and only if G = S(3, 2, 0). (vi) If n = 9, then ABC2(G)<(n~3)J^ + V3+Jl (20) with equality holding in (20) if and only if G = S(3, 3, 0). (vii) If 10 < n < 15, then ABC2(G)<(n~3)J^ + Jj + In—5 I n-S -^n-4 y]3(n- (21) 4) with equality holding in (21) if and only if G = S(n - 5, 2, 0). (viii) If n > 16, then ABC2(G) <(n~ 3+ V2 + (22) with equality holding in (22) if and only if G = Sin - 4,1,0). Proof: If n = 4, then G = C4 or G = S(1, 0, 0). We have ¿SC2(S(1,0,0)) - V2 + < 2V2 = ABC2(C4). From the above, we get the required result (15). Moreover, the equality holds in (15) if and only if G = C4 If n = 5, then G = C5 or G = C4'or G = S(1, 1, 0), or G = S(2, 0, 0), or G = C3'. Now we have ABCz(Cs) = J? + ~ + zj^ « 3.206 < 3.365 as ABCZ(S( 1,1,0)). From the above, we get the required result (16). Moreover, the equality holds in (16) if and only if G = S(1, 1, 0). Otherwise, n > 6. Let G be a connected unicyclic graph of order n with girth k. Then k > 3. We consider two cases (a) k > 4, (b) k = 3. Case (a): k > 4. In this case there are at most n - 4 pendent edges and at least 4 non-pendent edges in uni-cyclic graph G. Since k > 4, for each non-pendent edge vvvj e E(G), ni > 2 and nj > 2 as G is unicyclic graph. For each non-pendent edge vv. e E(G), by (12), ij ilTTT^ < ß< ß as n > 3. (23) -J n; nj njiiy -\J 2 -\jn-l Using the above result and by (13), we get ABC2(G) - IVlV/eE(C),dt*i !-+--—+ ZvtvjEEWAdj*! Jn. + Jj n.n. (24) Case (b): k =3. By Theorem 3.2, we get ABC2 (G) < ABC2 (S(m1, m2, m3)). Moreover, the equality holds if and only if G = S(m1, m2, m3). Without loss of generality, we can assume that m1 > m2 > m3 > 0, m1 + m2 + m3 = n - 3. Now we consider the following three subcases: Subcase (i): m1 > m2 > m3 > 1. For each non-pendent edge vjvj e E(S(m1, m2, m3)), ni > 2 and nj > 2. By (12), we get Using the above result, we get ABC2 (G) < ABC2 (S(m1, m2, m3)) I Vje£(s(m1(m2fm3))Jiii=i (25) Subcase (ii): m2 = m3 = 0. There are exactly one non-pendent edge vv e E(S(n - 3, 0, 0)) such that ni = 1 and nj = 1 and hence l± + ±—^ = 0. •J "i nj ntrtj Also there are exactly two non-pendent edaes v.v. e EiSin ij - 3, 0, 0)) such that ni = 1 and nj = n - 2 and hence 1 __J F^ nj Jll-3 n-Z nj ninj Thus we have ABC2 (G) < ABC2 iSin - 3, 0, 0)) ■ I I r, v j e g(S(n - 3,0 ,o)) .if 1 d j * l K1 2 4n' nJ n,tij 2 - TliTlj Subcase (iii): m3 = 0. In this subcase m1 + m2 = n - 3, m1 > m2 > 1, that is, (n - 3)/2 < m1 < n - 4. Now, ABC2 (G) < ABC2 (S(mv m2, 0)) I I i»(i'ye E(s (m m ¡ ,i:il ),£(, íí , * 1 ' 1 1 2 — +---+ n¡ Tij n¡ Tij i+i-i- I "j "y 1 1 + - Jm, 4-1 m2 + 1 (mx + l)(m2 + 1) From the above result and by Theorem 2.4, we get the following: If n = 6, then ABC2(G) < ABC2{S(mi, m2,0)) = ABC2{S(2,1,0)) = with equality holding if and only if G = S(2, 1, 0). If n = 7, then with equality holding if and only if G = S(2, 2, 0). If n = 8, then ABC2(C) < ABCz(S(m1, m2, 0)) < 5 JÏ+JÏ+JÏ+JI with equality holding if and only if G = S(3, 2, 0). If n = 9, then ABC2(G) < ABC2(S(m1, m2, 0)) < + V3 + J? with equality holding if and only if G = S(3, 3, 0). If 10 < n < 15, then ABC2(G) < ^5^(5(7«!, m2, 0)) < with equality holding if and only if G = S(n - 5, 2, 0). If n > 16, then ABC2(G) < ABC2(S(m„ m2, 0)) < with equality holding if and only if G = S(n - 4, 1, 0). Now we have by Theorem 2.4 and n > 10 and for n > 6. Using the above results, we get ABC2(G) < ABC2(S(2,1,0» - ABC2(G)^ < ABC2(S(2, 2, 0)) = ABC2(G) < ABC2(S(3,2,0)) = j4BC2(G) < /lfîC2(5(3,3,0)) = ABC2(G) < ABC2(S(n - 4, 1, 0)) - Using the above results with (24), (25) and (26), we get the required result. This completes the proof. 4. Conclusion Graovac et al.1 define the ABC2 index as a new version of the ABC index. In this paper we obtain the maximal unicyclic graphs with respect to new atom-bond connectivity index (ABC2). Maximal new atom-bond connectivity index in the case of bicyclic graphs and minimal atom-bond connectivity index in the case of trees, unicyclic graphs and bicyclic graphs, remains an open problem. Moreover, some extremal graphs with respect to new ABC index are still unknown which include certain chemical structure such as fullerene, benzenoid hydrocarbons, etc. And finding the chemical application of this new ABC index is more attractive in the near future. 5. Acknowledgement The authors would like to thank two anonymous referees for valuable comments and useful suggestions, which improved this work very much. K. Ch. D. thanks for support by the Faculty research Fund, Sungkyunkwan University, 2012 and Sungkyunkwan University BK21 Project, BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea. K. X. thanks for support of NNSF of China (No. 11201227). 6. References 1. A. Graovac, M. Ghorbani, Acta Chim. Slov. 2010, 57, 609612. 2. S. J. Cyvin, I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons, Lecture Notes in Chemistry, Vol. 46, Springer Verlag, Berlin, 1988. 3. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer Verlag, Berlin, 1986. 4. N. Trinajstic, I. Gutman, Croat. Chem. Acta, 2002, 75, 329356. 5. A. Graovac, I. Gutman, N. Trinajstic, Topological Approach to the Chemistry of Conjugated Molecules, Springer Verlag, Berlin, 1977. 6. R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000. 7. M. Randic, J. Am. Chem. Soc. 1975, 97, 6609-6615. 8. E. Estrada, L. Torres, L. Rodriguez, I. Gutman, Indian J. Chem. 1998, 37 A, 849-855. 9. E. Estrada, Chem. Phys. Lett. 2008, 463, 422-425. 10. K. C. Das, Discrete Appl. Math. 2010, 158, 1181-1188. 11. K. C. Das, N. Trinajstic, Chem. Phys. Lett. 2010, 497, 149151. 12. K. C. Das, I. Gutman, B. Furtula, Chem. Phys. Lett. 2011, 511, 452-454. 13. K. C. Das, I. Gutman, B. Furtula, Filomat, 2012, 26(4) 733738. 14. G. H. Fath-Tabar, B. Vaez-Zadeh, A. R. Ashrafi, A. Graovac, Discrete Applied Math. 2011, 159, 1323-1330. 15. B. Furtula, A. Graovac, D. Vukicevic, Discrete Applied Math. 2009, 157, 2828-2835. 16. I. Gutman, B. Furtula, M. Ivanovic, MATCH Commun. Math. Comput. Chem. 2012, 67(2), 467-482. 17. R. Xing, B. Zhou, Z. Du, Discrete Appl. Math. 2010, 157, 1536-1545. 18. R. Xing, B. Zhou, F. Dong, Discrete Applied Math. 2011, 159, 1617-1630. 19. B. Zhou, R. Xing, Z Naturforsch. 2011, 66 (a), 61-66. 20. J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Macmillan Press, New York, 1976. 21. T. Zivkovic, N. Trinajstic, R. Randic, Molecular Physics 1975, 30, 517-532. 22. W. Liu, Scientia Sinica 1979, 5, 539-554. 23. I. Gutman, J. Serb. Chem. Soc. 2005, 70, 441-456. Povzetek Koncept »atom-vez« indeksa povezanosti (ABC) je bil pred kratkim vpeljan v kemijsko teorijo grafov. »Atom-vez« indeks povezanosti (ABC) grafa G je definiran kot ^ ^^ ci + dj — 2 kjer je E(G) niz povezav in di stopnja vozlišča (točke) vi od G. Graovac in soavtorji je definiral novo verzijo ABC indeksa kot Zliii + n, — 2 r^T' BjPjiHC« N ' kjer ni predstavlja število vozlišč v G katerega razdalje do vozlišča vi so manjše od tistih do drugega vozlišča v. povezave e = vi v., ni pa je definiran analogno. V tem članku determiniramo maksimalne enociklične grafe glede na novi »atom-vez« indeks povezanosti (ABC2).