Filomat 8:8 (014), 1711 1717 DOI 10.98/FIL1408711D Publishe by Faculty of Sciences an Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On the Atom-Bon Connectivity Inex of Cacti Hawei Dong a, Xiaoxia Wu b a Department of Mathematics, Minjiang University, Fuzhou Fujian 350108, China b School of Mathematics an Statistics, Minnan Normal University, Zhangzhou Fujian 363000, China Abstract. The Atom-Bon Connectivity (ABC) inex of a connecte graph G is efine as ABC(G) = (u)+(v) uv E(G), where (u) is the egree of vertex u in G. A connecte graph G is calle a cactus if (u)(v) any two of its cycles have at most one common vertex. Denote by G 0 (n, r) the set of cacti with n vertices an r cycles an G 1 (n, p) the set of cacti with n vertices an p penent vertices. In this paper, we give sharp bouns of the ABC inex of cacti among G 0 (n, r) an G 1 (n, p) respectively, an characterize the corresponing extremal cacti. 1. Introuction Let G be a simple graph with vertex set V(G) an ege set E(G). For u V(G), the egree of u, enote by (u). The Atom-Bon Connectivity (ABC) inex of G is efine as [8] ABC(G) = uv E(G) (u) + (v). (1) (u)(v) The ABC inex was shown to be well correlate with the heats of formation of alkanes, an that it thus can serve for preicting their thermoynamic properties [8]. Various properties of the ABC inex have been establishe, see chemical literature [0][1]. In aition to this, Estraa [7] elaborate a novel quantum-theory-like justification for this topological inex, showing that it provies a moel for taking into account 1,-, 1,3-, an 1,4-interactions in the carbon-atom skeleton of saturate hyrocarbons, an that it can be use for rationalizing steric effects in such compouns. These results triggere a number of mathematical investigations of ABC inex [1]- [6], [9]- [17], []- [5]. 010 Mathematics Subject Classification. Primary 05C35 ; Seconary 05C07, 05C90 Keywors. Atom-Bon Connectivity inex, cactus, upper boun Receive: 8 July 013; Accepte: 0 July 014 Communicate by Francesco Belaro Research supporte by the Founation to the Eucational Committee of Fujian (JA166 an JA108), by the NSF of Fujian (01D140) an by the Scientific an Technological Startup Project of Minjiang University (No. YKQ1009). Email aress: onghawei@16.com (Hawei Dong)
H. Dong, X. Wu / Filomat 8:8 (014), 1711 1717 171 Let G be a graph. The neighborhoo of a vertex u V(G) will be enote by N(u), (G) = max(u) u V(G)} an δ(g) = min(u) u V(G)}. The graph that arises from G by eleting the vertex u V(G) will be enote by G u. Similarly, the graph G + uv arises from G by aing an ege uv between two non-ajacent vertices u an v of G. A penent vertex of a graph is a vertex with egree 1. If all of blocks of G are either eges or cycles, i.e., any two of its cycles have at most one common vertex, then G is calle a cactus. We use G 0 (n, r) to enote the set of cacti with n vertices an r cycles an G 1 (n, p) to enote the set of cacti with n vertices an p penent vertices. Obviously, G 0 (n, 0) are trees, G 0 (n, 1) are unicyclic graphs an G 1 (n, n 1) is star. The star with n vertices, enote by S n, is the tree with n 1 penent vertices. Let G(n, r) enote the cactus obtaine by aing r inepenent eges to the star S n (See Fig.1 (a)). Obviously, G(n, r) is a cactus with n vertices, r cycle, an n r 1 penent vertices. Note that G(n, r) G 0 (n, r) an G(n, r) G 1 (n, n r 1). G (n, p) enotes the cactus obtaine by aing (n p 1)/ inepenent eges to the star S n if n p is o an by aing (n p )/ inepenent eges to the start S an then inserting a egree -vertex in one of those inepenent eges if n p is even (See Fig.1(b)). Obviously, G (n, p) G 1 (n, p) an G (n, p) G(n, (n p 1)/) when n p is o. r n - p - n-r-1 G(n,r) p G (n,p) ( a ) ( b) Figure 1: G(n, r) an G (n, p) for n p is even It is straightforwar to compute ABC(G(n, r)) = 3r n + (n r 1) n 1, ABC(G (n, p)) = 3(n p 1) 3(n p ) n n + 1 i f n p is o, i f n p is even. For connectivity inex of cacti, Lu et al.[19] gave the sharp lower boun on the Ranić inex of cacti; Lin an Luo [18] gave the sharp lower boun of the Ranić inex of cacti with fixe penant vertices. In this paper, we use the techniques in [19] an [18] to give sharp upper bouns on ABC inex of cacti among G 0 (n, r) an cacti among G 1 (n, r), an characterize the corresponing extremal graphs, respectively. x+y For x, y 1, let h(x, y) = xy. Obviously, h(x, y) = h(y, x). This lemma will be useful in the following sections.
H. Dong, X. Wu / Filomat 8:8 (014), 1711 1717 1713 Lemma 1.1. [] h(x, 1) = x 1 x is strictly increasing with x, h(x, ) = x+y an h(x, y) = xy is strictly ecreasing with x for fixe y 3.. The Maximum ABC Inex of Ccacti with r Cycles In this section, we will stuy the sharp upper boun on ABC inex of cacti with n vertices an r cycles. Obviously, G 0 (n, 0) are trees with n vertices. Furtula et al [] prove that the star tree, S n, has the maximal ABC value among all trees with n( ) vertices. Lemma.1. [9] Let G G 0 (n, 0) an n, then ABC(G) () S n. n with equation if an only if G G(n, 0) Obviously, G 0 (n, 1) are unicyclic graphs. Xing et al [4] gave the upper boun for ABC inex of unicyclic graphs an characterize extremal graph. Lemma.. [3] Let G G 0 (n, 1) an n 3, then ABC(G) (n 3) G G(n, 1). n + 3 with equation if an only if Theorem.3. Let G G 0 (n, r), n 5. Then ABC(G) F(n, r) with equality if an only if G G(n, r), where F(n, r) = 3r n + (n r 1). Proof. By inuction on n + r. If r = 0 or r = 1, then the theorem hols clearly by Lemma.1 an.. Now, we assume that r, an n 5. If n = 5, then the theorem hols clearly by the facts that there is only one graph G(5, ) in G 0 (5, ) (see Fig. 1). Let G G 0 (n, r), n 6 an r. We consier the following two cases. Case 1. δ(g) = 1. Let u V(G) with (u) = 1 an uv E(G). Denote (v) = an N(v)\u} = x 1, x,, x 1 }. Then n 1. Assume, without loss of generality, that (x 1 ) = (x ) = = (x k 1 ) = 1 an (x i ) for k i 1, where k 1. Set G = G u x 1 x x k 1, then G G 0 (n k, r). By inuction assumption an Lemma 1.1, we have ABC(G) = ABC(G ) + k ABC(G ) + k 1 1 1 [h(, (x i )) h( k, (x i ))] + 1 (by Lemma 1.1) F(n k, r) + k (by inuctive assumption) n = F(n, r) (n r 1) + (n r 1 k) n k n k 1 + k n k = F(n, r) + (n r 1 k)( n k 1 n ) + k( 1 F(n, r). (by Lemma 1.1) 1 n ) The equality ABC(G) = F(n, r) hols if an only if equalities hol throughout the above inequalities, that is, if an only if = n 1, r = n k 1 an G G(n k, r). Thus we have ABC(G) = F(n, r) hols if an only if G G(n, r).
Case. δ(g). H. Dong, X. Wu / Filomat 8:8 (014), 1711 1717 1714 By the efinition of cactus an δ(g), there exist two eges u 0 u 1, u 0 u E(G) such that (u 0 ) = (u 1 ) = an (u ) = 3. We will finish the proof by consiering two subcases. Subcase 1. u 1 u E(G). Let G G u 0 + u 1 u. Then G G 0 (n 1, r). By inuction assumption an Lemma 1.1, we have ABC(G) = ABC(G ) + 1 F(n 1, r) + 1 (by inuctive assumption) = F(n, r) + 1 n (n r 1) + (n r ) n = F(n, r) + [ 1 n ] + [(n r ) n (n r ) n ] < F(n, r). (by Lemma 1.1) Subcase. u 1 u E(G). Let N(u )\u 0, u 1 } = x 1, x,, x }. (x i ) for 1 i, since δ(g). Let G = G u 0 u 1. Then G G 0 (n, r 1). By inuctive assumption an Lemma 1.1, we have ABC(G) = ABC(G ) + 3 + [h(, (x i )) h(, (x i ))] F(n, r 1) + 3 + [h(, (x i )) h(, (x i ))] (by inuctive assumption) = F(n, r) 3 n (n r 1) + (n r 1) n 4 + 3 + [h(, (x i )) h(, (x i ))] n 4 = F(n, r) + (n r 1)[ n ] + [h(, (x i )) h(, (x i ))] F(n, r). (by Lemma 1.1) The equality ABC(G) = F(n, r) hols if an only if equalities hol throughout the above inequalities, that is, if an only if G G(n, r 1), r = an (x i ) = for i = 1,,. Thus we have ABC(G) = F(n, r) hols if an only if G G(n, r). 3. The Maximum ABC Inex of Cacti with p Penent Vertices In this section, we will stuy the sharp upper boun on ABC inex of cacti with n vertices an p penent vertices. Denote by S 1, the tree obtaine by attaching one penent vertex to a penent vertex of the star S. Lemma 3.1. [5] Let T be a tree with n 4 vertices an T S n, then ABC(T) (n 3) n + with equality hols if an only if T S 1,. Theorem 3.. Let G G 1 (n, p), n 4. n (1) If p = n 1, then ABC(G) = (n 1) an G S n. () If p = n, then ABC(G) + (n 3) n with equality if an only if G S 1,. (3) If p n 3, then ABC(G) f (n, p) with equality if an only if G G (n, p) where 3(n p 1) f (n, p) = n i f n p is o, 3(n p ) n + 1 i f n p is even.
Proof. H. Dong, X. Wu / Filomat 8:8 (014), 1711 1717 1715 (1) If p = n 1, then G S n. So, the result is obvious. () Since G has n penent vertices, it is a tree an n 4. By Lemma 3.1, S 1, is the unique graph with the maximum ABC inex among trees with n( 4) vertices an n penent vertices. This result follows. (3) By inuction on n. If n = 4 an p n 3, then n = 4 an p = 0, that is, G C 4. Since C 4 G (4, 0), the theorem hols clearly for n = 4. Now, we assume that n 5. Case 1. p 1. Let u V(G) with (u) = 1 an uv E(G). Denote (v) = an N(v)\u} = x 1, x,, x 1 }. Then n 1. Assume, without loss of generality, that (x 1 ) = (x ) = = (x k 1 ) = 1 an (x i ) for k i 1, where k 1. If (G) = n 1, then (v) = n 1 an each block of G is either a triangle or an ege by the efinition of cactus. It follows that G G(n, n p 1 ) an n p 1 is even. So, we assume that (G) n. Set G = G u x 1 x x k 1, then G G 1 (n k, p k). Note that n p an n k (p k) have the same parity. By inuction assumption an Lemma 1.1, we have = = ABC(G) = ABC(G ) + k ABC(G ) + k 1 1 1 + (p k) 1 + [h(, (x i )) h( k, (x i ))] (by Lemma 1.1) f (n k, p k) + k (by inuctive assumption) n n k f (n, p) p n k 1 + k 1, i f n p is o f (n, p) p n + (p k) n k 3 n k + k 1, i f n p is even n k f (n, p) + (p k)( n k 1 n ) + k( 1 n ), n k 3 f (n, p) + (p k)( n k n ) + k( 1 n ), f (n, p). (by Lemma 1.1) i f n p is o i f n p is even The equality ABC(G) = f (n, p) hols if an only if equalities hol throughout the above inequalities, that is, if an only if = n 1, p = k an G G (n k, p k). Thus we have ABC(G) = f (n, p) hols if an only if G G (n, p). Case. p = 0. By the efinition of cactus an p = 0, there exist two eges u 0 u 1, u 0 u E(G) such that (u 0 ) = (u 1 ) = an (u ) = 3. We will finish the proof by consiering two subcases. Subcase 1. u 1 u E(G). Let G G u 0 + u 1 u. Then G G 1 (n 1, 0).
H. Dong, X. Wu / Filomat 8:8 (014), 1711 1717 1716 If n is o, then f (n 1, 0) = 3() + 1 an f (n, 0) = 3(). Thus, by inuctive assumption, we have ABC(G) = ABC(G ) + 1 f (n 1, 0) + 1 = f (n, 0) 1 < f (n, 0). If n is even, then f (n 1, 0) = 3(n ) an f (n, 0) = 3(n ) ABC(G) = ABC(G ) + 1 f (n 1, 0) + 1 = f (n, 0). + 1. Thus, by inuctive assumption, we have The equality ABC(G ) = f (n 1, 0) hols if an only if G G (n 1, 0). Hence, ABC(G) = f (n, 0) hols if an only if G G (n, 0) for n is even. Subcase. u 1 u E(G). Let N(u )\u 0, u 1 } = x 1, x,, x }. (x i ) for 1 i, since δ(g). Let G = G u 0 u 1. Then G G 1 (n, 0). By inuctive assumption an Lemma 1.1, we have ABC(G) = ABC(G ) + 3 + [h(, (x i )) h(, (x i ))] f (n, 0) + 3 + [h(, (x i )) h(, (x i ))] = f (n, 0) 3 + 3 + [h(, (x i )) h(, (x i ))] f (n, 0). (by Lemma 1.1) (by inuctive assumption) The equality ABC(G) = f (n, 0) hols if an only if equalities hol throughout the above inequalities, that is, if an only if G G (n, 0) an (x i ) = for i = 1,,. Thus we have ABC(G) = f (n, 0) hols if an only if G G (n, 0). References [1] J. Chen, X. Guo, Extreme atom-bon connectivity inex of graphs, MATCH Communications in Mathematical an in Computer Chemistry 65(011) 713 7. [] J. Chen, X. Guo, The atom-bon connectivity inex of chemical bicyclic graphs, Applie Mathematics 7 (01) 43 5. [3] J. Chen, J. liu, Q. li, The Atom-Bon Connectivity Inex of Cataconense Polyomino Graphs, Discrete Dynamics in Nature an Society, 013(013) 1 7. [4] K. C. Das, Atom-bon connectivity inex of graphs, Discrete Applie Mathematics 158 (010) 1181 1188. [5] K. C. Das, I. Gutman, B. Furtula, On atom-bon connectivity inex, Chemical Physics Letters 511 (011) 45 454. [6] K. C. Das, N. Trinajstić, Comparison between first geometric-arithmetic inex an atom-bon connectivity inex, Chemical Physics Letters 497 (010) 149 151. [7] E. Estraa, Atom-bon connectivity an the energetic of branche alkanes, Chemical Physics Letters 463 (008) 4 45. [8] E. Estraa, L. Torres, L. Roríguez an I. Gutman, An atom -bon connectivity inex: Moelling the enthalpy of form ation of alkanes, Inian Journal of Chemistry 37A (1998) 849 855. [9] B. Furtula, A. Graovac an D. Vuki ccević, Atom Cbon connectivity inex of trees, Discrete Applie Mathematics 157 (009) 88 835. [10] L. Gan, H. Hou, B. Liu, Some results on atom-bon connectivity inex of graphs, MATCH Communications in Mathematical an in Computer Chemistry 66(011) 669 680. [11] I Gutman, B Furtula, M Ivanovic, Notes on Trees with Minimal AtomCBon Connectivity Inex, MATCH Communications in Mathematical an in Computer Chemistry 67 (01) 467 48. [1] I. Gutman, B. Furtula, Trees with Smallest Atom-Bon Connectivity Inex, MATCH Communications in Mathematical an in Computer Chemistry 68 (01) 131 136. [13] B. Horolagva, I. Gutman, On some vertex-egree-base graph invariants, MATCH Communications in Mathematical an in Computer Chemistry 65(011) 73 730. [14] X. Ke, Atom-bon connectivity inex of benzenoi systems an fluoranthene congeners, Polycyclic Aromatic Compouns 3 (01) 7 35. [15] J. Li, B. Zhou, Atom-bon connectivity inex of unicyclic graphs with perfect matchings, Ars Combinatoria 109 (013) 31 36. [16] W. Lin, X. Lin, T. Gao, X. Wu, Proving a conjecture of Gutman concerning trees with minimal ABC inex, MATCH Communications in Mathematical an in Computer Chemistry 69 (013) 549 557. [17] W. Lin, T. Gao, Q, Chen, X. Lin, On The Minimal ABC Inex of Connecte Graphs with Given Degree Sequence, MATCH Communications in Mathematical an in Computer Chemistry 69 (013) 571 578.
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