South Asian Journal of Mathematics 01, Vol ( 5): 07 1 wwwsajm-onlinecom ISSN 51-15 RESEARCH ARTICLE Extremal Kirchhoff index of a class of unicyclic graph Shubo Chen 1, Fangli Xia 1, Xia Cai, Jianguang Yang 1, Lihui Yang 1 1 College of Mathematics, Hunan City University, Yiyang, Hunan 13000, PR China E-mail: shubochen@13com Received: July-15-01; Accepted: Sep-5-01 *Corresponding author Projects supported by Natural Science Foundation of Hunan Province(No 015JJ3031), Scientific Research Fund of Hunan Provincial Education Department (No 1A0, 1C05) Abstract The resistance distance between two vertices of a connected graph G is defined as the effective resistance between them in the corresponding electrical network constructed from G by replacing each edge of G with a unit resistor The Kirchhoff index of Kf(G) is the sum of resistance distances between all pairs of vertices of the graph G In this paper, we shall characterize a class of unicyclic graph with the maximum and minimum Kirchhoff index Key Words MSC 010 Resistance distance; Kirchhoff index; unicyclic graph 05C35, 05C0 1 Introduction All graphs considered here are both connected and simple if not stated in particular For any v V (G), we use N G (v) to denote the set of the neighbors of v, and let N G [v] = v N G (v), let d(v) be the number of edges incident with v If d(v) 3, then we call v a branched vertex The distance between vertices v i and v j, denoted by d(v i, v j ), is the length of a shortest path between them In 17, American Chemistry H Wiener in [1] defined the famous Wiener index as W(G) = {v i,v j} V (G) d(v i, v j ) (11) and in 13 Klein and Randić [] introduced a new distance function named resistance distance on the basis of electrical network theory They viewed a graph G as an electrical network N such that each edge of G is assumed to be a unit resistor Then, the resistance distance between the vertices v i and v j, are denoted by r(v i, v j ), is defined to be the effective resistance between nodes v i, v j N Analogous to the definition of the Wiener index, the Kirchhoff index Kf(G) of a graph G is defined as[, 3] Kf(G) = {v i,v j} V (G) r(v i, v j ) () Citation: Shubo Chen, Fangli Xia, Xia Cai, Jianguang Yang, Lihui Yang, Extremal Kirchhoff index of a class of unicyclic graph, South Asian J Math, 01, (5), 07-1
S Chen, et al : Extremal Kirchhoff index of a class of unicyclic graph If G is a tree, then r(u, v) = d(u, v) for any two vertices u and v Consequently, the Kirchhoff and Wiener indices of trees coincide The Kirchhoff index is an important molecular structure descriptor[], it has been well studied in both mathematical and chemical literatures For a general graph G, I Lukovits et al [5] showed that Kf(G) n 1 with equality if and only if G is complete graph K n, and P n has maximal Kirchhoff index Palacios [] showed that Kf(G) 1 (n3 n) with equality if and only if G is a path For more information on the Kirchhoff index, the readers are referred to recent papers[7-17] and references therein Let P n1 be the path with the vertices v 1, v,, v n1, the graph T(n, i, 1) is construct from P n1 by adding one pendant edge to the vertex v i ( i n ), the graph T(n, i, 1) is depicted in Figure 1 v n v 1 v v i v n1 Figure 1 The graph T(n, i, 1) A graph G is called a unicyclic graph if it contains exactly one cycle The unicyclic graph U(C k ; T(l, i, 1))(k 3, i l, k l = n 1) is the graph obtained from cycle C k by joining v 1 of T(l, i, 1) to a vertex of C k U(C k ; T(l, i, 1)) is showed in Figure v l C k v 1v v i v l1 Figure The graph U(C k ; T(l, i, 1)) The paper is organized as follows In Section we state some preparatory results, whereas in Section 3 we investigated the Kirchhoff index of U(C k ; T(l, i, 1)), and give an order for Kf(U(C k ; T(l, i, 1))) with respect to the values of i In section, we determine graph with the maximum Kirchhoff index in U(C k ; T(l, i, 1)), whereas in Section 5, we obtain graph with the minimum Kirchhoff index in U(C k ; T(l, i, 1)) Preliminary Results For a graph G with v V (G), G v denotes the graph resulting from G by deleting v (and its incident edges) For an edge uv of the graph G (the complement of G, respectively), G uv (G uv, respectively) denotes the graph resulting from G by deleting (adding, respectively) uv For a vertex u V (G), let Kf G (u) = r(u, v), then Kf(G) = 1 Kf G (u) v V (G) have r Cn (v i, v j ) = u V (G) Let C n be the cycle on n 3 vertices, for any two vertices v i, v j V (C n ) with i < j, by Ohm s law, we (j i)(n i j) For any vertex v V (G), one has Kf v (C n ) = n 1, Kf(C n ) = n n 3 n 08
South Asian J Math Vol No 5 Lemma 1([]) Let x be a cut vertex of a connected graph and a and b be vertices occurring in different components which arise upon deletion of x Then r G (a, b) = r G (a, x) r G (x, b) (1) Lemma ([7]) Let G 1 and G be two connected graphs with exactly one common vertex x, and G = G 1 G Then Kf(G) = Kf(G 1 ) Kf(G ) ( V (G 1 ) 1)Kf x (G ) ( V (G ) 1)Kf x (G 1 ) () 3 The Kirchhoff index of U(C k ; T(l, i, 1)) Theorem 31 Let T(l, i, 1) be the graph depicted in Figure 1 Then Kf(T(l, i, 1)) = 1 l3 5 l il i 1 (31) Proof The Kirchhoff and Wiener indices coincide for trees, it s ease to see that i li Kf(T(l, i, 1)) = W(T(l, i, 1)) = W(P l1 ) k k = 1 l3 5 l il i 1 k=1 k= since W(P l1 ) = 1 (l 1)(l l) The proof is completed Theorem 3 Let U(C k ; T(l, i, 1)) (k 3, i l, k l = n 1) be the graph depicted in Figure Then Kf(U(C k ; T(l, i, 1))) = i (n k)i 1 (n3 3k 3 nk k nk 0n 1k) Proof Let G = U(C k ; T(l, i, 1)), G 1 = C k, G = T(l, i, 1), then G 1 and G sharing the common vertex v 1 By Lemma, one has Kf(G) = Kf(C k ) Kf(T(l, i, 1)) (k 1)Kf v1 (T(l, i, 1)) (l 1)Kf v1 (C k ) It s easy to calculated out that Kf v1 (T(l, i, 1)) = 1 (l )(l 1) i Bearing in the mind that k l = n 1 Thus, Kf(G) = 1 (k3 k) 1 (l 1)(l l) (k 1) ( 1 (l )(l 1) i) (l 1) 1 (k 1) = i (k l 1)i 1 (k3 l 3 kl k l k l 18kl 11k l ) = i (n k)i 1 (n3 3k 3 nk k nk 0n 1k) 0
S Chen, et al : Extremal Kirchhoff index of a class of unicyclic graph This completes the proof Theorem 33 Let U(C k ; T(l, i, 1)) (k 3, i l, k l = n 1) be the graph depicted in Figure Then, (i) when k 3 n k, Kf(U(C k ; T(l,, 1))) < Kf(U(C k ; T(l, 3, 1))) < < Kf(U(C k ; T(l, n k 1, 1))) (ii) when n k, and n is even, Kf(U(C k ; T(l, n k 1, 1))) > Kf(U(C k ; T(l, n k, 1))) > > Kf(U(C k ; T(l, n k, 1))) = Kf(U(C k ; T(l,, 1))) > Kf(U(C k ; T(l, n k 1, 1))) = Kf(U(C k ; T(l, 3, 1))) > > Kf(U(C k ; T(l, n 1 k, 1))) (iii) when n k, and n is odd, Kf(U(C k ; T(l, n 1 k, 1))) > Kf(U(C k ; T(l, n k, 1))) > > Kf(U(C k ; T(l, n k 1, 1))) = Kf(U(C k ; T(l, 3, 1))) > Kf(U(C k ; T(l, n k, 1))) = Kf(U(C k ; T(l,, 1))) > > Kf(U(C k ; T(l, n 3 k, 1))) = Kf(U(C k ; T(l, n 1 k, 1))) Proof Let f(i) = i (nk)i 1 (n3 3k 3 nk k nk0n1k), i nk1 It s easy to see that (1) when n 1 k, ie, k 3 n k, f(i) is increasing in [, n k 1] Then Kf(U(C k ; T(l,, 1))) < Kf(U(C k ; T(l, 3, 1))) < < Kf(U(C k ; T(l, n k 1, 1))) holds () when n 1 k and n is even f(i) is decreasing in the interval [, n 1 k], and increasing in the interval [ n 1 k, n k 1] Then (ii) holds (3) when n n 1 1 k and n is odd, f(i) is decreasing in the interval [, k], and increasing in the interval [ n 3 k, n k 1] Then (iii) holds From above conclusion we arrive at Theorem 33 This completes the proof As a consequence of Theorem 33, we are able to determine the graph in U(C k ; T(l, i, 1))(k 3, i l, k l = n 1) with the maximum and minimum Kirchhoff index Corollary 3 Let G U(C k ; T(l, i, 1))(k 3, i l, k l = n 1), then 10
South Asian J Math Vol No 5 (i) when k 3 n k, (n 3)k (n )k n3 n ; (n 3)k (n 1)k n3 8n 18 The first equality holds if and only G = U(C k ; T(n 1 k,, 1)) and the second does if and only if G = U(C k ; T(n 1 k, n 1 k, 1)) (ii) when n k, and n is even, then (n 3)k (n 1)k n3 3n 8n ; (n 3)k (n 1)k n3 8n 18 The first equality holds if and only G = U(C k ; T(n 1 k, n 1 k, 1)) The second does if and only if G = U(C k ; T(n 1 k, n 1 k, 1)) (iii) when n k, and n is odd, then (n 3)k (n 3)k (n 1)k (n 1)k n3 3n 8n ; n3 8n 18 The first equality holds if and only G = U(C k ; T(n 1 k, n 1 k, 1)) The second does if and only if G = U(C k ; T(n 1 k, n 1 k, 1)) The maximum Kirchhoff index of U(C k ; T(l, i, 1)) Theorem 1 max {Kf(U(C k; T(l, i, 1)))} = Kf(U(C 3 ; T(n, n, 1))) 3 k n3 Proof From above analysis, one can see that U(C k ; T(n 1 k, n 1 k, 1)) has the maximum Kirchhoff index when k is given In the following, we shall investigate the maximum value of Kf(U(C k ; T(n 1 k, n 1 k, 1))) Let f(k) : = Kf(U(C k ; T(n 1 k, n 1 k, 1))) = k3 (n 3)k (n 1)k n3 8n 18 In what follows, we shall find the maximum value of f(k) on I := [3,,, n 3] The first derivative of f(k) is f (k) = 3 k n 3 k n 1 3 The roots of f (k) = 0 are k 1, = (n ) 1n n 11
S Chen, et al : Extremal Kirchhoff index of a class of unicyclic graph It is easy to see that for n 3, k 1 < n (n 1) = 3, k > n (1 n) In the following, we will show that f(3) is the maximum value of f(k) on I For n, it s easy to verify that k n 3 Then, one has (i) when k [3, k ), f (k) < 0, which indicates that f(k) is decreasing on [3, k ); (ii) when k [k, n 3], f (k) > 0, which indicates that f(k) is increasing on [k, n 3] So, the maximum value of f(k) must occurred between f(3) and f(n 3) It s suffice to see that f(3) f(n 3) = 1 n3 n 7n 15 > 0 Thus, f(3) > f(n 3) for n Then This completes the proof = 3 max {Kf(U(C k; T(l, i, 1)))} = f(3) = Kf(U(C 3 ; T(n, n, 1))) 3 k n3 From Theorem 1, it is suffice to see that, Theorem Let G U(C k ; T(l, i, 1)), k 3, k l = n 1, i l Then Kf(G) n3 17n, with the equality holds if and only if G = U(C 3 ; T(n, n, 1)) 5 The minimum Kirchhoff index of U(C k ; T(l, i, 1)) In this section, we shall determine graph in U(C k ; T(l, i, 1)) with the minimum Kirchhoff index Theorem 51 min Kf(U(C k; T(l, i, 1))) 3 k n3 1 (11n3 n 80n 1), 3 k [ n ], n is even; = 1 (11n3 n 3n 81), 3 k [ n ], n is odd; Kf ( U(C k ; T(n 1 k,, 1)) ), [ n ] k n 3, 8 n 1; n 3 n 5n, [n ] k n 3, n 1 where k = n 1n n 07, [ n ] stands for the integer part of n Proof By Corollary 3, one can see that (1) If k3 n k, U(C k ; T(n1k,, 1)) has the minimum Kirchhoff index in U(C k ; T(l, i, 1)) () If n k, and n is even, U(C k ; T(n 1 k, n 1 k, 1)) has the minimum Kirchhoff index in U(C k ; T(l, i, 1)) (3) If n k, and n is odd, U(C k ; T(n 1 k, n 1 k, 1)) has the minimum Kirchhoff index in U(C k ; T(l, i, 1))
South Asian J Math Vol No 5 Based on above results, one arrives at (1) When 3 k [ n ], k 3 (n 3)k Kf(G) k 3 (n 3)k (n 1)k (n 7)k n3 3n 8n, n is even; n3 0n 3, n is odd () When [ n ] k n 3, (n 3)k (n )k n3 n In the following, our discussion are divided into two cases: Case 1 3 k [ n ] Let h(k) = k3 (n 3)k (n 1)k n3 3n 8n Then h (k) = 3k (n 3)k 3 n 1 Let h (k) = 0, then k 1, = n 1n n It is easy to see that k 1 < 3, k > [ n for 3 k [ n ] (11) n is even ] Thus h (k) < 0 for 3 k [ n ], h(k) is decreasing min Kf(G) = Kf ( U(C n 3 k [ n ] 1 ; T( n,, 1))) = 1 ( 11n 3 n 80n 1 ) () n is odd min Kf(G) = Kf ( U(C n3; T( n 5,, 1)) ) = 1 ( 11n 3 n 3n 81 ) 3 k [ n ] Case [ n ] k n 3, Let I(k) = k3 (n 3)k (n )k I (k) = 3k n3 n, n k n 3 Then (n 3)k 3 n Let I (k) = 0, then k 1, = n 1n n 07 (i) when 8 n 1 It is easy to verify that k < n 3, thus I (k) < 0 for [ n ] k k and I (k) > 0 for k k n 3 That is to say that I(k) is decreasing for [ n ] k k and increasing for k k n 3 min Kf(G) = Kf ( U(C k ; T(n 1 k,, 1)) ) [ n ] k n3 (ii) when n 1 13
S Chen, et al : Extremal Kirchhoff index of a class of unicyclic graph It is easy to verify that k > n 3, thus I (k) < 0 for [ n ] k n 3, I(k) is decreasing for [ n ] k n 3 min Kf(G) = Kf ( U(C n3 ; T(,, 1)) ) = n3 [ n ] k n3 n 5n This completes the proof References 1 H Wiener, Structural determination of paraffin boiling points, J Amer Chem Soc17, : 17-0 D J Klein and M Randić, Resistance distance, J Math Chem 13, : 81-5 3 D Bonchev, A T Balaban, X Liu, D J Klein, Molecular cyclicity and centricity of polycyclic graphs I Cyclicity based on resistance distances or reciprocal distances, Int J Quantum Chem 1, 50: 1-0 J L Palacios, Foster s Formulas via Probability and the Kirchhoff index, Methodology and Computing in Applied Probability 00, : 381-387 5 I Lukovits, S Nikolić, N Trinajstić, Resistance distance in regular graphs, Int J Quantum Chem 1, 71: 17-5 J L Palacios, Resistance distance in graphs and random walks, Int J Quantum Chem 001, 81: -33 7 Y Yang, H Zhang, Unicyclic graphs with extremal Kirchhoff index, MATCH Commun Math Comput Chem 008, 0: 107-0 8 B Zhou, N Trinajstić, A note on Kirchhoff index, Chem Phys Lett 008, 5: 0-3 B Zhou, N Trinajstić, The kirchhoff index and the matching number, Int J Quantum Chem 00, 10: 78-81 10 W Zhang, H Deng, The second maximal and minimal Kirchhoff indices of unicyclic graphs, MATCH Commun Math Comput Chem 00, 1: 83-5 11 Q Guo, H Deng, D Chen, The extremal Kirchhoff index of a class of unicyclic graphs, MATCH Commun Math Comput Chem 00, 1: 713-7 H Zhang, X Jiang, Y Yang, Bicyclic graphs with extremal Kirchhoff index, MATCH Commun Math Comput Chem 00, 1: 7-7 13 H Deng, On the minimum Kirchhoff index of graphs with a given number of cut-edges, MATCH Commun Math Comput Chem 010, 3: 171-180 1 J L Palacios and J M Renom, Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks, Int J Quantum Chem 010, 110: 137-11 15 H Wang, H Hua, D Wang, Cacti with minimum, second minimum, and third-minimum Kirchhoff indices, Mathematical Communications 010, 15: 37-358 1 R Li, Lower bounds for the Kirchhoff index, MATCH Commun Math Comput Chem, 013, 70: 13-17 17 L Feng, G Yu, K Xu, and Z Jiang, A note on the Kirchhoff index of bicyclic graphs, Ars Combinatoria 01,11: 33-0 18 Xia Cai, Z Guo, S Wu, L Yang, Unicyclic graphs with the second maximum Kirchhoff index, South Asian J Math 01, (): 181-18 1