A lower bound for the harmonic index of a graph with minimum degree at least two

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1 Filomat 7:1 013), DOI 10.98/FIL W Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: A lower bound for the harmonic index of a graph with minimum degree at least two Renfang Wu a, Zikai Tang a, Hanyuan Deng* a a College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing Ministry of Education of China), Hunan Normal University, Changsha, Hunan , P. R. China Abstract. The harmonic index HG) of a graph G is defined as the sum of the weights of all edges uv du)dv) of G, where du) denotes the degree of a vertex u in G. We give a best possible lower bound for the harmonic index of a graph a triangle-free graph, respectively) with minimum degree at least two and characterize the extremal graphs. 1. Introduction In this work, we consider the harmonic index. For a simple graph or a molecular graph) G = V, E), the harmonic index HG) is defined in [1] as HG) = uv EG) du)dv), where du) denotes the degree of a vertex u in G. Favaron et al. [] considered the relation between harmonic index and the eigenvalues of graphs. Zhong [3] found the minimum and maximum values of the harmonic index for simple connected graphs and trees, and characterized the corresponding extremal graphs. Deng, Balachandran, Ayyaswamy, Venkatakrishnan [4] considered the relation relating the harmonic index HG) and the chromatic number χg) and proved that χg) HG) by using the effect of removal of a minimum degree vertex on the harmonic index. It strengthens a result relating the Randić index and the chromatic number conjectured by the system AutoGraphiX and proved by Hansen et al. in [5], since we always have HG) RG) for any graph G. Deng, Tang, Zhang [6] considered the harmonic index HG) and the radius rg) and strengthened some results relating the Randić index and the radius in [7] [8] [9]. Deng, Balachandran, Ayyaswamy, Venkatakrishnan [10] determined the trees with the second-the sixth maximum harmonic indices, and unicyclic graphs with the second-the fifth maximum harmonic indices, and bicyclic graphs with the first-the fourth maximum harmonic indices. For other related results see [11] [1] [13] [14]. Here we will establish a best possible lower bound for the harmonic index of a graph, a triangle-free graph, respectively, with n vertices and minimum degree at least two and characterize the extremal graphs.. A lower bound for the harmonic index of a graph with minimum degree at least two In the section, we will establish a best possible lower bound for the harmonic index of a graph with minimum degree at least two and characterize the extremal graphs. 010 Mathematics Subject Classification. Primary 05C07; Secondary 05C15 Keywords. Graph, the harmonic index, the minimum degree Received: 10 May 01; Accepted: 6 September 01 Communicated by Dragan Stevanović Research supported by Hunan Provincial Natural Science Foundation of China 13JJ3053) and the Program Excellent Talent Hunan Normal University ET13101) address: *Corresponding author: hydeng@hunnu.edu.cn Hanyuan Deng*)

2 R. Wu, T. Tang, H. Deng / Filomat 7:1 013), For an edge e = uv of a graph G, its weight is defined to be du)dv). The harmonic index of G is the sum of weights over all its edges. Lemma.1. If e is an edge with maximal weight in G, then HG e) < HG). Proof. Let e = uv. Since uv is an edge with maximal weight in G, we have dw) dv) for w Nu) and dw) du) for w Nv). Note that 1 x 1 x 1 is increasing for x > 1. HG) HG e) = du)dv) du)dw) du)dw) 1 ) w Nu)\{v} dv)dw) dv)dw) 1 ) w Nv)\{u} du)dv) 1 ) dv) 1) dv)du) dv)du) 1 ) du)dv) du) 1) du)dv) = du)dv) 1 du)dv) > 0 which proves the result. Let K a,b be the complete bipartite graph with a and b vertices in its two partite sets, respectively. For n 4, let K,n be the graph obtained from K,n by joining an edge between the two non-adjacent vertices of degree n. Obviously, HK,n ) = h 1n) = n1. Let δg) be the minimum degree of the graph G. Theorem.. Let G be a graph with n 3 vertices and δg). Then HG) h 1 n) with equality if and only if G = K,n. Proof. It is easy to check that the assertion is true for n = 4. Suppose it holds for 4 k < n; we next show that it also holds for n. Let G be a graph with n > 4 vertices. If δg) 3, then by Lemma 1, the deletion of an edge with maximal weight yields a graph G of minimal degree at least two such that HG ) < HG). So, we only need to prove the result is true for G with δg) =. Case 1. Every pair of adjacent vertices of degree two has a common neighbor. Let u 1 and u be a pair of adjacent vertices with degree two in G which has a common neighbor u 3. Obviously, du 3 ) n 1. Subcase 1.1. If du 3 ) =, let G 1 = G {u 1, u, u 3 }, then HG 1 ) h 1 n 3) by the induction hypothesis, and HG) = HG 1 ) 3 h 1n 3) 3 > h 1n). Subcase 1.. If du 3 ) 4, let G = G {u 1, u }, then HG ) h 1 n ) by the induction hypothesis. Note that 1 x 1 x is increasing for x >. HG) = HG ) 1 4 du 3 ) du 3 )dv) du 3 )dv) ) v Nu 3 )\{u 1,u } du 3 ) ) HG ) 1 4 du 3 ) du 3) ) du 3 ) = HG ) 1 4 du 3 ) 4 du 3 ) h 1 n ) 1 4 du 3 ) 4 du 3 ) h 1 n ) n1 > h 1n). Subcase 1.3. If du 3 ) = 3, let u 4 be the neighbor of u 3 in G different from u 1 and u, where du 4 ) n 3. i) Suppose that du 4 ) =. Denote by u 5 the neighbor of u 4 in G different from u 3, where du 5 ) n 4. Let G 3 = G u 4 u 3 u 5, then HG 3 ) h 1 n 1) by the induction hypothesis. Note that 1 x 1 x1 is decreasing for x > 0. HG) = HG 3 ) 5 du 5 ) du 5 )3 HG 3 ) 5 n h 1 n 1) 5 n > h 1n).

3 R. Wu, T. Tang, H. Deng / Filomat 7:1 013), ii) Suppose that 3 du 4 ) n 3. Let G 4 = G u 1 u u 3, then HG 4 ) h 1 n 3) by the induction hypothesis. Note that x 6 x1 4 x is decreasing for x > 0. HG) = HG 4 ) du 4 )3 du 4 )dv) du 4 )dv) 1 ) v Nu 4 )\{u 3 } du 4 )1 ) HG 4 ) du 4 )3 du 4) 1) du 3 ) = HG 4 ) du 4 )3 6 du ) 4 du 4 )1 HG 4 ) n 6 4 n h 1 n 3) n 6 4 n > h 1n). Case. There is a pair of adjacent vertices of degree two without common neighbor. Let u 1 and u be a pair of adjacent vertices with degree two in G which has no common neighbor. Denote by u 3 the neighbor of u 1 in G different from u. Let G 5 = G u 1 u u 3, then HG 5 ) h 1 ) by the induction hypothesis, and HG) = HG 5 ) 1 h 1n 1) 1 > h 1n). Case 3. There is no pair of adjacent vertices of degree two. Let u be a vertex of degree two with neighbors v and w in G. Subcase 3.1. vw E, where 3 dv) n and 3 dw) n. Let G 6 = G u vw, then HG 6 ) h 1 n 1) by the induction hypothesis. Note that f x, y) = x y xy 3 x n and 3 y n, since f < 0 and f < 0. HG) = HG 6 ) dv) dw) dv)dw) HG 6 ) f n, n ) h 1 n 1) 4 n 1 n > h 1n). f n, n ) for Subcase 3.. vw E, where 3 dv) n 1 and 3 dw) n 1. Let G 7 = G u, then HG 7 ) h 1 n 1) by the induction hypothesis. Note that x, y) = xy 6 x1 6 y1 xy 6 x 6 y n 1, n 1) for 3 x n 1 and 3 y n 1, since ) < 0 and x,3) < 0, and ) < 0 and 3,y) < 0. HG) = HG 7 ) dv) dw) dv)dw) dv)dz) dv)dz) 1 ) dw)dz) dw)dz) 1 ) z Nv)\{u,w} z Nw)\{u,v} HG 7 ) dv) dw) dv)dw) dv) ) dv)) dv)1 ) dw) ) dw) dw)1 ) with equality if and only if dz) = for all z Nv) Nw) \ {u, v, w}) = HG 7 ) dv)dw) 6 dv)1 6 dw)1 dv)dw) ) 6 dv) ) 6 dv) 6 dw) ) HG 7 ) n 1, n 1) with equality if and only if dv) = dw) = n 1) h 1 n 1) 1 1 n 1 n 1 n1 with equality if and only if G 7 = K,n 3 ) = h 1 n) with equality if and only if G = K,n. Hence, the assertion is true for all n A lower bound for the harmonic index of a triangle-free graph with minimum degree at least two In the section, we will give a best possible lower bound for the harmonic index of a triangle-free graph with minimum degree at least two and characterize the extremal graphs. Theorem 3.1. Let G be a triangle-free graph of order n 4 with δg). Then HG) h n) = 4 8 n with equality if and only if G = K,n.

4 R. Wu, T. Tang, H. Deng / Filomat 7:1 013), Proof. It is easy to check that the assertion is true for n = 4. Suppose it holds for 4 k < n; we next show that it also holds for n. Let G be a graph with n > 4 vertices. If δg) 3, then by Lemma 1, the deletion of an edge with maximal weight yields a graph G of minimal degree at least two such that HG ) < HG). So, we only need to prove the result is true for G with δg) =. Case 1. There exists a vertex u of degree two such that the neighbors of u have degree at least three. Let Nu) = {u 1, u } and 3 du i ) n for i = 1,, then δg u) and G u is triangle-free. HG u) h n 1) by the induction hypothesis. HG) = HG u) du 1 ) du ) du 1 )dv) du 1 )dv) 1 ) v Nu 1 )\{u} v Nu )\{u} du )dv) du )dv) 1 ) HG u) du 1 ) du ) du 1) 1) du 1 ) du 1 )1 ) du ) 1) du ) du )1 ) with equality if and only if dv) = for all v Nu 1 ) Nu ) \ {u}) = HG u) 4 du 1 )1 4 du 1 ) 4 du )1 4 du ) HG u) 4 4 n 4 4 n with equality if and only if du 1 ) = du ) = n ) h n 1) 8 8 n with equality if and only if G u = K,n 3) = h n) with equality if and only if G = K,n. Case. Every vertex u of degree two has a neighbor of degree two in G. Let Nu) = {u 1, u } and du 1 ) =, du ) ; Nu 1 ) = {u, v}. Subcase.1. v is not a neighbor of u. Let G 1 = G u u 1 u, then δg 1 ) and G 1 is triangle-free. HG 1 ) h n 1) by the induction hypothesis. HG) = HG 1 ) 1 h n 1) 1 > h n). Subcase.. v is also a neighbor of u. I) If dv) = du ) =, let G = G u v u 1 u, then δg ) and G is triangle-free, implying n 8. HG ) h n 4) by the induction hypothesis. HG) = HG ) h n 4) > h n). II) If none of v, u has degree two, then 3 dv) n 3 and 3 du ) n 3 since G is triangle-free. Let G 3 = G u u 1, then δg 3 ) and G 3 is triangle-free, implying n 6. HG 3 ) h n ) by the induction hypothesis. Note that tx, y) = xy xy 6 x1 6 y1 6 x 6 y tn 3, n 3) for 3 x n 3 and 3 y n 3, since t ) = 4 4 < 0 and t xy) 3 xy ) 3 tx,3) = x3 1x 60x49) < 0, and t x1) x) x3) < 0, similarly. HG) = HG 3 ) 1 w Nv)\{u 1,u } dv) du ) dv)du ) dv)du ) dw)dv) dw)dv) 1 ) du )dw) du )dw) 1 ) w Nu )\{u,v} HG 3 ) 1 dv) du ) dv)du ) dv)du ) dv) ) dv) dv)1 ) du ) ) du ) = HG 3 ) 1 tdv), du )) HG 3 ) 1 tn 3, n 3) h n ) 1 tn 3, n 3) > h n). du )1 )

5 R. Wu, T. Tang, H. Deng / Filomat 7:1 013), III) If exactly one of v, u has degree two, without loss of generality, assume du ) =, then 3 dv) n 3 since G is triangle-free. i)if dv) 4, let G 4 = G u u 1 u, then δg 4 ) and G 4 is triangle-free, implying n 7. HG 4 ) h n 3) by the induction hypothesis. HG) = HG 4 ) 1 4 dv) dw)dv) dw)dv) ) w Nv)\{u 1,u } dv) ) HG 4 ) 1 4 dv) dv) ) dv) = HG 4 ) 1 4 dv) 4 dv) HG 4 ) 1 4 h n 3) 1 4 > h n) n 3 4 n 3 4 ii)if dv) = 3, denote by u 3 the neighbor of v in G different from u 1 and u. a) If du 3 ) =, let u 4 be the neighbor of u 3 in G different from v and G 5 = G u 3 vu 4, then δg 5 ) and G 5 is triangle-free. HG 5 ) h n 1) by the induction hypothesis. And HG) = HG 5 ) 5 du 4 ) du 4 )3 HG 5 ) 5 3 = HG 5 ) 1 h n 1) 1 > h n). b) If du 3 ) 3, then du 3 ) n 5 as G is triangle-free. Let G 6 = G u v u 1 u, we have δg 6 ) and G 6 is triangle-free, implying n 8. HG 6 ) h n 4) by the induction hypothesis. Note that x3 6 x 4 x1 is decreasing for x 0. HG) = HG 6 ) du 3 )3 du 3 )dw) du 3 )dw) 1 ) w Nu 3 )\{v} du 3 )1 ) HG 6 ) 9 5 du 3 )3 du 3) 1) du 3 ) = HG 6 ) 9 5 du 3 )3 6 du 3 ) 4 du 3 )1 HG 6 ) 9 5 h n 4) 9 5 > h n). The proof of our theorem is completed. n 6 n 3 4 n 4 n 6 n 3 4 n 4 References [1] S. Fajtlowicz, On conjectures of Graffiti-II, Congr. Numer ) [] O. Favaron, M. Mahio, J. F. Saclé, Some eigenvalue properties in graphs Conjectures of Graffiti-II), Discrete Math ) [3] L. Zhong, The harmonic index for graphs, Appl. Math. Lett. 5 01) [4] H. Deng, S. Balachandran, S. K. Ayyaswamy, Y. B. Venkatakrishnan, On the harmonic index and the chromatic number of a graph, preprint. [5] P. Hansen, D. Vukicević, Variable neighborhood search for extremal graphs. 3. On the Randić index and the chromatic number, Discrete Math ) [6] H. Deng, Z. Tang, J. Zhang, On the harmonic index and the radius of a graph, preprint. [7] G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs 1: The Autographix system, Discrete Math ) [8] B. Liu, I. Gutman, On a conjecture in Randić indices, MATCH Commun. Math. Comput. Chem ) [9] Z. You, B. Liu, On a conjecture of the Randić index, Discrete Appl. Math ) [10] H. Deng, S. Balachandran, S. K. Ayyaswamy, Y. B. Venkatakrishnan, On harmonic indices of trees, unicyclic graphs and bicyclic graphs, preprint. [11] S. Wang, B. Zhou, N. Trinajstić, On the sum-connectivity index, Filomat 53) 011) 9 4. [1] X. Xu, Relationships between harmonic index and other topological indices, Applied Mathematical Sciences 6 01) [13] A. Ilić, Note on the harmonic index of a graph, Arxiv preprint arxiv: , 01. [14] H. Deng, et al. On the harmonic index and the girth of a graph, preprint.

THE HARMONIC INDEX OF UNICYCLIC GRAPHS WITH GIVEN MATCHING NUMBER

Kragujevac Journal of Mathematics Volume 38(1 (2014, Pages 173 183. THE HARMONIC INDEX OF UNICYCLIC GRAPHS WITH GIVEN MATCHING NUMBER JIAN-BO LV 1, JIANXI LI 1, AND WAI CHEE SHIU 2 Abstract. The harmonic