THE HARMONIC INDEX OF SUBDIVISION GRAPHS. Communicated by Ali Reza Ashrafi. 1. Introduction

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1 Transactions on Combinatorics ISSN print: , ISSN on-line: Vol. 6 No. 07, pp c 07 University of Isfahan toc.ui.ac.ir THE HARMONIC INDEX OF SUBDIVISION GRAPHS BIBI NAIME ONAGH Communicated by Ali Reza Ashrafi Abstract. The harmonic index of a graph G is defined as the sum of the weights deg G u+deg G v of all edges uv of G, where deg G u denotes the degree of a vertex u in G. In this paper, we study the harmonic index of subdivision graphs, t-subdivision graphs and also, S-sum and S t -sum of graphs.. Introduction Throughout this paper, all graphs are finite, simple, undirected and connected. Let G be a graph with vertex set V G and edge set EG. As usual, the maximum and minimum degrees of G are denoted by and δ, respectively. We will use P n, C n and K n to denote the path, the cycle and the complete graph of order n, respectively. For a graph G, the subdivision graph SG is a graph obtained from G by replacing each edge of G by a path of length. The t-subdivision graph S t G of G is a graph obtained from G by replacing each edge of G by a path of length t +. Obviously, S G = SG. Let G and G be two graphs. The S-sum G + S G is a graph with vertex set V G EG V G in which two vertices u, v and u, v of G + S G are adjacent if and only if [ u = u V G and v v EG ] or [ v = v and u u ESG ] []. P 3 + S P is shown in Fig.. MSC00: Primary: 05C07; Secondary: 05C35. Keywords: Harmonic index, subdivision, S-sum, inverse degree, Zagreb index. Received: 07 September 06, Accepted: April 07. 5

2 6 Trans. Comb. 6 no B. N. Onagh Fig.. P 3 + S P For a graph G, the harmonic index HG is defined as HG = deg G u + deg G v. In recent years, this vertex-degree-based topological index has been extensively studied. For example, Zhong [8, 9, 0] gave the minimum and maximum values of the harmonic index for simple graphs, trees, unicyclic graphs and graphs with girth at least k k 3 and characterized the corresponding extremal graphs, respectively. Deng et al. [3] considered the relation between the harmonic index of a graph and its chromatic number. Lv et al. [9, 0] established the relationship between the harmonic index of a graph and its matching number. Shwetha et al. [5] derived expressions for the harmonic index of the join, corona product, Cartesian product, composition and symmetric difference of graphs. Recently, Onagh investigated the harmonic index of product graphs, vertex-semitotal graphs, edgesemitotal graphs, total graphs and F -sum of graphs, where F {R, Q, T } [, 3, ]. More results on this index can been found in [8, 6, 7,, ]. In this paper, we present some sharp bounds for the harmonic index of subdivision and t-subdivision graphs and characterize graphs for which these bounds are best possible. Also, we obtain some upper bounds for the harmonic index of S-sum and S t -sum of graphs.. Lower bounds for the harmonic index of subdivision graphs In this section, we give some lower bounds for the harmonic index of subdivision and t-subdivision graphs and obtain some inequalities between the harmonic index of these graphs and some topological indices such as the inverse degree and the first Zagreb index. Hereafter, we deal with nontrivial graphs. Let G be a graph of size m. By definition of the harmonic index, we have HSG = deg G u + +., + deg G v and HS t G deg G u + + t times {}}{ deg G v deg G u t m + deg G v = HSG + t m.

3 Trans. Comb. 6 no B. N. Onagh 7 It is easy to see that HG < HSG and then HG < HS t G. Example.. i For any n, HSP n = 3 + n and HS tp n = 3 + n + t n, ii for any n 3, HSC n = n and HS t C n = n + t n, iii for any n, HSK n = nn n + and HS t K n = nn n + + t nn. Example.. Let G be a k-regular graph of order n. Then HSG = kn k + and HS tg = kn k + + t kn. In the following, simpler formulas are given for the harmonic index of subdivision graphs. Theorem.3. Let G be a graph of order n. Then HSG = n u V G deg G u +. Proof. Let u be a vertex of G. For each neighbor of u, deg G u + +. Thus, + deg G v deg G u + appears exactly once in. So, HSG = HSG = u V G u V G deg G u times {}}{ deg G u deg G u + = n deg G u + u V G u V G, as desired. deg G u + deg G u deg G u +. Corollary.. Let G be a graph of order n and size m. Then HS t G = n + t m u V G deg G u +. Remark.5. Let G and G be two graphs with the same degree sequence. Then HSG = HSG and HS t G = HS t G. A graph G is called a, δ-bidegreed if whose vertices have degree either or δ δ. We will use Schweitzer s inequality [, 7] in order to present a sharp lower bound for the harmonic index of subdivision graphs. Schweitzer s inequality. m x i M. Then Let x,..., x n be positive real numbers such that for i n holds n n x i n m + M x i mm. i= i=

4 8 Trans. Comb. 6 no B. N. Onagh Equality holds if and only if x = = x n = m = M or n is even, x = = x n x n + = = x n = M, where m < M and x x n. = m and Theorem.6. Let G be a graph of order n and size m. Then.3 n n δ + + n + mδ + + HSG, with equality if and only if G is a regular graph or a, δ-bidegreed graph. Proof. For every vertex u in G, we have δ + deg G u + +. Using m + n and Schweitzer s inequality, we get u V G Now, Theorem.3 implies the desired inequality. deg G u + n δ + + 8n + mδ + +. u V G deg G u + = Moreover, by Schweitzer s inequality, equality holds in.3 if and only if δ = or n vertices of G have degree δ and the remaining n vertices have degree, i.e., G is regular or, δ-bidegreed. Corollary.7. Let G be a graph of order n and size m. Then n n δ + + n + mδ t m HS tg, with equality if and only if G is a regular graph or a, δ-bidegreed graph. The first Zagreb index of a graph G is originally defined as M G = deg Gu. This topological index can be also expressed as M G = Theorem.8. Let G be a graph of size m. Then u V G degg u + deg G v [, 6, ]. m M G < HSG. Proof. Let fx =. Since f is a convex function on 0, +, by Jensen s inequality, for every edge x uv EG, we have. deg G u deg G v 8 + deg G u + deg G v, with equality if and only if deg G u = deg G v.

5 Trans. Comb. 6 no B. N. Onagh 9 So, by Bernoulli s inequality HSG = = > = m deg G u deg G v 8 + deg G u + deg G v + deg Gu + deg G v deg Gu + deg G v deg G u + deg G v = m M G. It completes the proof. Corollary.9. Let G be a graph of size m. Then m M G + t m < HS tg. The inverse degree of a graph G is defined as rg = Theorem.0. Let G be a graph of order n. Then u V G.5 3 n rg HSG, with equality if and only if G is a disjoint union of cycles. Proof. Similar to., for every vertex u V G, we have deg G u + with equality if and only if deg G u =. Thus, u V G deg G u + u V G By Theorem.3, we have the required inequality. deg G u + deg G u [5]., deg G u + n = rg + n. Furthermore, equality holds in.5 if and only if for every vertex u V G, deg G u =, i.e., G is the disjoint union of cycles. Corollary.. Let G be a graph of order n and size m. Then 3 n + t m rg HS tg, with equality if and only if G is a disjoint union of cycles.

6 0 Trans. Comb. 6 no B. N. Onagh 3. Upper bounds for the harmonic index of subdivision graphs In this section, we present some sharp upper bounds for the harmonic index of subdivision and t-subdivision graphs and characterize the corresponding extremal graphs. The following corollary is an immediate consequence of Theorem.3. Corollary 3.. Let G be a graph of order n. Then HSG nn n + and HS t G In both cases, equality holds if and only if G = K n. nn n + + t nn. In the following, some upper bounds for the harmonic index of subdivision graphs are given in terms of order and size. Theorem 3.. Let G be a graph of order n and size m. Then 3. HSG n + m, with equality if and only if G is a disjoint union of cycles. Proof. For every vertex u V G, we have with equality if and only if deg G u =. By., we get as desired. HSG u V G + deg G u deg G u deg G u + + deg Gu, = n + u V G deg G u = n + m = n + m, Moreover, equality holds in 3. if and only if for every vertex u V G, deg G u =, i.e., G is the disjoint union of cycles. Corollary 3.3. Let G be a graph of order n and size m. Then 3. HS t G n + tm, with equality if and only if G is a disjoint union of cycles. Theorem 3.. Let G be a graph of order n and size m. Then 3.3 with equality if and only if G is a regular graph. HSG nm n + m,

7 Trans. Comb. 6 no B. N. Onagh Proof. By Cauchy-Schwarz inequality, we have degg u + deg G u + u V G u V G with equality if and only if all the deg G u s are equal. Since deg G u + = n + m, we get u V G this yields the required inequality. u V G u V G degg u +, degg u + deg G u + n. By Theorem.3, n + m Furthermore, equality holds in 3.3 if and only if all the deg G u s are equal, i.e., G is regular. Corollary 3.5. Let G be a graph of order n and size m. Then 3. with equality if and only if G is a regular graph. HS t G nm n + m + t m, Remark 3.6. The inequalities 3.3 and 3. are always better than the inequalities 3. and 3., respectively. Let G be a graph of order n and size m. If m = n, n and n+ then G is called a tree, a unicyclic graph and a bicyclic graph, respectively. Corollary 3.7. Let G be a tree of order n. Then HSG nn n and HS t G In both cases, equality holds if and only if G = P. nn n + t n. Corollary 3.8. Let G be a unicyclic graph of order n. Then HSG n and HS t G t + n. In both cases, equality holds if and only if G = C n. Corollary 3.9. Let G be a bicyclic graph of order n. Then HSG < nn + n + and HS t G < nn + n + + t n +. A pendant vertex in a graph is a vertex of degree. Now, an upper bound for the harmonic index of subdivision graphs is given in terms of order, the maximum degree and the number of pendant vertices. Theorem 3.0. Let G be a graph of order n with p pendant vertices. Then 3.5 HSG n n p p 3 +, with equality if and only if G is a regular graph or a, -bidegreed graph.

8 Trans. Comb. 6 no B. N. Onagh Proof. Note that deg Gu u V G p times {}}{ deg G u + = = 3 p + u V G, deg G u> n p times {}}{ 3 p = 3 p + n p +. Now, by Theorem.3, we get the desired inequality. u V G, deg G u> deg G u + deg G u + Moreover, equality holds in 3.5 if and only if for every non-pendant vertex u V G, deg G u =. If p = 0 then for every vertex u V G, deg G u =, i.e., G is -regular, where n. Now, suppose that p > 0. If there is no non-pendant vertex in G then G = K and otherwise, G is, -bidegreed. Conversely, one can see easily that the equality holds in 3.5 for regular graph or, -bidegreed graph. Corollary 3.. Let G be a graph of order n and size m with p pendant vertices. Then HS t G n n p p t m, with equality if and only if G is a regular graph or a, -bidegreed graph. Corollary 3.. Let G be a graph of order n and size m with δ >. Then HSG n n + and HS t G n n + + t m. In both cases, equality holds if and only if G is a -regular graph, where n. Let G be a graph. For two vertices u and v in G, the distance d G u, v from u to v is the length of a shortest path connecting u and v. For a vertex u in G, the eccentricity ϵu of u is max v V G d G u, v. The minimum eccentricity among the vertices of G is the radius of G, denoted by radg, and the maximum eccentricity is its diameter diamg. A vertex u in G is a central vertex if ϵu = radg. A graph G is self-centered if ϵu = radg for all vertices u V G. For any even n, the cocktail party graph CP n is the unique regular graph with n vertices of degree n ; it is obtained from K n by removing n Theorem 3.3. Let G be a graph of order n. Then 3.6 with equality if and only if G = K n or G = CP n. disjoint edges. HSG n n radg n radg +,

9 Trans. Comb. 6 no B. N. Onagh 3 Proof. Let n i u be the number of vertices at distance i from the vertex u in G. One can see that deg G u n ϵu, with equality if and only if ϵu = and deg G u = n or ϵu and n u = n 3 u = = n ϵu u =. Thus, for every vertex u V G, we have This implies that n. Now, Theorem.3 yields the required in- n radg + equality. u V G deg G u + n ϵu + n radg +. deg G u + Suppose that equality holds in 3.6. Then G is self-centered and for every vertex u V G, equality holds in. If ϵu = for some vertex u V G then deg G u = n and ϵv for all vertices v u. Since G is self-centered, ϵu = for all vertices u V G. Thus G = K n. Now, suppose that ϵu for all vertices u V G. If ϵv 3 for some vertex v, then diamg = 3 otherwise, there exist at least two neighbors at distance for the central vertex and G = P. This contradicts that G is self-centered. So, ϵu = for all vertices u V G and then deg G u = n for all vertices u V G. It implies that G = CP n. It can be easily seen that the upper bound is attained for K n or CP n. Corollary 3.. Let G be a graph of order n and size m. Then HS t G n n radg 3.7 n radg + + t m, with equality if and only if G = K n or G = CP n.. The harmonic index for S-sum and S t -sum of graphs In the following theorem, we give an upper bound for the harmonic index of G + S G in terms of HSG, HG, rg and rg. Theorem.. Let G and G be two graphs. Then HG + S G < n HSG + n HG + m rg + m rg, where n i = V G i and m i = EG i, i =,. Proof. Let degu, v = deg G + S G u, v be the degree of a vertex u, v in G + S G. By definition of the harmonic index, we have HG + S G u V G v v EG + v V G u u ESG : +. degu, v + degu, v degu, v + degu, v

10 Trans. Comb. 6 no B. N. Onagh Then, = u V G v v EG u V G v v EG degg u + deg G v + deg G u + deg G v deg G u + deg G v + deg G v. By Jensen s inequality, for every vertex u V G and every edge v v EG, we have. deg G u + deg G v + deg G v deg G u +, deg G v + deg G v with equality if and only if deg G u = deg G v + deg G v. So, = = u V G v v EG u V G m deg G u + + deg G u m rg + n HG. u V G v v EG u V G HG deg G v + deg G v Also, = v V G u u ESG v V G u u ESG deg SG u + deg G v + deg SG u degsg u + deg SG u + deg G v. Similarly, for every vertex v V G and every edge u u ESG, we have. degsg u + deg SG u + deg G v deg SG u + deg SG u +, deg G v with equality if and only if deg SG u + deg SG u = deg G v. Thus, = v V G u u ESG v V G HSG + deg SG u + deg SG u + v V G = n HSG + m rg. m deg G v v V G u u ESG deg G v Therefore,.3 HG + S G n HSG + n HG + m rg + m rg.

11 Trans. Comb. 6 no B. N. Onagh 5 Now, suppose that there exist two graphs G and G such that equality holds in.3. Then, the inequalities. and. must be equalities. So, G and G are k -regular and k -regular graphs, respectively, such that k = k + k and k + = k, a contradiction. This completes the proof. Example.. For any n, HP n + S P = + 3 n n = n. 5 In the following, we extend the definition of S-sum to S t -sum of graphs. Definition.3. Let G and G be two graphs. The S t -sum G + St G is a graph with vertex set V S t G V G in which two vertices u, v and u, v of G + St G are adjacent if and only if [ u = u V G and v v EG ] or [ v = v and u u ES t G ]. Note that G + St G has V G copies of the graph S t G and we can label these copies by vertices of G. The vertices in each copy have two situations: the vertices in V G black vertices and the vertices in V S t G V G white vertices. We join only black vertices with the same name in S t G in which their corresponding labels are adjacent in G. P 3 + S P is shown in Fig.. Fig.. P 3 + S P Now, we obtain an upper bound for the harmonic index of G + St G in terms of HS t G, HG, rg and rg. Theorem.. Let G and G be two graphs. Then HG + St G < n HSG + n HG + m rg + m rg + t n m, where n i = V G i and m i = EG i, i =,. Proof. Let degu, v = deg G + St G u, v be the degree of a vertex u, v in G + St G. By definition of the harmonic index, we have HG + St G u V G v v EG + v V G u u ES t G : +. degu, v + degu, v degu, v + degu, v Similar to the proof of Theorem., we can get m rg + n HG. It suffices to compute the value of. Note that

12 6 Trans. Comb. 6 no B. N. Onagh = v V G + v V G : +. u u ES tg u V G, u V S tg V G u u ES t G u,u V S tg V G degu, v + degu, v degu, v + degu, v By similar argument in the proof of Theorem., we can show that n HSG + m rg. On the other hand, Hence, Therefore, HG + St G v V G v V G v V G u u ES t G u,u V S t G V G u u ES t G u,u V S t G V G degu, v + degu, v + t m = t n m. n HSG + m rg + t n m. n HSG + n HG + m rg + m rg + t n m. One can see that equality in above inequality cannot occur. This completes the proof. Example.5. For any n, HP n + St P = + 3 n n + t n = n + t n. 5 Acknowledgments The author would like to thank the anonymous referees for carefully reading the manuscript and their valuable comments and suggestions that improve the the presentation of this work. References [] A. Astaneh-Asl and G. H. Fath-Tabar, Computing the first and third Zagreb polynomials of Cartesian product of graphs, Iranian J. Math. Chem., no [] P. S. Bullen, A dictionary of inequalities, Addison-Wesley Longman, 998.

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