THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS, TOTAL GRAPHS AND RELATED SUMS

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1 Kragujevac Journal of Mathematics Volume 08, Pages 7 8. THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS, TOTAL GRAPHS AND RELATED SUMS B. N. ONAGH Abstract. For a connected graph G, there are several related graphs such as line graph LG, subdivision graph SG, vertex-semitotal graph RG, edge-semitotal graph QG and total graph T G [I. Gutman, B. Furtula, Ž. K. Vukićević and G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 7 05, 5 6, W. Yan, B. -Y. Yang and Y. -N. Yeh, The behavior of Wiener indices and polynomials of graphs under five graph decorations, Appl. Math. Lett , 90 95]. Let F be one of symbols S, R, Q or T. The F -sum G F G of two connected graphs G and G is a graph with vertex set V G EG V G in which two vertices u, v and u, v of G F G are adjacent if and only if [u = u V G and v v EG ] or [v = v and u u EF G] [M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math , ]. In this paper, we investigate the harmonic index of edge-semitotal graphs, total graphs and F -sum of graphs, where F = Q or T.. 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 degree of a vertex u in G is denoted by deg G u. We will use P n to denote the path of order n. The edge-semitotal graph QG is obtained from G by inserting a new vertex into each edge of G, then joining with edges those pairs of new vertices on adjacent edges of G see Figure. The total graph T G has as its vertices the edges and vertices of G; adjacency in T G is defined as adjacency or incidence for the corresponding elements of G see Figure. Key words and phrases. Harmonic index, line graph, total graph, edge-semitotal graph, F -sum. 00 Mathematics Subject Classification. Primary: 05C07. Secondary: 05C76. Received: November, 06. Accepted: February, 07. 7

2 8 B. N. ONAGH QP T P Figure. QP and T P Let G and G be two graphs and F {Q, T }. Then, the F -sum G F G has V G copies of the graph F 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 EG white vertices. Now, we join only black vertices with the same name in F G in which their corresponding labels are adjacent in G see Figure. P Q P P T P Figure. P Q P and P T P The first Zagreb index, the inverse degree and the harmonic index of a graph G are important vertex-degree-based indices related to G, where denoted by M G, rg and HG, respectively, and defined as follows [3, ]: M G = u V G deg Gu, rg = u V G deg G u, HG = uv EG deg G u deg G v. The first Zagreb index can be expressed as M G = uv EG deg G u deg G v [8]. In this paper, we consider the harmonic index. In recent years, this topological index has been extensively studied. Shwetha et al. [] 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, subdivision graphs, t-subdivision graphs and F -sum of graphs, where F {S, S t } [9, 0]. More results on the harmonic index can been found in [, 6, 7, 5]. In this paper, we study the harmonic index of edge-semitotal graphs, total graphs and F -sum of graphs, where F = Q or T.

3 THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS The harmonic index of edge-semitotal and total graphs In this section, we give some upper bounds for the harmonic index of edge-semitotal and total graphs. Hereafter, we deal with nontrivial graphs. Note that deg QG u = deg G u for all u V G and deg QG w = deg G u deg G v = deg LG w for all w V QG V G, where w is the vertex inserted into the edge uv of G. Theorem.. Let G be a graph of order n and size m. Then HQG < HG HLG 6 M G n 8 m. Proof. By definition of the harmonic index, we have So, HQG = deg QG u deg QG x deg QG x deg QG v uv EG where x is the vertex inserted into the edge uv of G deg ww ELG QG w deg QG w = deg G u deg G u deg G v deg G u deg G v deg G v uv EG ww ELG = uv EG ww ELG :=. deglg w deg LG w deg G u deg G u deg G v deg G u deg G v deg G v deg LG w deg LG w By Jensen s inequality, for every uv EG, we have deg G u deg G u deg G v < deg G u deg G u deg G v, deg G u deg G v deg G v < deg G u deg G v deg G v. < deg uv EG G u deg G v uv EG deg G u deg G v

4 0 B. N. ONAGH = deg u V G G u deg Gu HG = n HG. Similarly, for every ww ELG, deg LG w deg LG w 8 deg LG w deg LG w, with equality if and only if deg LG w deg LG w =. This implies that Therefore, 8 ELG HLG = 8 M G m HLG = 6 M G 8 m HLG. HQG < HG HLG 6 M G n 8 m. This completes the proof. Example.. For any n, 9 HQP n = 5 3 n 3 7 n = n 3 n. Note that deg T G u = deg G u for all u V G and deg T G w = deg G u deg G v = deg LG w for all w V T G V G, where w is the vertex inserted into the edge uv of G. Theorem.. Let G be a graph of order n and size m. Then HT G HG HLG 6 M G n 8 m, with equality if and only if G = C n. Proof. Note that HT G = deg T G u deg T G v uv EG uv EG deg T G u deg T G x deg T G x deg T G v where x is the vertex inserted into the edge uv of G

5 THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS... ww ELG = deg T G w deg T G w deg uv EG G u deg G v deg uv EG G u deg G u deg G v deg G u deg G v deg G v deglg w deg LG w ww ELG = HG deg uv EG G u deg G u deg G v deg G u deg G v deg G v deg LG w deg LG w ww ELG := HG. By Jensen s inequality, for every uv EG, we have. deg G u deg G u deg G v deg G u deg G u deg G v, with equality if and only if deg G u = deg G v. Also,. 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. Thus, deg G u deg G v uv EG = n HG. uv EG deg G u deg G v On the other hand,.3 HLG 6 M G 8 m,

6 B. N. ONAGH with equality if and only if deg LG w deg LG w = for all ww Therefore, ELG. HT G HG HLG 6 M G n 8 m. One can see that equality holds in above inequality if and only if the inequalities.,. and.3 be equalities, i.e., G is a k-regular such that deg LG w deg LG w = for all ww LG. Since LG is k -regular, by, we can get k =. This completes the proof. Example.. For any n, HT P n = 3 n 3 = n 3 n n 3 7 n 3. The harmonic index for Q-sum and T -sum of graphs In the following theorem, we give an upper bound for the harmonic index of G Q G in terms of HQG, HLG, HG, M G, rg and rg. Theorem 3.. Let G and G be two graphs. Then HG Q G < n HQG 3 6 n HLG n HG where n i = V G i and m i = EG i, i =,. 3 6 n M G m rg m rg 3 3 n m, Proof. Let degu, v = deg G Q G u, v be the degree of a vertex u, v in G Q G. Then, Note that = HG Q G = u V G v v EG = u V G v v EG u V G v v EG v V G :=. u u EQG degu, v degu, v degu, v degu, v degqg u deg G v deg QG u deg G v deg QG u deg G v deg G v

7 THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS... 3 = u V G v v EG deg G u deg G v deg G v. By Jensen s inequality, for every u V G and every v v EG, we have 3. 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. Thus, u V G v v EG deg G u u V G v v EG = m HG u V G deg G u u V G Also, = = m rg n HG. v V G u u EQG u V G, u V QG V G = v V G v V G u u EQG u,u V QG V G u u EQG u V G, u V QG V G v V G u u EQG u,u V QG V G degu, v degu, v degu, v degu, v deg G v deg G v degqg u deg G v deg QG u deg QG u deg QG u. For every v V G and every u u EQG with u V G and u V QG V G, we have 3. degqg u deg QG u deg G v deg QG u deg QG u deg G v, with equality if and only if deg QG u deg QG u = deg G v. Thus, u u EQG u V G, u V QG V G degqg u deg G v deg QG u

8 B. N. ONAGH = u u EQG u V G, u V QG V G u u EQG u V G, u V QG V G u u EG deg QG u deg QG u deg G v deg QG u deg QG u deg QG u deg QG u where u is the vertex inserted into the edge u u of G m deg G v = HQG ww ELG m deg QG w deg QG w deg G v. So, v V G v V G v V G HQG m deg G v u u EQG u,u V QG V G = n HQG m rg n ww ELG n u u ELG ww ELG deg QG w deg QG w deg QG u deg QG u deg QG w deg QG w deg QG u deg QG u = n HQG m rg 3 n = n HQG m rg 3 n ww ELG ww ELG deg LG w deg LG w. deg QG w deg QG w

9 THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS... 5 On the other hand, 3.3 ww ELG deg LG w deg LG w HLG 6 M G 8 m, with equality if and only if deg LG w deg LG w = for all ww ELG. Therefore, HG Q G n HQG 3 6 n HLG n HG 3 6 n M G m rg m rg 3 3 n m. Now, suppose that there exist two graphs G and G such that equality holds in above inequality. Then, the inequalities 3., 3. and 3.3 must be equalities. So, G and G are k -regular and k -regular graphs, respectively, such that k = k k, k k k = k and = k k, a contradiction. This completes the proof. Example 3.. For any n, HP n Q P = 3 n 5 7 n 3 7 n = n 8 7 n 3 n. Now, we obtain an upper bound for the harmonic index of G T G in terms of HT G, HLG, HG, M G, rg and rg. Theorem 3.. Let G and G be two graphs. Then HG T G < n HT G 3 6 n HLG n HG where n i = V G i and m i = EG i, i =,. 3 6 n M G 8 m rg 5 m rg 3 3 n m, Proof. Let degu, v = deg G T G u, v be the degree of a vertex u, v in G T G. By definition of harmonic index, we have HG T G = u V G v v EG v V G :=. u u ET G degu, v degu, v degu, v degu, v

10 6 B. N. ONAGH Note that = u V G = u V G v v EG v v EG degt G u deg G v deg T G u deg G v deg T G u deg G v deg G v = u V G v v EG deg G u deg G v deg G v. By similar argument as in the proof of Theorem 3., one can show that 3. 8 m rg n HG, with equality if and only if deg G u = deg G v deg G v for all u V G and all v v EG. On the other hand, = degu, v degu, v v V G v V G = v V G v V G v V G = v V G v V G v V G u u ET G u,u V G u u ET G u V G, u V T G V G u u ET G u,u V T G V G u u ET G u,u V G degu, v degu, v degu, v degu, v degt G u deg G v deg T G u deg G v u u ET G u V G, u V T G V G u u ET G u,u V T G V G u u ET G u,u V G degt G u deg G v deg T G u deg T G u deg T G u degt G u deg T G u deg G v u u ET G u V G, u V T G V G degt G u deg T G u deg G v

11 THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS... 7 v V G u u ELG = v V G u u EG v V G v V G deg T G u deg T G u degg u deg G u deg G v u u ET G u V G, u V T G V G u u ELG degt G u deg T G u deg G v deg T G u deg T G u. One can prove that 8 n HG m rg n HT G 8 n HG m rg 3 n HLG 6 M G 8 m. Equality holds in above inequality if and only if 3.5 for all v V G and all u u EG, 3.6 deg G u deg G u = deg G v, deg T G u deg T G u = deg G v, for all v V G and all u u ET G with u V G and u V T G V G, and 3.7 for all ww ELG. Therefore, deg LG w deg LG w =, HG T G n HT G 3 6 n HLG n HG n M G 8 m rg 5 m rg 3 3 n m. Moreover, equality holds in 3.8 if and only the inequalities 3., 3.5, 3.6 and 3.7 be equalities, i.e., G and G are k -regular and k -regular graphs, respectively, such that k = k k, k k = k, k k k = k and k k =, a contradiction. This completes the proof. Example 3.. For any n, HP n T P = 3 5 n 5 n n 3 7 n = n 5 5 n 3 n.

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