The Forgotten Topological Index of Four Operations on Some Special Graphs Sirous Ghobadi 1,a *, Mobina Ghorbaninejad 2,b

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1 Bulletin of Mathematical Sciences Applications Submitted: ISSN: , Vol. 16, pp Accepted: doi: / Online: SciPress Ltd., Switzerl The Forgotten Topological Index of Four Operations on Some Special Graphs Sirous Ghobadi 1,a *, Mobina Ghorbaninejad 2,b 1,2 Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran. a GhobadiMath46@gmail.com, b Ghorbani325@gmail.com Keywords: F index, F-sums, Degree, operation on graphs Abstract. For a graph, the forgotten topological index (F index) is defined as the sum of cubes of degrees of vertices. In 2009, Eliasi Taeri [M. Eliasi, B. Taeri, Four new sums of graphs their wiener indices, Discrete Appl. Math. 157 (2009) ] introduced four new sums (F sums) of graphs. In this paper we study the F index for the F sums of some special well-known graphs. 1. Introduction For a graph G = (V, E) with vertex set V = V(G) edge set E = E(G), the degree of a vertex v in G is the number of edges incident to v denoted by d G (v). In chemical graph theory, a topological index is a number related to a graph which is structurally invariant. One of the oldest most popular extremely studied topological indices are well known Zagreb indices first introduced in 1972 by Gutman Trinajestic [6] as follows: For a graph G with a vertex set V(G) an edge set E(G), the first second Zagreb indices are defined as M 1 (G) = d 2 G (v) = [d G (u) + d G v V(G) uv E(G) M 2 (G) = d G (u) d G (v) uv E(G) respectively. In [6], beside the first Zagreb index, another topological index defined as F(G) = d 3 G (v) = [d 2 G (u) + d 2 G. v V(G) uv E(G) However this index, except (implicitly) in a few works about the general first Zagreb index [9,10] the Zeroth Order general Ric index [8], was not further studied till then, except in a recent article by Furtula Gutman [5], where they reinvestigated this index studied some basic properties of this index. They proposed that F(G) be named the forgotten topological index, or shortly the F index. The extremal trees that maximize or minimize the F index is obtained by Abdo et. al. in [1]. De N. et. al. studied behavior of F index under several operations applied their results to find the F index of different chemically interesting molecular graphs nano Structures [3]. In this work we will study the F index of four operations on Paths, Cycles, Stars Complete graphs. For this purpose we recall four related graphs as follows: (a) S(G) is the graph obtained by inserting an additional vertex in each edge of G. Equivalently, each edge of G is replaced by a Path of length 2. (b) R(G) is obtained from G by adding a new vertex corresponding to each edge of G, then joining each new vertex to the end vertices of the corresponding edge. (c) Q(G) 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. SciPress applies the CC-BY 4.0 license to works we publish:

2 90 Volume 16 (d) T(G) has as its vertices the edges vertices of G. Adjacency in T(G) is defined as adjacency or incidence for corresponding elements of G S(P 5 ) R(P 5 ) P Q(P 5 ) T(P 5 ) Fig1. P 5, S(P 5 ), R(P 5 ), Q(P 5 ), T(P 5 ) The graph P 5 The graph P a b (2,a) (3,a) (4,a) (2,a) (3,a) (4,a) P 5 + S P 2 P 5 + Q P 2 (2,a) (3,a) (4,a) (2,a) (3,a) (4,a) P 5 + R P 2 P 5 + T P 2 Fig 2. Graphs P 5 + F P 2 The graph S(G) T(G) are called the subdivision total graph of G, respectively. For more details on these operations we refer the reader to [2]. If G is P 5, then S(P 5 ), R(P 5 ), Q(P 5 ) T(P 5 ) are shown in Fig 1. Suppose that G 1 G 2 are two connected graphs. Based on these operations above, Eliasi Taeri [4] introduced four new operations on these graphs in the following: Let F {S, R, Q, T}. The F sum of G 1 G 2, denoted by G 1 + F G 2 is a graph with the set of vertices V(G 1 + F G 2 ) = (V(G 1 ) E(G 1 )) V(G 2 ) two vertices (u 1, u 2 ) (v 1, v 2 ) of

3 Bulletin of Mathematical Sciences Applications Vol G 1 + F G 2 are adjacent if only if [u 1 = v 1 V(G 1 ) u 2 v 2 E(G 2 )] or[u 2 = v 2 V(G 2 ) u 1 v 1 E(F(G 1 ))]. P 5 + S P 2, P 5 + R P 2, P 5 + Q P 2 P 5 + T P 2 are shown in Fig 2. In [4], Eliasi Taeri obtained the expression for the wiener index W(G 1 + F G 2 ) in terms of W(F(G 1 )) W(G 2 ). In [7] Hanyuan Deng et. al. obtained the Zagreb indices of four operations on connected graphs. Here, we will study the F index for the F sums of Paths, Cycles, Stars Complete graphs. 2. The F index for F sums of some special graphs In the following four Theorems let G 1 G 2 be two Path of order n 1 n 2 respectively with G 1 = e 1 G 2 = e 2. At first we consider the case F = S. Theorem 1. F(G 1 + S G 2 ) = 2e 2 M 1 (G 1 ) + 6e 1 M 1 (G 2 ) + n 1 F(G 2 ) + n 2 F(G 1 ) + 8e 1 n e 2 n 1 24e 2 Proof. Let d(u, v) = d G1 + S G 2 (u, v) be the degree of vertex (u, v) in the graph G 1 + S G 2. F(G 1 + S G 2 ) = [d 2 (u 1, v 1 ) + d 2 (u 2, v 2 )] (u 1,v 1 )(u 2,v 2 ) E(G 1 + S G 2 ) = [d 2 (u, v 1 ) + d 2 (u, v 2 )] u V(G 1 ) u 1 u 2 E(S(G 1 )) = I 1 + I 2 Then I 1 = [d 2 (u, v 1 ) + d 2 (u, v 2 )] u V(G 1 ) = [2d 2 G1 (u) + 2d G1 (u)(d G2 (v 1 ) + d G2 (v 2 )) + (d 2 G2 (v 1 ) + d 2 G2 (v 2 ))] u V(G 1 ) = [2e 2 d 2 G1 (u) + 2d G1 (u)m 1 (G 2 ) + F(G 2 )] u V(G 1 ) = 2e 2 M 1 (G 1 ) + 4e 1 M 1 (G 2 ) + n 1 F(G 2 ) I 2 = [d 2 (u 1, v) + d 2 (u 2, v)] u 1 u 2 E(S(G 1 )) u 1 V(G 1 ),u 2 V(S(G 1 )) V(G 1 ) = [(d S(G1 )(u 1 ) + d G2 (v)) + d S(G1 )(u 2 )] u 1 u 2 E(S(G 1 )) u 1 V(G 1 ),u 2 V(S(G 1 )) V(G 1 )

4 92 Volume 16 = [(d S(G1 )(u 1 ) + d S(G1 )(u 2 )) + d 2 G2 (v) +2d S(G1 )(u 1 )d G2 u 1 u 2 E(S(G 1 )) u 1 V(G 1 ),u 2 V(S(G 1 )) V(G 1 ) = [F(S(G 1 )) + 2e 1 d 2 G2 (v) + 2(4n 1 6)d G2 = n 2 F(S(G 1 )) + 2e 1 M 1 (G 2 ) + 4e 2 (4n 1 6) Hence F(G 1 + S G 2 ) = 2e 2 M 1 (G 1 ) + 4e 1 M 1 (G 2 ) + n 1 F(G 2 ) + n 2 F(S(G 1 )) + 2e 1 M 1 (G 2 ) + 4e 2 (4n 1 6) Note that F(S(G 1 )) = F(G 1 ) + 8e 1. We then have the proof. In the next three Theorems let X Y be the sets of endvertices of Paths G 1 G 2 respectively. Then X = Y = 2 d G1 (x) = d G2 (y) = 1 for x X y Y. Theorem 2. F(G 1 + R G 2 ) = 10e 1 M 1 (G 2 ) + 8e 2 M 1 (G 1 ) + 4(e 2 + 1)F(G 1 ) + (e 1 + 1)F(G 2 ) +112e 1 e e 1 56e 2 24 Proof. F(G 1 + R G 2 ) = [d 2 (u 1, v 1 ) + d 2 (u 2, v 2 )] (u 1,v 1 )(u 2,v 2 ) E(G 1 + R G 2 ) = [d 2 (u, v 1 ) + d 2 (u, v 2 )] u X + [d 2 (u, v 1 ) + d 2 (u, v 2 )] u V(G 1 ) X u 1,u 2 V(G 1 ) v Y u 1 V(G 1 ),u 2 V(R(G 1 )) V(G 1 ) Y u 1 V(G 1 ),u 2 V(R(G 1 )) V(G 1 ) = I 1 + I 2 + I 3 + I 4 + I 5. Then I 1 = [d 2 (u, v 1 ) + d 2 (u, v 2 )] u X

5 Bulletin of Mathematical Sciences Applications Vol = [8d 2 G1 (u) + (d 2 G2 (v 1 ) + d 2 G2 (v 2 )) + 4d G1 (u) (d G2 (v 1 ) + d G2 (v 2 ))] u X = [8e 2 d 2 G1 (u) + F(G 2 ) + 4d G1 (u)m 1 (G 2 )] u X = 16e 2 + 2F(G 2 ) + 8M 1 (G 2 ). Similar to the case I 1 we have I 2 = [8e 2 d 2 G1 (u) + F(G 2 ) + 4d G1 (u)m 1 (G 2 )] u V(G 1 ) X Since for u V(G 1 ) X there are n 1 2 vertices of order 2 then I 2 = (n 1 2)(32e 2 + F(G 2 ) + 8M 1 (G 2 )). Now we have I 3 = [d 2 (u 1, v) + d 2 (u 2, v)] u 1, u 2 V(G 1 ) = [d R(G1 )(u 1 ) + d R(G1 )(u 2 ) + 2d 2 G2 (v) u 1, u 2 V(G 1 ) +2d G2 (v) (d R(G1 )(u 1 ) + d R(G1 )(u 2 ))] Note that u 1, u 2 V(G 1 ) u 1 u 2 E(R(G 1 )) iff u 1 u 2 E(G 1 ) d R(G1 )(u i ) = 2d G1 (u i ), i = 1,2. Then I 3 = [4 (d 2 G1 (u 1 ) + d 2 G1 (u 2 )) + 2d 2 G2 (v) + 4d G2 (v) (d G1 (u 1 ) + d G1 (u 2 ))] u 1 u 2 E(G 1 ) u 1, u 2 V(G 1 ) = [4F(G 1 ) + 2e 1 d 2 G2 (v) + 4d G2 (v)m 1 (G 1 )] v v(g 2 ) = 4n 2 F(G 1 ) + 2e 1 M 1 (G 2 ) + 8e 2 M 1 (G 1 ). I 4 = [d R(G1 )(u 1 ) + d R(G1 )(u 2 ) + d 2 G2 (v) + 2d R(G1 )(u 1 )d G2 v Y u 1 V(G 1 ),u 2 V(R(G 1 )) v(g 1 ) 2 if u X Since d R(G1 )(u) = { 4 if u V(G 1 ) X then d R(G1 )(u 1 ) = = 2 + (2n 1 4) = 8n d R(G1 )(u 1 ) then = 32n 1 56 I 4 = [(32n 1 56) + 8e 1 + 2e 1 d 2 G2 (v) + 2(8n 1 12)d G2 v Y

6 94 Volume 16 = 2(32n 1 56) + 16e 1 + 4e 1 + 4(8n 1 12) = 96n e Similar to the case I 4, we have I 5 = [(32n 1 56) + 8e 1 + 2e 1 d 2 G2 (v) + 2(8n 1 12)d G2 Y = (n 2 2)[(32n 1 56) + 8e 1 + 8e 1 + 4(8n 1 12)] = (n 2 2) (64n e 1 104). Hence the proof. Theorem 3. F(G 1 + Q G 2 ) = (e 1 + 1)F(G 2 ) + (e 2 + 1)(M 1 (G 1 ) + 2M 2 (G 1 )) +108e 1 e 2 82e e 1 70 Proof. F(G 1 + Q G 2 ) = [d 2 (u 1, v 1 ) + d 2 (u 2, v 2 )] (u 1,v 1 )(u 2,v 2 ) E(G 1 + Q G 2 ) = [d 2 (u, v 1 ) + d 2 (u, v 2 )] u V(G 1 ) v Y u 1 u 2 E(Q(G 1 )) u 1 V(G 1 ),u 2 V(Q(G 1 )) V(G 1 ) Y u 1 u 2 E(Q(G 1 )) u 1 V(G 1 ),u 2 V(Q(G 1 )) V(G 1 ) u 1 u 2 E(Q(G 1 )) u 1,u 2 V(Q(G 1 )) V(G 1 ) With respect to the above Theorems its easy to see that I 1 = 24e 1 e 2 4e 2 8e 1 + n 1 F(G 2 ) I 4 = n 2 M 1 (G 1 ) + 2n 2 M 2 (G 1 ) + 20n 1 n 2 56n 2 For the case I 2 I 3 we have = I 1 + I 2 + I 3 + I 4 I 2 = [d Q(G1 )(u 1 ) + d Q(G1 )(u 2 ) + d 2 G2 (v) + 2d G1 (u 1 )d G2 v Y u 1 u 2 E(Q(G 1 )) u 1 V(G 1 ),u 2 V(Q(G 1 )) V(G 1 ) = 2(8n 1 14) + 2(32n 1 60) + 4e 1 + 4(4n 1 6) = 96n 1 + 4e I 3 = 56n 1 n 2 98n 2 112n 1 + 8e 1 n 2 16e Note that n 1 = e n 2 = e we then have

7 Bulletin of Mathematical Sciences Applications Vol F(G 1 + Q G 2 ) = I 1 + I 2 + I 3 + I 4 = (e 1 + 1)F(G 2 ) + (e 2 + 1)(M 1 (G 1 ) + 2M 2 (G 1 )) +108e 1 e 2 82e e d G1 + R G 2 (u, v) for u V(G 1 ) v V(G 2 ) Since d G1 + T G 2 (u, v) = { d G1 + Q G 2 (u, v) for u V(T(G 1 )) V(G 1 ) v V(G 2 ) we can get the following Theorem by the proofs of Theorems 2 3. Theorem 4. F(G 1 + T G 2 ) = n 1 F(G 2 ) + 4n 2 F(G 1 ) + M 1 (G 1 ) (n 2 + 8e 2 ) + 2M 1 (G 2 ) (e 1 + 4n 1 4) +2n 2 M 2 (G 1 ) + 32n 1 e 2 + 8n 2 e n 1 n 2 196n 2 12e 1 48e 2 Applying the above four Theorems we have the following Theorems. Theorem 5. For n 3 m 2, (a) F (C n + S P m ) = n(72m 74) (b) F (C n + R P m ) = n(224m 182) (c) F (C n + Q P m ) = n(128m 74) (d) F (C n + T P m ) = n(280m 182) Theorem 6. For n 4 m 2 (a) F (K 1,n 1 + S P m ) = m(n 3 + 3n n 34) 2(3n n 18) (b) F (K 1,n 1 + R P m ) = 8m(n 3 + 9n 9) 2(12n n 36) (c) F (K 1,n 1 + Q P m ) = m(n 4 + 3n n 26) 2(3n n 18) (d) F (K 1,n 1 + T P m ) = m(n 4 + 7n n 64) 2(12n n 36) Theorem 7. For n 3 m 2, (a) F (K n + S P m ) = 2n 4 + 4mn(n 1) + n(n + 1) 3 (m 2) (b) F (K n + R P m ) = 2n(2n 1) 3 + 4mn(n 1) + 8n 4 (m 2) (c) F (K n + Q P m ) = 2n 4 + 4mn(n 1) 4 + n(n + 1) 3 (m 2) (d) F (K n + T P m ) = 1 2 mn(n 1)(2n 2)3 + 2n(2n 1) 3 + 8n 4 (m 2) References [1] H. Abdo, D. Dimitrov I. Gutman, On extremal trees with respect to the F index, CORR abs/ (2015). [2] D. M. Cvetkocic, M. Doob, H. Sachs, Spectra of graphs theory application, Academic press, New York, [3] N. De, SM. Nayeem, A. Pal, F index of some graph operations. Discrete Math. Algorithm Appl. doi: / (2016). [4] M. Eliasi B. Taeri, Four new sums of graphs their wiener indices, Discrete Appl. Math. 157 (2009) [5] B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) [6] I. Gutman, N. Trinajstic, Graph theory molecular orbitals total π electron energy of alternant hydrocarbons, Chem. Phys. Let. 17 (1972) [7] Hanyuan Deng, D. Sarala, S. K. Ayyaswamy, S. Balachran, The Zagreb indices of four operations on graphs, Appl. Math. Computation. 275 (2016) [8] Y. Hu, X. Li, Y. Shi, T. Xu, I. Gutman, On molecular graphs with Smallest greatest zeroth order general Ric index, MATCH Commun. Math. Comput. Chem. 54 (2005) [9] X. Li, J. Zheng, A unified approach to the entremal trees for different indices, MATCH Commun. Math. Comput. Chem. 54 (2005) [10] X. Li, H. Zhao, Trees with the first three smallest largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50 (2004)