EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 5, 2020, 1057-1071 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Special Issue Dedicated to Professor Hari M. Srivastava On the Occasion of his 80th Birthday Transmission and Reciprocal Transmission Based Topological Co-Indices of Graphs Harishchandra S. Ramane1, Saroja Y. Talwar1, Ismail Naci Cangul2,∗ 1 Department of Mathematics, Karnatak University, 580003 Dharwad, India 2 Department of Mathematics, Bursa Uludag University, 16059 Bursa, Turkey Abstract. The transmission of a vertex u in a connected graph G is defined as the sum of the distances between u and all other vertices of a graph G. The reciprocal transmission of a vertex u in a connected graph G is defined as the sum of the reciprocal of distances between u and all other vertices of a graph G. In this paper, we introduce and study new topological co-indices based on the transmission and reciprocal transmission of a vertex, such as transmission and re- ciprocal transmission sum-connectivity co-indices, transmission and reciprocal transmission atom bond connectivity co-indices, transmission and reciprocal transmission geometric-arithmetic co- indices, transmission and reciprocal transmission augmented Zagreb co-indices, and transmission and reciprocal transmission arithmetic-geometric co-indices. Further we obtain general formulae for some graphs. 2020 Mathematics Subject Classifications: 05C07, 05C30, 05C69 Key Words and Phrases: Transmission of a vertex, reciprocal transmission of a vertex, topo- logical index, graph distance 1. Introduction Topological indices are proved to be very useful in chemistry, biochemistry and nan- otechnology in isomer discrimination, structure-property relationship, structure-activity relationship and pharmaceutical drug design. According to the International Academy of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3724 Email addresses: cangul@uludag.edu.tr (I. N. Cangul) hsramane@yahoo.com (H. S. Ramane), sarojaytalwar@gmail.com (S. Y. Talwar) https://www.ejpam.com 1057 c© 2020 EJPAM All rights reserved. H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1058 Mathematical Chemistry, to identify whether any topological index is useful for prediction of chemical properties, the correlation between the values of that topological index for dif- ferent octane isomers and parameter values related to certain physicochemical property of them should be considered. Generally octane isomers are convenient for such studies, because the number of the structural isomers of octane is large enough to make the sta- tistical conclusion reliable. Furtula and Gutman [5] showed that for octane isomers both M1 and F yield correlation coefficient greater than 0.95 in case of entropy and acentric factor. They also improved the predictive ability of these index by considering a simple linear model in the form (M1 + λF ), where λ varies from −20 to 20. For graph theoretical parameters, we refer the book [10]. Let G be a graph having n vertices and m edges. Let V (G) be the vertex set and E(G) be the edge set of G. The edge e joining the vertices u and v is denoted by e = uv. e is said to be incident to u and v and u and v are called adjacent. The degree of a vertex u is the number of edges incident to it and is denoted by d(u). The distance between the vertices u and v is the length of the shortest path joining u and v and is denoted by d(u, v). The diameter of G is the maximum distance between all pair of vertices of G and is denoted by diam(G). A topological index is a numerical invariant of a given graph [13]. Particular topo- logical indices include the Zagreb indices, ABC index, GA index, Balaban index, Harary index, molecular topological index and Wiener index. Unless otherwise stated, hydrogen atoms are usually ignored in the computation of such indices as organic chemists usually do when they write a benzene ring as a hexagon. In the literature, several degree based topological indices have been introduced and studied [6]. The most studied degree based topological indices are the family of Zagreb indices, [1–4, 7, 12, 17, 18, 20–24, 26–28]. The first and second Zagreb indices of a graph G are defined by M1(G) = ∑ uv∈E(G) [d(u) + d(v)] and M2(G) = ∑ uv∈E(G) d(u)d(v), see [8]. The transmission (or status) of a vertex u ∈ V (G), [9, 16], denoted by σ(u), is defined by σ(u) = ∑ v∈V (G) d(u, v). The reciprocal transmission (or reciprocal status) of a vertex u ∈ V (G), denoted by rs(u), is defined by rs(u) = ∑ v∈V (G) 1 d(u, v) . H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1059 The oldest transmission based topological index is the Wiener index [25] defined by W (G) = ∑ {u,v}⊆V (G) d(u, v) = 1 2 ∑ u∈V (G) σ(u). The Wiener index is also called as gross status or total status, [9]. The transmission sum-connectivity index of a graph G, [19], denoted by TSC(G), is defined by TSC(G) = ∑ uv∈E(G) 1√ σ(u) + σ(v) . The transmission geometric-arithmetic index of a graph G, [11], denoted by TGA(G), is defined by TGA(G) = ∑ uv∈E(G) 2 √ σ(u)σ(v) σ(u) + σ(v) . The transmission arithmetic-geometric index of a graph G, [15], denoted by TAG(G), is defined by TAG(G) = ∑ uv∈E(G) σ(u) + σ(v) 2 √ σ(u)σ(v) . The transmission atom-bond connectivity index of a graphG, [15], denoted by TABC(G), is defined by TABC(G) = ∑ uv∈E(G) √ σ(u) + σ(v)− 2 σ(u)σ(v) . The transmission augmented Zagreb index of a graph G, [15], denoted by TAZ(G), is defined by TAZ(G) = ∑ uv∈E(G) [ σ(u)σ(v) σ(u) + σ(v)− 2 ]3 . The reciprocal transmission arithmetic-geometric index of a graph G, [14], is denoted by RTAG(G) and it is defined by RTAG(G) = ∑ uv∈E(G) rs(u) + rs(v) 2 √ rs(u)rs(v) . (1) The reciprocal transmission geometric-arithmetic index of a graph G, [14], is denoted by RTGA(G) and it is defined by RTGA(G) = ∑ uv∈E(G) 2 √ rs(u)rs(v) rs(u) + rs(v) . (2) H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1060 The reciprocal transmission sum-connectivity index of a graph G, [14], is denoted by RTSC(G) and it is defined by RTSC(G) = ∑ uv∈E(G) 1√ rs(u) + rs(v) . (3) The reciprocal transmission atom-bond connectivity index of a graphG, [14], is denoted by RTABC(G) and it is defined by RTABC(G) = ∑ uv∈E(G) √ rs(u) + rs(v)− 2 rs(u)rs(v) . (4) The reciprocal transmission augmented Zagreb index of a graph G, [14], is denoted by RTAZ(G) and it is defined by RTAZ(G) = ∑ uv∈E(G) [ rs(u)rs(v) rs(u) + rs(v)− 2 ]3 . (5) In the next section, we obtain bounds for the other transmission and reciprocal transmission- based topological co-indices. Now we define the following transmission and reciprocal transmission based topological co-indices of graphs. The transmission sum-connectivity co-index of a graph G, denoted by TSC(G), is defined by TSC(G) = ∑ uv/∈E(G) 1√ σ(u) + σ(v) . The transmission geometric-arithmetic co-index of a graph G, denoted by TGA(G), is defined by TGA(G) = ∑ uv/∈E(G) 2 √ σ(u)σ(v) σ(u) + σ(v) . The transmission arithmetic-geometric co-index of a graph G, denoted by TAG(G), is defined by TAG(G) = ∑ uv/∈E(G) σ(u) + σ(v) 2 √ σ(u)σ(v) . The transmission atom-bond connectivity co-index of a graph G, denoted by TABC(G), is defined by TABC(G) = ∑ uv/∈E(G) √ σ(u) + σ(v)− 2 σ(u)σ(v) . H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1061 The transmission augmented Zagreb co-index of a graph G, denoted by TAZ(G), is defined by TAZ(G) = ∑ uv/∈E(G) [ σ(u)σ(v) σ(u) + σ(v)− 2 ]3 . The reciprocal transmission arithmetic-geometric co-index of a graph G is denoted by RTAG(G) and it is defined by RTAG(G) = ∑ uv/∈E(G) rs(u) + rs(v) 2 √ rs(u)rs(v) . (6) The reciprocal transmission geometric-arithmetic co-index of a graph G is denoted by RTGA(G) and it is defined by RTGA(G) = ∑ uv/∈E(G) 2 √ rs(u)rs(v) rs(u) + rs(v) . (7) The reciprocal transmission sum-connectivity co-index of a graph G is denoted by RTSC(G) and it is defined by RTSC(G) = ∑ uv/∈E(G) 1√ rs(u) + rs(v) . (8) The reciprocal transmission atom-bond connectivity co-index of a graph G is denoted by RTABC(G) and it is defined by RTABC(G) = ∑ uv/∈E(G) √ rs(u) + rs(v)− 2 rs(u)rs(v) . (9) The reciprocal transmission augmented Zagreb co-index of a graph G is denoted by RTAZ(G) and it is defined by RTAZ(G) = ∑ uv/∈E(G) [ rs(u)rs(v) rs(u) + rs(v)− 2 ]3 . (10) We have already obtained explicit formulae for transmission and reciprocal transmis- sion based topological coindices in terms of order and size. In the following, we obtain bounds for the above defined transmission and reciprocal transmission based topological co-indices. H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1062 2. Bounds for transmission and reciprocal transmission based topological co-indices Now we obtain inequalities for transmission and reciprocal transmission based topo- logical coindices: Theorem 1. Let G be a connected graph with n vertices and let D = diam(G). Then∑ uv/∈E(G) 1√ 2D(n− 1)− (D − 1)(d(u) + d(v)) ≤ TSC(G) ≤ ∑ uv/∈E(G) 1√ 4n− 4− (d(u) + d(v)) . (11) Equality holds on both sides if and only if diam(G) ≤ 2. Proof. For any vertex u of G, there are d(u) vertices which are at distance 1 from u. Further, the distance between u and remaining n − 1 − d(u) vertices is at least 2 and at most D. Therefore σ(u) ≤ d(u) +D(n− 1− d(u)) = D(n− 1)− (D − 1)d(u) and σ(u) ≥ d(u) + 2(n− 1− d(u)) = 2n− 2− d(u) with equality in both cases if and only if D = 2. Therefore, 4n− 4− (d(u) + d(v)) ≤ σ(u) + σ(v) ≤ 2D(n− 1)− (D − 1)(d(u) + d(v)). Hence TSC(G) = ∑ uv/∈E(G) 1√ σ(u) + σ(v) ≥ ∑ uv/∈E(G) 1√ 2D(n− 1)− (D − 1)(d(u) + d(v)) and TSC(G) = ∑ uv/∈E(G) 1√ σ(u) + σ(v) ≤ ∑ uv/∈E(G) 1√ 4n− 4− (d(u) + d(v)) . Equality holds in both cases if and only if D = 2. Theorems 2, 3, 4 and 5 can be proved analogously to Theorem 1. Theorem 2. Let G be a connected graph with n vertices and let D = diam(G). Then∑ uv/∈E(G) √ 2D(n− 1)− (D − 1)(d(u) + d(v))− 2 (D(n− 1))2 −D(n− 1)(D − 1)(d(u) + d(v)− (D − 1)2d(u)d(v) ≤ TABC(G) ≤ ∑ uv/∈E(G) √ 4n− 6− (d(u) + d(v)) 4n2 − 8n− 4 + (2− 2n)(d(u) + d(v)) + d(u)d(v) . (12) H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1063 Theorem 3. Let G be a connected graph with n vertices and let D = diam(G). Then∑ uv/∈E(G) 2D(n− 1)− (D − 1)(d(u) + d(v)) 2 √ (D(n− 1))2 −D(n− 1)(D − 1)(d(u) + d(v)) + (D − 1)2d(u)d(v) ≤ TAG(G) ≤ ∑ uv/∈E(G) 4n− 4− (d(u) + d(v)) 2 √ (2n− 2− d(u))(2n− 2− d(v)) . (13) Theorem 4. Let G be a connected graph with n vertices and let D = diam(G). Then ∑ uv/∈E(G) [ (D(n− 1))2 −D(n− 1)(D − 1)(d(u) + d(v))− (D − 1)2d(u)d(v) 2D(n− 1) + (D − 1)(d(u) + d(v))− 2 ]3 ≤ TAZ(G) ≤ ∑ uv/∈E(G) [ 4n2 − 8n− 4 + (2− 2n)(d(u) + d(v)) + d(u)d(v) 4n− 6− (d(u) + d(v)) ] . (14) Theorem 5. Let G be a connected graph with n vertices and let D = diam(G). Then, ∑ uv/∈E(G) 2 √ (D(n− 1)− (D − 1)d(u)) (D(n− 1)− (D − 1)d(v)) 2D(n− 1)− (D − 1)(d(u) + d(v)) ≤ TGA ∑ uv/∈E(G) 2 √ (2n− 2− d(u)) (2n− 2− d(v)) 4n− 4− (d(u) + d(v)) . (15) Theorem 6. Let G be a connected graph with n vertices and let diam(G) = D. Then,∑ uv/∈E(G) 1√ (n− 1) + 1 2(d(u) + d(v)) ≤ RTSC(G) ≤ ∑ uv/∈E(G) 1√ 2(n−1) D + ( 1− 1 D ) (d(u) + d(v)) . (16) Proof. Lower bound: For any vertex u of G there are d(u) vertices which are at distance 1 from u and remaining n− 1− d(u) vertices are at distance at least 2. Therefore rs(u) ≤ 1 2(n− 1 + d(u)) and rs(u) + rs(v) ≤ (n− 1) + 1 2(d(u) + d(v)). We have RTSC(G) = ∑ uv/∈E(G) 1√ rs(u) + rs(v) H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1064 ≥ ∑ uv/∈E(G) 1√ (n− 1) + 1 2(d(u) + d(v)) . Upper bound: For any vertex u of G there are d(u) vertices which are at distance 1 from u and remaining n− 1− d(u) vertices are at distance at most D. Therefore rs(u) ≥ 1 D (n− 1) + (1− 1 D )d(u) and rs(u) + rs(v) ≥ 2(n−1) D + ( 1− 1 D ) (d(u) + d(v)). We have RTSC(G) ≤ ∑ uv/∈E(G) 1√ 2(n−1) D + ( 1− 1 D ) (d(u) + d(v)) . Theorems 7, 8, 9 and 10 can be proved analogously to Theorem 6: Theorem 7. Let G be a connected graph with n vertices and let diam(G) = D. Then ∑ uv/∈E(G) √ 2 (2(n− 1) + d(u) + d(v)− 4) (n− 1)2 + (n− 1)(d(u) + d(v)) + d(u)d(v) ≤ RTABC(G) ≤ ∑ uv/∈E(G) √√√√ 2(n−1) D + ( 1− 1 D ) (d(u) + d(v))− 2( 1 D (n− 1) )2 + ( 1− 1 D ) d(u)d(v) + (n−1) D ( 1− 1 D ) (d(u) + d(v)) . (17) Theorem 8. Let G be a connected graph with n vertices and let diam(G) = D. Then ∑ uv/∈E(G) [ (n− 1)2 + (n− 1)(d(u) + d(v)) + d(u)d(v) 2 (2(n− 1) + d(u) + d(v)− 4) ]3 ≤ RTAZ(G) ≤ ∑ uv/∈E(G) [( 1 D (n− 1) )2 + ( 1− 1 D ) d(u)d(v) + n−1 D ( 1− 1 D ) (d(u) + d(v)) 2(n−1) D + ( 1− 1 D ) (d(u) + d(v))− 2 ]3 . (18) Theorem 9. Let G be a connected graph with n vertices and let diam(G) = D. Then∑ uv/∈E(G) 2(n− 1) + (d(u) + d(v)) 2 √ (n− 1 + d(u))(n− 1 + d(v)) ≤ RTAG(G) ≤ ∑ uv/∈E(G) 2(n−1) D + ( 1− 1 D ) (d(u) + d(v)) 2 √( 1 D (n− 1) )2 + 1 D (n− 1) ( 1− 1 D ) (d(u) + d(v)) + ( 1− 1 D )2 d(u)d(v) . (19) H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1065 Theorem 10. Let G be a connected graph with n vertices and let diam(G) = D. Then∑ uv/∈E(G) 2 √ (n− 1 + d(u))(n− 1 + d(v)) 2(n− 1) + (d(u) + d(v)) ≤ RTGA(G) ≤ ∑ uv/∈E(G) 2 √( 1 D (n− 1) )2 + 1 D (n− 1) ( 1− 1 D ) (d(u) + d(v)) + ( 1− 1 D )2 d(u)d(v) 2(n−1) D + ( 1− 1 D ) (d(u) + d(v)) . (20) 3. Transmission and reciprocal transmission based topological co-indices of some graphs For any vertex u of a complete graph Kn, we have σ(u) = n − 1. Hence we get the following result: Proposition 1. For a complete graph Kn on n vertices, TSC(Kn) = 0, TABC(Kn) = 0, TAG(Kn) = 0, TAZ(Kn) = 0, TGA(Kn) = 0, RTSC(Kn) = 0, RTABC(Kn) = 0, RTAG(Kn) = 0, RTAZ(Kn) = 0, RTGA(Kn) = 0. The vertex set of a complete bipartite graph Kp,q can be partitioned into two sets V1 and V2 such that every edge of Kp,q has one end in V1 and other end in V2, where |V1| = p and |V2| = q. If the vertex u ∈ V1 and v ∈ V2, then d(u) = p and d(v) = q. Recall that the graph Kp,q has n = p+ q vertices and m = pq edges. Also diam(Kp,q) ≤ 2. Therefore by the equality part of Theorems 1, 2, 3 and 4, we get the following result: Proposition 2. For a complete bipartite graph Kp,q, we have TSC(Kp,q) = (( p+ q 2 ) − pq ) 1√ 3(p+ q)− 4 , TABC(Kp,q) = (( p+ q 2 ) − pq )√ 3(p+ q)− 6 (p+ 2(q − 1))(q + 2(p− 1)) , TAG(Kp,q) = (( p+ q 2 ) − pq )( 3(p+ q)− 4 2 √ (p+ 2(q − 1))(q + 2(p− 1)) ) , TGA(Kp,q) = (( p+ q 2 ) − pq )( 2 √ (p+ 2(q − 1))(q + 2(p− 1)) 3(p+ q)− 4 ) , TAZ(Kp,q) = (( p+ q 2 ) − pq )( (p+ 2(q − 1))(q + 2(p− 1)) 3(p+ q)− 6 )3 , RTSC(Kp,q) = (( p+ q 2 ) − pq ) 1√ 3 2(p+ q)− 1  , H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1066 RTABC(Kp,q) = (( p+ q 2 ) − pq )√ 3 2(p+ q)− 3 (p+ 2(q − 1))(q + 2(p− 1)) , TAG(Kp,q) = (( p+ q 2 ) − pq )( 3 2(p+ q)− 1 2 √ (p+ 2(q − 1))(q + 2(p− 1)) ) , RTGA(Kp,q) = (( p+ q 2 ) − pq )( 2 √ (p+ 2(q − 1))(q + 2(p− 1)) 3 2(p+ q)− 1 ) , RTAZ(Kp,q) = (( p+ q 2 ) − pq )( (p+ 2(q − 1))(q + 2(p− 1)) 3 2(p+ q)− 3 )3 . For any vertex u of a cycle Cn on n ≥ 3 vertices, we have σ(u) =  2 [ 1 + 2 + · · ·+ n−1 2 ] + n 2 = n2 4 , if n is even 2 [ 1 + 2 + · · ·+ n−1 2 ] = n2−1 4 , if n is odd and rs(u) =  2 ∑n−2 2 i=1 1 i + 2 n , if n is even 2 ∑n−1 2 i=1 1 i , if n is odd. Proposition 3. For a cycle Cn on n ≥ 3 vertices, we have TSC(Cn) =  (( n 2 ) − n )√ 2 n2 , if n is even(( n 2 ) − n )√ 2 n2−1 , if n is odd TABC(Cn) =  (( n 2 ) − n )√8(n2−4) n4 , if n is even(( n 2 ) − n )√8(n2−5) (n2−1)2 , if n is odd TAZ(Cn) =  (( n 2 ) − n ) ( n4 8(n2−4) )3 , if n is even(( n 2 ) − n ) ( (n2−1)2 8(n2−5) )3 , if n is odd TAG(Cn) = (( n 2 ) − n ) for any value of n. TGA(Cn) = (( n 2 ) − n ) for any value of n. H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1067 RTSC(Cn) =  (( n 2 ) − n ) 1 2 √∑n−2 2 i=1 1 i + 1 n , if n is even(( n 2 ) − n ) (1) 2 √∑n−1 2 i=1 1 i , if n is odd RTABC(Cn) =  (( n 2 ) − n )√√√√ ∑n−2 2 i=1 1 i + 1 n −1(∑n−2 2 i=1 1 i + 1 n )2 , if n is even (( n 2 ) − n )√√√√ 2 ∑n−1 2 i=1 1 i −1 2 (∑n−1 2 i=1 1 i )2 , if n is odd RTAZ(Cn) =  (( n 2 ) − n ) (∑n−2 2 i=1 1 i + 1 n )2 (∑n−2 2 i=1 1 i + 1 n ) −1 3 , if n is even (( n 2 ) − n )2 (∑n−1 2 i=1 1 i )2 2 ∑n−1 2 i=1 1 i −1 3 , if n is odd RTAG(Cn) = (( n 2 ) − n ) for any value of n RTGA(Cn) = (( n 2 ) − n ) for any value of n A wheel Wn+1 is a graph obtained from the cycle Cn, n ≥ 3, by adding a new vertex and making it adjacent to all the vertices of Cn. The degree of a central vertex of Wn+1 is n and the degree of all other vertices is 3. Hence Proposition 4. For a wheel Wn+1, n ≥ 3, TSC (Wn+1) = (( n+ 1 2 ) − 2n ) 1√ 4n− 6 , TABC (Wn+1) = (( n+ 1 2 ) − 2n )√ 4n− 8 4n2 − 12n+ 9 , TAZ (Wn+1) = (( n+ 1 2 ) − 2n )( 4n2 − 12n+ 9 4n− 8 )3 , H. S. Ramane, S. Y. Talwar, I. N. Cangul / Eur. J. Pure Appl. Math, 13 (5) (2020), 1057-1071 1068 TGA (Wn+1) = (( n+ 1 2 ) − 2n )( 2 √ 4n2 − 12n+ 9 4n− 6 ) , TAG (Wn+1) = (( n+ 1 2 ) − 2n )( 4n− 6 2 √ 4n2 − 12n+ 9 ) , RTAG (Wn+1) = (( n+ 1 2 ) − 2n ) n+ 3 2 √( 3 + 1 2(n− 3) )2  , RTGA (Wn+1) = (( n+ 1 2 ) − 2n )2 √( 3 + 1 2(n− 3) )2 n+ 3  , RTSC (Wn+1) = (( n+ 1 2 ) − 2n ) 1√ n+ 3 , RTABC (Wn+1) = (( n+ 1 2 ) − 2n )√ n+ 1( 3 + 1 2(n− 3) )2 , RTAZ (Wn+1) = (( n+ 1 2 ) − 2n )(( 3 + 1 2(n− 3)2 ) n+ 1 )3 . A friendship graph (or Dutch windmill graph) Fn, n ≥ 2, is a graph that can be constructed by coalescence n copies of the cycle C3 of length 3 with a common vertex. It has 2n+ 1 vertices and 3n edges. The degree of a coalescence vertex of Fn is 2n and the degree of all other vertices is 2. Proposition 5. For a friendship graph Fn, n ≥ 2, TSC(Fn) = (( 2n+ 1 2 ) − 3n ) 1 2 √ 2n− 1 , TABC(Fn) = (( 2n+ 1 2 ) − 3n )√ 4n− 3 8n2 − 8n+ 2 , TAZ(Fn) = (( 2n+ 1 2 ) − 3n )( 8n2 − 8n+ 2 4n− 3 )3 , REFERENCES 1069 TGA(Fn) = (( 2n+ 1 2 ) − 3n )(√ 4n2 − 4n+ 1 2n− 1 ) , TAG(Fn) = (( 2n+ 1 2 ) − 3n )( 2n− 1√ 4n2 − 4n+ 1 ) , RTAG(Fn) = ( 2n+ 1 2 ) − 3n, RTGA(Fn) = ( 2n+ 1 2 ) − 3n, RTSC(Fn) = (( 2n+ 1 2 ) − 3n ) 1√ 2(n+ 1) , RTABC(Fn) = (( 2n+ 1 2 ) − 3n ) √ 2n n+ 1 , RTAZ(Fn) = (( 2n+ 1 2 ) − 3n )( (n+ 1)2 2n )3 . Acknowledgements The first author HSR is thankful to University Grants Com- mission (UGC), New Delhi, for the support through grant under UGC-SAP DRS-III, 2016-2021: F.510/3/DRS-III /2016 (SAP-I). The second author SYT is thankful to Min- istry of Tribal Affairs, Govt. of India, New Delhi for awarding National Fellowship for Higher Education No. 2017 18-NFST-KAR-01182. References [1] B Basvanagoud, V R Desai, and I N Cangul. Four new tensor products of graphs and their zagreb indices and coindices. Electronic Journal of Mathematical Analysis and Applications, 8(1):209–219, 2020. [2] A R Bindusree, I N Cangul, V Lokesha, and A S Cevik. Zagreb Polynomials of three Graph Operators. Filomat, 30(7):1979–1986, 2016. [3] K C Das, N Akgunes, M Togan, A Yurttas, I N Cangul, and A S Cevik. On the first Zagreb index and multiplicative Zagreb coindices of graphs. Analele Stiintifice ale Universitatii Ovidius Constanta, 24(1):153–176, 2016. REFERENCES 1070 [4] K C Das, A Yurttas, M Togan, I N Cangul, and A S Cevik. The multiplicative zagreb indices of graph operations. Journal of Inequalities and Applications., 90:1–14, 2013. [5] B Furtula and I Gutman. A forgotten topological index. J Math Chem, 53(4):1184– 1190, 2015. [6] I Gutman. Degree-based topological indices. Analele Stiintifice ale Universitatii Ovidius Constanta, 86:351–361, 2013. [7] I Gutman and K C Das. The first Zagreb index 30 years after. MATCH Commu. Math. Comput. Chem., 50:83–92, 2004. [8] I Gutman and N Trinajstić. Graph theory and molecular orbitals, Total π-electron energy of alternant hydrocarbons. Chem. Phys. Lett., 17:535–538, 1972. [9] F Harary. Status and contrastatus. Sociometry, 22(1):23–43, 1959. [10] F Harary. Graph Theory. Narosa Publishing House, New Delhi, 1999. [11] K P Narayankar and D Selvan. Geometric arithmetic status index of graphs. Int. J. Math. Arch., 8:230–233, 2017. [12] S Nikolić, G Kovac̆ević, A Milic̆ević, and N Trinajstić. The Zagreb indices 30 years after. Croat. Chem. Acta, 76:113–124, 2003. [13] D Plavi, S Nikoli, N Trinajsti, and Z Mihali. On the Harary Index for the Character- ization of Chemical Graphs. J. Math. Chem., 12:235–250, 1993. [14] H S Ramane and S Y Talwar. Reciprocal transmission based topological indices of graphs and its applications in chemistry. Preprint. [15] H S Ramane and S Y Talwar. Transmission based topological indices of graphs and its regression analysis with some molecular properties. Preprint. [16] H S Ramane and A S Yalnaik. Status connectivity indices of graphs and its appli- cations to the boiling point of benzenoid hydrocarbons. J. Appl. Math. Comput., 55:609–627, 2017. [17] P S Ranjini, V Lokesha, and I N Cangul. On the Zagreb Indices of the Line Graphs of the Subdivision Graphs. Applied Mathematics and Computation, 218(3):699–702, 2011. [18] P Sarkar, N De, I N Cangul, and A Pal. The (a,b)-Zagreb index of some derived networks. Journal of Taibah University for Science, 13(1):79–86, 2019. [19] R Sharafdini and T Reti. On the transmission-based graph topological indices. Kragu- jevac J. Math., 44:41–63, 2020. REFERENCES 1071 [20] M Togan, A Yurttas, and I N Cangul. All versions of Zagreb indices and coindices of subdivision graphs of certain graph types. Advanced Studies in Contemporary Mathematics, 26(1):227–236, 2016. [21] M Togan, A Yurttas, and I N Cangul. Zagreb and multiplicative Zagreb indices of r-subdivision graphs of double graphs. Scientia Magna, 12(1):115–119, 2017. [22] M Togan, A Yurttas, and I N Cangul. Inverse problem for the first entire Zagreb index. Advanced Studies in Contemporary Mathematics, 29(2):161–169, 2019. [23] M Togan, A Yurttas, A S Cevik, and I N Cangul. Effect of Edge Deletion and Addition on Zagreb Indices of Graphs. In: Ta, K., Baleanu, D., Machado, J. (eds), Mathematical Methods in Engineering, Theoretical Aspects. Nonlinear Systems and Complexity, 23:191–201, 2019. [24] M Togan, A Yurttas, A S Cevik, and I N Cangul. Indices and Multiplicative Zagreb Indices of Double Graphs of Subdivision Graphs. Turkic World of Mathematical Society, Journal of Applied and Engineering Mathematics, 9(2):404–412, 2019. [25] H Wiener. Structural determination of paraffin boiling points. J. Am. Chem. Soc., 69:17–20, 1947. [26] A Yurttas, M Togan, and I N Cangul. Zagreb indices and multiplicative Zagreb indices of subdivision graphs of double graphs. Advanced Studies in Contemporary Mathematics, 26(3):407–416, 2016. [27] A Yurttas, M Togan, and I N Cangul. Zagreb Indices of Graphs with Added Edges. Proceedings of the Jangjeon Mathematical Society, 21(3):385–392, 2018. [28] A Yurttas, M Togan, V Lokesha, I N Cangul, and I Gutman. Inverse problem for Zagreb indices. Journal of Mathematical Chemistry, 57:609–615, 2019.