CCO Commun. Comb. Optim.

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1 Communcatons n Combnatorcs and Optmzaton Vol. 2 No. 2, 2017 pp DOI: /CCO CCO Commun. Comb. Optm. Reformulated F-ndex of graph operatons Hamdeh Aram 1 and Nasrn Dehgard 2 1 Department of Mathematcs Garezaeddn Center, Khoy Branch, Islamc Azad Unversty, Khoy, Iran hamdeh.aram@gmal.com 2 Department of Mathematcs and Computer Scence Srjan Unversty of Technology, Srjan, Iran n.dehgard@srjantech.ac.r Receved: 18 March 2017; Accepted: 24 June 2017 Publshed Onlne: 2 July 2017 Communcated by Ivan Gutman Abstract: The frst general Zagreb ndex s defned as M1 λ(g) = v V (G) d G(v) λ where λ R {0, 1}. The case λ = 3, s called F-ndex. Smlarly, reformulated frst general Zagreb ndex s defned n terms of edge-drees as EM1 λ(g) = e E(G) d G(e) λ and the reformulated F-ndex s RF (G) = e E(G) d G(e) 3. In ths paper, we compute the reformulated F-ndex for some graph operatons. Keywords: Frst general Zagreb ndex, reformulated frst general Zagreb ndex, F- ndex, reformulated F-ndex. AMS Subject classfcaton: 05C12, 05C07 1. Introducton We follow Bondy and Murty [3] for termnology and notaton not defned here and consder fnte smple connected graphs only. In 1972, Gutman and Trnajstć [7] explored the study of total π-electron energy on the molecular structure and ntroduced a vertex degree-based graph nvarants. These nvarants are defned as M 1 (G) = v V (G) d G (v) 2 = uv E(G) (d G (u) + d G (v)) Correspondng Author c 2017 Azarbajan Shahd Madan Unversty. All rghts reserved.

2 88 Reformulated F-ndex of graph operatons and M 2 (G) = uv E(G) (d G (u).d G (v)). In the same paper, where frst Zagreb ndex was ntroduced, Gutman and Trnajstć ndcated that another term of the form v V (G) d G(v) 3 nfluences the total ϕ-electron energy. But ths remaned unstuded by the researchers for a long tme, except for a few occasons [8, 9, 12] untl publcaton of an artcle by Furtula and Gutman n 2015 and so they named t forgotten topologcal ndex or F-ndex n short [6]. Thus F-ndex of a graph G s defned as F (G) = d G (v) 3 = (d G (u) 2 + d G (v) 2 ). v V (G) uv E(G) In 2004, Mlčevć et al. [13] reformulated frst Zagreb ndex n terms of edge degrees nstead of vertex degrees, where the degree of an edge e = uv s defned as d G (e) = d G (u) + d G (v) 2. Thus, the reformulated frst Zagreb ndex of a graph G s defned as EM 1 (G) = d G (e) 2. e E(G) In 2004 and 2005, L et al. [9, 12] ntroduced the concept of the frst general Zagreb ndex M1 λ (G) of G as follows: M1 λ (G) = d G (v) λ = d G (u) λ 1 + d G (v) λ 1 for λ R {0, 1}. v V (G) e=uv E(G) The reformulated verson of the general frst Zagreb ndex s defned as EM1 λ (G) = d G (e) λ for λ R {0, 1}. e E(G) For the specal case λ = 3, RF (G) s called reformulated F-ndex. Graph operatons play an mportant role n chemcal graph theory. Some chemcally mportant graphs can be obtaned from some graphs by dfferent graph operatons, such as some nanotorus or Hammng graph, that s Cartesan product of complete graphs. Many authors computed some ndces for some graph operatons (see, for nstance [2, 4, 5, 10, 11] and the references cted theren). We [1] defned and studed the general Zagreb condces for λ R as and M 1 λ (G) = M 2 λ (G) = uv / E(G) uv / E(G) (d G (u) λ 1 + d G (v) λ 1 ) (d G (u) λ d G (v) λ ). In ths paper, we compute the reformulated F-ndex for some graph operatons.

3 H. Aram and N. Dehgard The jon of graphs The jon G+H of graphs G and H wth dsjont vertex sets V (G) and V (H) and edge sets E(G) and E(H) s the graph unon G H together wth all the edges between V (G) and V (H). Obvously, V (G + H) = V (G) + V (H) and E(G + H) = E(G) + E(H) + V (G) V (H). Theorem 1. Let G be a graph of order n and sze m for = 1, 2. Then RF (G 1 + G 2) = RF (G 1) + RF (G 2) + 7n 2F (G 1) + 7n 1F (G 2) + 12n 2M 2(G 1) + 12n 1M 2(G 2) + 3(5n n 2 + 2m 2 + n 1n 2)M 1(G 1) + 3(5n n 1 + 2m 1 + n 1n 2)M 1(G 2) + (8n n n 2)m 1 + (8n n n 1)m 2 + n 1n 2(n 1 + n 2 2) m 1m 2(n 1 + n 2 2) + (6m 1n 2 + 6n 1m 2)(n 1 + n 2 2) 2. Proof. By defnton, RF (G 1 + G 2 ) = (d G1+G 2 (e)) 3. e=uv E(G 1+G 2) We partton the edge set of G 1 + G 2 nto three subsets E 1 = E(G 1 ), E 2 = E(G 2 ) and E 3 = {e = uv u V (G 1 ), v V (G 2 )}. For any vertex v V (G 1 ), we have d G1+G 2 (v) = d G1 (v) + n 2 and for each vertex u V (G 2 ) we have d G1+G 2 (u) = d G2 (u) + n 1. It follows that and (d G1+G 2 (uv)) 3 = (d G1 (u) + d G1 (v) 2 + 2n 2 ) 3 uv E 1 uv E 1 = (d G1 (u) + d G1 (v) 2) 3 + uv E 1 6n 2 (d G1 (u) + d G1 (v) 2) 2 + uv E 1 12n 2 2(d G1 (u) + d G1 (v) 2) + 8n 3 2 uv E 1 uv E 1 =RF (G 1 ) + 6n 2 F (G 1 ) + 12n 2 M 2 (G 1 )+ (12n n 2 )M 1 (G 1 ) + 8m 1 (n 3 2 3n n 2 ), (1) e E 2 (d G1+G 2 (e)) 3 =RF (G 2 ) + 6n 1 F (G 2 ) + 12n 1 M 2 (G 2 ) + (12n n 1 )M 1 (G 2 )+ 8m 2 (n 3 1 3n n 1 ). (2)

4 90 Reformulated F-ndex of graph operatons If e = uv E 3 where u V (G 1 ) and v V (G 2 ), then we have (d G1+G 2 (uv)) 3 = (d G1 (u) + d G2 (v) + n 1 + n 2 2) 3 uv E 3 u V (G 1),v V (G 2) = (d G1 (u) + d G2 (v)) 3 + (n 1 + n 2 2) 3 + 3(d G1 (u) + d G2 (v)) 2 (n 1 + n 2 2)+ 3(d G1 (u) + d G2 (v))(n 1 + n 2 2) 2 = n 2 F (G 1 ) + n 1 F (G 2 ) + 6m 2 M 1 (G 1 ) + 6m 1 M 1 (G 2 )+ n 1 n 2 (n 1 + n 2 2) 3 + 3n 2 (n 1 + n 2 2)M 1 (G 1 ) + 3n 1 (n 1 + n 2 2)M 1 (G 2 )+ 24m 1 m 2 (n 1 + n 2 2)+ (6m 1 n 2 + 6n 1 m 2 )(n 1 + n 2 2) 2. (3) We conclude from Equatons (1), (2) and (3) that RF (G 1 + G 2 ) = RF (G 1 ) + RF (G 2 ) + 7n 2 F (G 1 ) + 7n 1 F (G 2 ) + 12n 2 M 2 (G 1 ) + 12n 1 M 2 (G 2 ) + 3(5n n 2 + 2m 2 + n 1 n 2 )M 1 (G 1 ) + 3(5n n 1 + 2m 1 + n 1 n 2 )M 1 (G 2 ) + (8n n n 2 )m 1 + (8n n n 1 )m 2 + n 1 n 2 (n 1 + n 2 2) m 1 m 2 (n 1 + n 2 2) + (6m 1 n 2 + 6n 1 m 2 )(n 1 + n 2 2) The corona product of graphs The corona product G H of graphs G and H wth dsjont vertex sets V (G) and V (H) and edge sets E(G) and E(H) s the graph obtaned by one copy of G and V (G) copes of H and jonng the -th vertex of G to every vertex n -th copy of H. Obvously, V (G H) = V (G) + V (G) V (H) and E(G H) = E(G) + V (G) E(H) + V (G) V (H). Theorem 2. If G s a graph of order n and of sze m for = 1, 2, then RF (G 1 G 2) = RF (G 1) + 7n 2F (G 1) + n 1F (G 2) + 12n 2M 2(G 1) + (15n n 2 + 6m 2)M 1(G 1) + n 1RF (G 2) + 6n 1EM 1(G 2) + 12n 1(M 1(G 2) 16m 2) + (3n 1(n 2 1) + 6m 1)M 1(G 2) + 24m 1m 2(n 2 1) + (6m 1n 2 + 6n 1m 2)(n 2 1) 2.

5 H. Aram and N. Dehgard 91 Proof. Suppose V (G 1 ) = {v 1,..., v n1 } and V (G 2 ) = {u 1,..., u n2 }. Let E 1 = E(G 1 ), E2 = E(G 2) and E3 = {v u j 1 j n 2 } for 1 n 1. Then E(G 1 G 2 ) = E 1 ( n1 =1 E 2) ( n1 =1 E 3) s a partton of E(G 1 G 2 ). Usng an argument smlar to that descrbed n the proof of Theorem 1, we have e E 1 (d G1 G 2 (uv)) 3 =RF (G 1 ) + 6n 2 F (G 1 ) + 12n 2 M 2 (G 1 )+ (12n n 2 )M 1 (G 1 ) + 8m 1 (n 3 2 3n n 2 ). (4) For any edge uv E 2, we have d G1 G 2 (uv) = d G2 (u) + d G2 (v) and hence n 1 n 1 (d G1 G 2 (uv)) 3 = (d G2 (u) + d G2 (v) 2 + 2) 3 =1 uv E2 =1 uv E2 n 1 = (d G2 (u) + d G2 (v) 2) 3 + =1 uv E2 n 1 =1 uv E2 n 1 =1 uv E2 6(d G2 (u) + d G2 (v) 2) 2 + n 1 12(d G2 (u) + d G2 (v) 2) + 8 =1 uv E2 =n 1 RF (G 2 ) + 6n 1 EM 1 (G 2 ) + 12n 1 (M 1 (G 2 ) 16m 2 ). (5) On the other hand, we have n 1 =1 uv E3 n 1 n 2 (d G1 G 2 (uv)) 3 = (d G1 G 2 (v u j )) 3 =1 j=1 n 1 n 2 = (d G1 (v ) + d G2 (u j ) + n 2 1) 3 =1 j=1 n 1 n 2 = (d G1 (v ) + d G2 (u j )) 3 + =1 j=1 3(d G1 (v ) + d G2 (u j )) 2 (n 2 1)+ 3(d G1 (v ) + d G2 (u j ))(n 2 1) 2 (n 2 1) 3 + =n 2 F (G 1 ) + n 1 F (G 2 ) + 6m 2 M 1 (G 1 ) + 6m 1 M 1 (G 2 )+ n 1 n 2 (n 2 1) 3 + 3n 2 (n 2 1)M 1 (G 1 )+ 3n 1 (n 2 1)M 1 (G 2 ) + 24m 1 m 2 (n 2 1)+ (6m 1 n 2 + 6n 1 m 2 )(n 2 1) 2. (6)

6 92 Reformulated F-ndex of graph operatons Summng up the Equaltes (4), (5) and (6) we obtan, RF (G 1 G 2 ) = RF (G 1 ) + 7n 2 F (G 1 ) + n 1 F (G 2 ) + 12n 2 M 2 (G 1 ) + (15n n 2 + 6m 2 )M 1 (G 1 ) + n 1 RF (G 2 ) + 6n 1 EM 1 (G 2 ) + 12n 1 (M 1 (G 2 ) 16m 2 ) + (3n 1 (n 2 1) + 6m 1 )M 1 (G 2 ) + 24m 1 m 2 (n 2 1) + (6m 1 n 2 + 6n 1 m 2 )(n 2 1) The Cartesan product of graphs The Cartesan product G H of graphs G and H has the vertex set V (G H) = V (G) V (H) and (u, x)(v, y) s an edge of G H f uv E(G) and x = y, or u = v and xy E(H). Obvously, V (G H) = V (G) V (H) and E(G H) = E(G) V (H) + V (G) E(H). If G 1, G 2,..., G n are arbtrary graphs, then we denote G 1 G n by n =1 G. Lemma 1. (M.H. Khalfeh et al. [11]) Let G 1 = (V 1, E 1),..., G n = (V n, E n) be graphs and let G = n =1G and V = V ( n =1G ). Then M 1( n =1G ) = V n =1 M 1(G ) V + 4 V n j,,j=1 E E j V V j. In partcular, M 1(G n ) = n V (G) n 2 (M 1(G) V (G) + 4(n 1) E(G) 2 ). Lemma 2. (M.H. Khalfeh et al. [11]) Let G 1 = (V 1, E 1),..., G n = (V n, E n) be graphs and let G = n =1G, V = V ( n =1G ) and E = E( n =1G ). Then In partcular, M 2( n =1G ) = V 4 V n =1 M 2(G ) V n,j,k=1, j, k,j k + 3M E V E 1(G )( V V ) + 2 E E j E k V V j V k. M 2(G n ) = n V (G) n 3 ( V (G) 2 M 2(G)+3(n 1) E(G) V (G) M 1(G)+4(n 1)(n 2) E(G) 3 ). Lemma 3. (De Dlanjan et al. []) Let G 1 = (V 1, E 1),..., G n = (V n, E n) be graphs and let G = n =1G, V = V ( n =1G ) and E = E( n =1G ). Then F ( n =1G ) = V n =1 M 1(G ) V + 4n n,j=1, j E(G ) V (G ). E(Gj) V (G j).

7 H. Aram and N. Dehgard 93 Now we determne the reformulated F-ndex of the Cartesan product of graphs. Theorem 3. Let G be a graph of order n and sze m for = 1, 2. Then RF (G 1 G 2) =n 1RF (G 2) + n 2RF (G 1) + 12(m 2EM 1(G 1) + m 1EM 1(G 2))+ 24M 1(G 2)M 1(G 1) 24(m 1M 1(G 2) + m 2M 1(G 1))+ 8(m 1F (G 2) + m 2F (G 1)). Proof. Suppose V (G 1 ) = {u 1,..., u n1 } and V (G 2 ) = {v 1,..., v n2 }. Set E = {(u, v )(v, v ) uv E(G 1 )} for 1 n 2 and L j = {(u j, x)(u j, y) xy E(G 2 )} for 1 j n 1. Clearly ( n2 =1 E ) ( n1 j=1 L j) s a partton of V (G 1 G 2 ). Snce d G1 G 2 (u, v j ) = d G1 (u ) + d G2 (v j ) for each, j, we have n 2 =1 uv E(G 1) Smlarly, we have n 1 n 2 (d G1 G 2 ((u, v )(v, v ))) 3 = j=1 uv E(G 2) =1 uv E(G 1) n 2 = =1 uv E(G 1) n 2 =1 uv E(G 1) n 2 =1 uv E(G 1) n 2 =1 uv E(G 1) (d G1 (u) + d G1 (v) 2 + 2d G2 (v )) 3 (d G1 (u) + d G1 (v) 2) 3 + 6(d G1 (u) + d G1 (v) 2) 2 d G2 (v )+ 12(d G1 (u) + d G1 (v) 2)d 2 G 2 (v )+ 8d 3 G 2 (v ) = n 2 RF (G 1 ) + 12m 2 EM 1 (G 1 )+ 12M 1 (G 2 )(M 1 (G 1 ) 2m 1 ) + 8m 1 F (G 2 ). (d G1 G 2 ((u j, u)(u j, v))) 3 =n 1 RF (G 2 ) + 12m 1 EM 1 (G 2 )+ Summng up Equatons (7) and (8), we obtan 12M 1 (G 1 )(M 1 (G 2 ) 2m 2 )+ (7) 8m 2 F (G 1 ). (8) RF (G 1 G 2 ) = n 1 RF (G 2 ) + n 2 RF (G 1 ) + 12(m 2 EM 1 (G 1 ) + m 1 EM 1 (G 2 ))+ 24M 1 (G 2 )M 1 (G 1 ) 24(m 1 M 1 (G 2 ) + m 2 M 1 (G 1 )) + 8(m 1 F (G 2 )+ m 2 F (G 1 )).

8 94 Reformulated F-ndex of graph operatons Example 1. As regards, for natural numbers n, m, M 1(C n) = EM 1(C n) = 4n, F (C n) = RF (C n) = 8n, M 1(P m) = 4m 6, EM 1(P m) = 4m 10, F (P m) = 8m 14 and RF (P m) = 8m 22, 1. RF (C n C m) = 432mn. 2. RF (C n P m) = 432mn 702n. 3. RF (P n P m) = 432mn 582(m + n) Example 2. Let T = T [p, q] be the molecular graph of a nanotorus (Fgure 1). Then V (T ) = pq and E(T ) = 3 pq. Obvously, M1(T ) = 9pq, EM1(T ) = 24pq, F (T ) = 2 27pq and RF (T ) = 96pq. Therefore for a q-mult-walled nanotorus G = P n T we have RF (P n T ) = 1280npq 1738pq. Theorem 4. Let G 1, G 2,, G n be graphs wth V = V (G ) and E = E(G ) for 1 n, and V = V ( n =1G ) and E = E( n =1G ). Then RF ( n =1G ) = n RF (G ) =1 n j=1,j V j + 12( E n EM 1( n 1 =1 G ) + E( n 1 =1 G ) EM 1(G n)) 24( E n M 1( n 1 =1 G ) + E( n 1 =1 G ) M 1(G n)) + 8( E n F ( n 1 =1 G ) + E( n 1 =1 G ) F (G n)) + n 3 j 24M 1(G n)m 1( n 1 =1 G ) + 24 n 1 j j=1 =1 j=0 =0 V n M 1(G n 1)M 1( n 2 =1 G ) + V n +1 [ E n (12EM 1( n 1 =1 G ) 24M 1( n 1 =1 G )) + 8F ( n 1 =1 G ) + E( n 1 =1 G ) (12EM 1(G n ) 24M 1(G n ) + 8F (G n ))]. Proof. The proof s by nducton on n. Accordng on Theorem 3, the statement holds for n = 2. Clearly, E( n =1 G ) = V n E =1 V and V ( n =1 G ) = n =1 V. One can see that for every graph G, E(M 1 (G)) = F (G) + 2M 2 (G) + 4m 4M 1 (G). Now wth applyng the Theorem 3 for n =1 G = n 1 =1 G G n and nducton, the proof s completed. 5. The composton product of graphs The composton G[H] of graphs G and H has the vertex set V (G[H]) = V (G) V (H) and (u, x)(v, y) s an edge of G[H] f (uv E(G)) or (xy E(H) and u = v). Obvously, V (G[H]) = V (G) V (H) and E(G[H]) = E(G) V (H) 2 + E(H) V (G).

9 H. Aram and N. Dehgard 95 Fgure 1. The graph of a nanotorus Theorem 5. Let G 1 be a graph of order n 1 and sze m 1 and let G 2 be a graph of order n 2 and sze m 2. Then RF (G 1[G 2]) = RF (G 1)n m 1(RF (G 2) + RF (G 2)) + 8m 1n n 2m 1(EM 1(G 2) + EM 1(G 2)) + 8n 2 2m 1(M 1(G 2) + M 1(G 2) 2) + 2n 2 2EM 1(G 1)(M 1(G 2) + M 1(G 2) 2 + 2n 2) + 2n 2(M 1(G 1) 2m 1)(EM 1(G 2) + EM 1(G 2) + 4n 2 2) + 8m 2n 3 2F (G 1) + RF (G 2) + 12n 2 2M 1(G 1)(M 1(G 2) 2m 2) + 12n 2m 1EM 1(G 2). Proof. Suppose V (G 1 ) = {u 1,..., u n1 } and V (G 2 ) = {v 1,..., v n2 }. We partton the edges of G 1 [G 2 ] nto two subsets E 1 and E 2, as follows: Set L = {e = (u, x)(v, y) uv E(G 1 )} and E = {(u, x)(u, y) xy E(G 2 )} for 1 n 1. Clearly ( n1 =1 E ) L s a partton of V (G 1 [G 2 ]). Let e = (u, x)(v, y) L. Then d G1[G 2](u, x) = n 2 d G1 (u) + d G2 (x) and d G1[G 2](u, x)(v, y) = n 2 (d G1 (u) + d G2 (v)) 2 + d G2 (x) + d G2 (y). In ths case we notce that xy can be adjacent n G 2 or not.

10 96 Reformulated F-ndex of graph operatons e=(u,x)(v,y), e L d 3 G 1 [G 2 ] (e) = e=(u,x)(v,y) e L [n 2 (d G1 (u) + d G1 (v) 2) 2 + d G2 (x) + d G2 (y) + 2n 2 ] 3 = [n 3 2 (d G 1 (u) + d G1 (v) 2) 3 + e=(u,x)(v,y) e L (d G2 (x) + d G2 (y) 2) 3 + 8n n 2 (d G2 (x) + d G2 (y) 2) 2 + 8n 2 2 (d G 2 (x) + d G2 (y) 2) + 2n 2 2 (d G 1 (u) + d G1 (v) 2) 2 (d G2 (x) + d G2 (y) 2 + 2n 2 ) + 2n 2 (d G1 (u) + d G1 (v) 2)((d G2 (x) + d G2 (y) 2) 2 + 4n 2 2)] = RF (G 1 )n m 1(RF (G 2 ) + RF (G 2 )) + 8m 1 n n 2 m 1 (EM 1 (G 2 ) + EM 1 (G 2 )) + 8n 2 2 m 1(M 1 (G 2 ) + M 1 (G 2 ) 2) + 2n 2 2EM 1 (G 1 )(M 1 (G 2 ) + M 1 (G 2 ) 2 + 2n 2 ) + 2n 2 (M 1 (G 1 ) 2m 1 )(EM 1 (G 2 ) + EM 1 (G 2 ) + 4n 2 2 ). Let e = (u, x)(u, y) E. Then d G1[G 2](u, x)(u, y) = 2n 2 d G1 (u ) 2 + d G2 (x) + d G2 (y) and we have e=(u,x)(u,y), e E d 3 G 1[G 2] (e) = = e=(u,x)(u,y) e E [8n e=(u,x)(u,y) e E [2n 2 d (u) 2 + d (x) + d (y)] 3 G1 G2 G2 3 2d 3 G 1 (u) + (d G2 (x) + d G2 (y) 2) n 2 2d 2 G 1 (u)(d G2 (x) + d G2 (y) 2) + 6n 2 d G1 (u)(d G2 (x) + d G2 (y) 2) 2 = 8m 2 n 3 2F (G 1 ) + RF (G 2 ) + 12n 2 2M 1 (G 1 )(M 1 (G 2 ) 2m 2 ) + 12n 2 m 1 EM 1 (G 2 ).

11 H. Aram and N. Dehgard 97 Therefore RF (G 1 [G 2 ]) = (d G1[G 2](e)) 3 + (d G1[G 2](e)) 3 e L e E = RF (G 1 )n m 1 (RF (G 2 ) + RF (G 2 )) + 8m 1 n n 2 m 1 (EM 1 (G 2 ) + EM 1 (G 2 )) + 8n 2 2m 1 (M 1 (G 2 ) + M 1 (G 2 ) 2) + 2n 2 2EM 1 (G 1 )(M 1 (G 2 ) + M 1 (G 2 ) 2 + 2n 2 ) + 2n 2 (M 1 (G 1 ) 2m 1 )(EM 1 (G 2 ) + EM 1 (G 2 ) + 4n 2 2) + 8m 2 n 3 2F (G 1 ) + RF (G 2 ) + 12n 2 2M 1 (G 1 )(M 1 (G 2 ) 2m 2 ) + 12n 2 m 1 EM 1 (G 2 ). 6. The tensor product of graphs The Tensor Product G H of graphs G and H has the vertex set V (G H) = V (G) V (H) and (u, x)(v, y) s an edge of G H f uv E(G) and xy E(H). Obvously, V (G H) = V (G) V (H), E(G H) = 2 E(G) E(H) and d G H (u, x) = d G (u).d H (x). Theorem 6. If G s a graph of order n and sze m for = 1, 2, then RF (G 1 G 2) = M 4 1 (G 1 G 2) 6F (G 1 G 2) + 12M 1(G 1 G 2) + 3M 2(G 1 G 2)M 1(G 1 G 2) 12M 2(G 1 G 2) 8m 1m 2. Proof. For any edge e = (u, x)(v, y) E(G 1 G 2 ), we have d G1 G 2 (e) = d G1 (u).d G2 (x) + d G1 (v).d G2 (y) 2. By defnton, we have RF (G 1 G 2 ) = (d G1 G 2 ((u, x)(v, y))) 3 uv E(G 1 ) xy E(G 2 ) = (d G1 (u).d G2 (x) + d G1 (v).d G2 (y) 2) 3 uv E(G 1 ) xy E(G 2 ) = [d 3 G 1 (u)d 3 G 2 (x) + d 3 G 1 (v)d 3 G 2 (y) 6d 2 G 1 (v)d 2 G 2 (y) + uv E(G 1 ) xy E(G 2 ) 12d G1 (v)d G2 (y) 8 + 3d 2 G 1 (u)d 2 G 2 (x)d G1 (v)d G2 (y) 6d 2 G 1 (u)d 2 G 2 (x) + 3d G1 (u)d G2 (x)d 2 G 1 (v)d 2 G 2 (y) 12d G1 (u)d G2 (x)d G1 (v)d G2 (y) + 12d G1 (u)d G2 (x)] = M1 4 (G 1 G 2 ) 6F (G 1 G 2 ) + 12M 1 (G 1 G 2 ) + 3M 2 (G 1 G 2 )M 1 (G 1 G 2 ) 12M 2 (G 1 G 2 ) 8m 1 m 2.

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