ZAGREB INDICES OF A NEW SUM OF GRAPHS Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 9, No. 1, 2023, pp. 4-17

DOI: 10.15826/umj.2023.1.001

ZAGREB INDICES OF A NEW SUM OF GRAPHS

Liju Alex

Department of Mathematics, Bishop Chulaprambil Memorial College(B.C.M),

Kottayam - 686001, India

Marthoma College, Thiruvalla, Pathanamthitta - 689103, India

lij [email protected]

Indulal Gopalapillai

Department of Mathematics, St.Aloysius College, Edathua, Alappuzha - 689573, India

[email protected]

Abstract: The first and second Zagreb indices, since its inception have been subjected to extensive research in the physio-chemical analysis of compounds. In [5], Hanyuan Deng et al. computed the first and second Zagreb indices of four new operations on a graph defined by M. Eliasi, B. Taeri [6]. Motivated by [6], in this paper we define a new operation on graphs and compute the first and second Zagreb indices of the resultant graph. We illustrate the results with some examples.

Keywords: First Zagreb index Mi(G), Second Zagreb index M2(G), F* sum.

1. Introduction

A graph without loops and also without parallel edges is called a simple graph and if all the pairs of vertices of the graph are connected by a path then it is said to be connected. Throughout our discussion, we consider only connected simple graphs. The degree-based structural descriptors have been a subject of detailed study since their induction from the first degree-based topological index in 1972 by I. Gutman, N. Trinajstic [11]. Later, in 1975 I. Gutman, B. Ruscic, N. Trinajstic, C.F. Wilcox [12] defined another degree based index in connection with studying physical properties of chemical compounds. At first, both these indices were named as Zagreb group indices [3], but later I. Gutman named them as first and second Zagreb indices. The first Zagreb index M1(G) is defined as the sum of squares of degrees of all the vertices and the second Zagreb index M2(G) is defined as the sum of product of degrees of end vertices of all the edges. That is,

M1 (G) = ^ dG(u)2, M2(G) = ^ dc(u)dG(v).

ueV(G) uveE(G)

Various physical applications of these indices can be found in [8-10, 13, 19, 20]. A more unified and general approach on degree based indices of graphs were considered by X. Li, H. Zhao in [17, 18] which lead in defining generalized Zagreb index as

Ma (G)= £ dG (U)a.

«ev (G)

Various particular cases for this generalized Zagreb index were considered separately, one among them is the Forgotten index F(G) (when a = 3) defined in 1972 [11] but resurged in 2015 through the works of B. Furtula, I. Gutman [7]. For more works on topological indices, see [2, 15, 17, 18, 21]. The degree based topological indices of graph operations have been a subject of detailed study recently [1, 5]. In [16], M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi computed the first and second Zagreb indices of graph operations such as cartesian product, composition, join, disjunction and symmetric difference of graphs. In [6], M. Eliasi, B. Taeri defined four new operations of graphs related to subdivisions and computed the Wiener index. Motivated by [6], in this paper we define a new sum related to the four subdivison graphs and compute the first and second Zagreb indices of the new sum. We also find the Zagreb indices of some chemical structures and some classes of bridge graphs using the expressions obtained. We refer to this new sum as F* sums of graphs.

2. F* sums of graphs

Let Gi, G2 be two graphs with vertex set V, V2 and edge set E1, E2 respectively. The four subdivision graphs S(G1), R(G1), Q(G1), T(G1) are defined as follows in [4]:

1. S(G1) is the graph obtained from G1 by replacing each edge ej of G1 with a vertex and making the new vertex adjacent to the corresponding end vertices of ej for each ej € E1. That is, S(G1) is a graph with vertex set V(S(G1)) = V (J V* where V* is the collection of new vertices and the edge set

E(S(G1)) = {(v,h), (u,h) : e = vu € € V**}.

2. R(G1) is the graph obtained from G1 by replacing each edge ej of G1 with a vertex and making new vertex adjacent to the corresponding end vertices of ej for each ej € E1 also keeping every edge in G1 as well. That is, R(G1) is a graph with vertex set V(R(G1)) = V1 U V* where V* is the collection of new vertices and edge set

E(R(G1)) = {(v,h), (u,h) : e = vu € € V*} U E.

3. Q(G1) is the graph obtained from G1 by replacing each edge ej of G1 with a vertex and making new vertex adjacent to the corresponding end vertices of ej for each ej € E1 along with edges joining vertex in the ith copy of V* to the vertex in the jth copy of V* whenever ej adjacent to ej in G 1. That is, Q(G 1) is a graph with vertex set V(Q(G 1)) = V (J V* where V* is the collection of new vertices and edge set

E(Q(G{)) = {(v, h), (u, h) : e = vu € € V^} U E* = {(uj,ui) : ej adjacent to ej in E1,uj ,uj € V*},

where uj,uj- are the vertices corresponding to the edges ej,ej- € E1.

4. T(G 1) is the graph obtained from G 1 by replacing each edge ej of G 1 with a vertex and making new vertex adjacent to the corresponding end vertices of ej for each ej € E along with edges joining vertex in the ith copy of V/ to the vertex in the jth copy of V/ whenever ej adjacent to ej in G 1 and keeping every edge of G 1 as well. That is, T(G 1) is a graph with vertex set V(T(G 1)) = V11J V/ where V/ is the collection of new vertices and edge set

E(T(G 1)) = {(v, h), (u, h) : e = vu € E1, h € V*} U E*, E* = {(uj,uj) : ej adjacent to ej in E1,uj,uj- € V/} U E1,

where uj,uj- are the vertices corresponding to the edges ej,ej- € E1.

In each of these new subdivision graphs the vertices V1 can be termed as black vertices and the vertices V* can be termed as white vertices. In [6], M. Eliasi, B. Taeri defined four new sums called F sums with the operation cartesian product on black vertices on copies of subdivison graphs. Motivated by this we define a sum on copies of white vertices related to the cartesian product. Let F be any one of the symbols S, R, Q,T, then the F* sum of two graphs G1 and G2 is denoted by G1 *f G2, is a graph with the vertex set V(G1 G2) = V(F(G1)) x V2 and the edge set

E(Gi G2) = {(a,b)(c,d) : a = c € V* and bd € E2 or ac € E(F(Gi)) and b = d € V2}.

Fig. 1 is an example with G1 = P4, G2 = P6.

/MVK

/vv\

/tvtvts

AAA

/ V V

/ V V

/ V V \

/ V V \

/ V V \

(a.)

(b.)

(c.)

(d)

Figure 1. (a) P4 Pe, (b) P4 Pe, (c) P4 *q Pe, (d) P4 *t Pe.

3. Zagreb index of F* sum

In this section we compute the first and second Zagreb indices of F* sums of graphs. Theorem 1. Let G1 and G2 be two connected graphs, then

(a) Mi(Gi *s G2) = |V2|Mi(Gi) + |Ei|Mi(G2) + 4|Ei| (2|E2| + |V21),

(b) M2(Gi *s G2) = 2|Ei|Mi(G2) + |Ei|M2(G2) + 4|Ei| (2|E21 + |V2|).

Proof. From the definition of first Zagreb index, we have

Mi (Gi G2) = Y1 (d(Gi*sG2)(a, b))2

(a,b)£V (Gi*s G2)

= Y1 (d(Gi*sG2)(u,v)+ d(Gi*sG2) (x,y))

(u,v)(x,y)eE(G 1 *s G2)

= Y1 Y1 (d(Gi *sG2)(u,v)+ d(Gi *sG2)(u,y)) ueVj* vy€E2

+ ^ Y1 (d(Gi*sG2)(u,v) + d(Gi*sG2)(x,v^ . veV2 uxeE(S(Gi)

Now we separately find the values of the each parts in the sum. Firstly we consider the sum in which u € V/ and vy € E2

Y1 Y1 (d(G 1 *s G2)(u,v)+ d(G 1 *sG2}(u,y)) «eV* vyeE

= E [dS(G i )(u)+ dG2 (v) + (dS(Gi )(u)+ dG2 (y))]

«eV-,* vyeE2

= E X) [2dS(Gi)(u)+ dG2 (v)+ dG2 (y)] uev* vyeE2

= ^ [4|E2| + M1 (G2)] =4|E1 ||E2| + |E1 |M 1 (G2). uev,*

Now for each edge ux € E(S(G 1)), v € V2

E E (d(Gi*s G2)(u,v)+ d(Gi*s G2)(x,v)) veV2 uxeE(S(Gi))

= E E [dS(Gi)(u) + (dG2 (v)+ dS(Gi) (x))] veV2 «xeE(S(Gi) uevi.xevi*

= ^ (2|E1 |dG2(v) + M1 (G 1) +4|E11) = 4|E11|E21 + |V>|M1 (G 1) +4|E||V>|.

veV2

From the expressions we obtain

M1 (G 1 *s G2) =|Vi|M1 (G 1) + |E1 |M1 (G2) + 4|E11 (2|E2| + |V>|).

Next consider

M2(G1 G2) = ^ (dGi*SG2(u,v)dGi*SG2 (x,y))

(u,v)(x,y)eE(Gi*s G2)

= E (dGi*sG2 (u,v)dGi*sG2 (u,y»+£ Y^ (dGi*sG2 (u,v)dGi*sG2 (x,v))

ueV* vyeE2 veV2 «xeE(S(Gi)

= [dS(Gi)(u)+ dG2 (v)] [dS(Gi)(u)+ dG2 (y)]

uev* vyeE2

+ E X) [dS(Gi) (u) (dG2 (v) + dS(Gi) (x))] veV2 uxeE(S(Gi)), uevi.xev*

= E E [4 + 2(dG2 (v)+ dG2 (y))+ dG2 (v)dG2 (y)]^^(2(|E |) dG2 (v)+4|E |)

«eV* vyeE2 vev2

= 4|E1 ||E2| + 2|E1 |M1 (G2) + |E1 |M2(G2) + 4|E| (|E2| + |V>|).

Thus,

M2(G 1 G2) = 2|E1 |M1 (G2) + |E 1 |M2(G2) + 4|E11 (2|E21 + |V21).

□

Theorem 2. Let G1 and G2 be two connected graphs, then

(a) M1(G1 *r G2) = 4|V> |M1(G1) + EM^) + 4|E1|(2|E21 + | V21),

(b) M2(G1 *r G2) = 4M1(G1 )(1 + |E21) + M2(G2)(4|V21 + |E^) + 2|E1|M1(G2) + 4|E1||E2|.

Proof. We have

M1(G1 G2) = (d(Gi*RG2)(a, b))2

(a,b)€V (Gi*R G2)

= E (d(Gi*flG2) (u,v)+ d(Gi *flG2)(x,y))

(u,v)(x,y)eE(Gi G2)

= E E (d(Gi *R G2)(u,v)+d(Gi *r G2)(u,y«^ E (d(Gi*flG2)(u,v)+d(Gi *flG2)(x,v)) . «eV* vyeE2 veV2 «xeE(R(Gi)

Now we separately find the values of each part in the sum. First we consider the sum in which u € V! and vy € E2

E E (d(Gi *R G2)(u,v)+ d(Gi G2 )(u,y)) «eV* vyeE2

^ ^ [(dR(Gi)(u) + dG2 (v)) + (dR(Gi)(u)+ dG2 (y))] «eV* vyeE2

= E E [4 + dG2(v) + dG2(y)]= E [4|E21 + M1 (G2)]=4|E1||E21 + EM^).

«eV* vyeE2 «eV*

Now for each edge ux € E(R(G1)), v € V2

E E (d(Gi *SG2)(u,v)+ d(Gi *sG2) (x,v)) veV2 «xeE(R(Gi))

= E E (d(Gi *sG2)(u,v)+ d(Gi *SG2 )(x,v)) veV2 «xeE(R(Gi)), «,xeVi

+ E E (d(Gi*sG2)(u,v) + d(Gi *SG2) (x,v)) . veV2 «xeE(R(Gi)), «eVi,xeVi*

Now we calculate the each sum separately

E E (d(Gi *RG2)(u,v)+ d(Gi*RG2)(x,v)) veV2 «xeE(R(Gi)), «,xeVi

= E E (dR(Gi) (u) + dR(Gi) (x)) veV2 «xeE(R(Gi)), «,xeVi*

= E E 2(dGi (u)+ dGi (x)) =2|V2 |M1(G1).

veV2 «xeE(R(Gi)), «,xeVi

By considering the case where ux € E(R(G1)), u € Vi, x € V*

(d(Gi *RG2)(u,v)+d(Gi *rG2) (x,v))=E dR(Gi )(u) + (dG2(v) +2)

veV2 uxeE(R(Gi)), veV2 ux€E(R(Gi)),

uevi*,xevi* ueVi.xeV!

= E E 2dGi (u) + dG2 (v) + 2 = 2|V2|Mi(Gi) + 4|Ei|(|E2| + |Vi|).

veV2 ux€E(R(G i)),

uevi.xev;*

Thus we obtain

Mi(Gi *RG2) = 4|V>|Mi(Gi) + |Ei|Mi(G2) + 4|^i|(2|E3| + |V>|).

Similarly,

M2(Gi G2) = ^ (dGi*rG2(u,v)dGi*rG2(x,y))

(u,v)(x,y)eE(G i *rG2)

= E X) (dGi*RG2 (u,v)dG i*RG2 (u,y))^ ^ (d(Gi*RG2 (u,v)d(Gi *RG2 (x,v)) . ueVj* vyeE2 veV2 ux€E(R(G i)

Now we find the sums separately

(dGi *RG2 (u,v)dGi *RG2 (u,y))^ ^ [(dR(G i )(u)+ dG2 (v)) (dR(Gi) (u) + dG2 (^))] ueVj* vyeE2 ueV]* vy€E2

= E E [dR(G i )(u)2 + dR(G i )(u)(dG2 (v)+ dG2 (y))+ dG2 (v)dG2 (y)] ueVj* vyeE2

= E E [4 + 2(dG2(v) + dG2(y))+ dG2(v)dG2(y)] = 4|Ei||E2| + 2|Ei|Mi(G2) + |Ei|M2(G2). uev/ vyeE2

Also,

E E (dG i*RG2 (u,v)dGi*RG2 (x,v)) = E E (dG i*RG2 (u,v)dGi*RG2 (x,v)) veV2 ux€E(R(G i)) veV2 ux€E(R(G i)),

u,x€Vi

+ E E (dG i*rG2 (u,v)dGi *RG2 (x,v)) . veV2 ux€E(R(G i)),

ueVi.xev;* Finding the sums separately, we get

E E (dGi*RG2 (u,v)dG i*RG2 (x,v)) = E E (dR(G i)(u)dR(Gi)(x)) veV2 ux€E(R(G i)) veV2 ux€E(R(G i))

u,x€Vi u,x€Vi

^ ^ 4dGl (u)dGl (x) =4|V2|M2(G2).

v€V2 MxeE(R(G 1))

Now,

E E (dG 1 *rG2 (u,v)dGi *RG2 (x,v)) = E E dR(Gi )(uKdG2 (v) + dR(Gi )(x))

' 1*rG21 *

veV2 uxeE(R(G 1)) veV2 «xeE(ß(G 1))

Y, E 4dG1 (u) + 2dG 1 (u)dG2 (v) =4Mi(Gi) + 4|E2|Mi(Gi).

»I

v€V2 MxeE(R(G 1)) «ev^xevj*

Now collecting all the previous terms, we get

M2(G1 *r G2) = 4M1(G1)(1 + |E2|) + M2(G2)(4|V2| + |E1|) + 2|E1 |M1(G2) + 4|E1||E21.

Theorem 3. Let Gi and G2 be two connected graphs, then

(a) Mi(Gi *q G2) = (|V21 + 2|E2|)Mi(Gi) + |Ei|Mi(G2) + 2|V>|M2(Gi)

+ |V2|F (Gi) + 2|E2|(2|E(Q(Gi ))| + 3|Ei|),

(b) M2(Gi G2) = |E2|M2(Gi) + |Ei|M2(G2) + M2(Gi)M2(G2) + 2|E2|Mi(Gi)

+ \ [|^2|M4(Gi) + (2\E2\ + |F2|F(GI))]

□

+ |V2 ^ Y1 rij (ui )dGi (uj)+ Y1 (uj)2 Y1 (ui ^ '

ui,uj Uj €Vi «¿€Vi ,UiUj €Ei

where rjj denotes the number of neighbouring common vertices adjacent to both uj and uj. P r o o f. We have

M1 (G1 g2) = Y1 (d(Gi*QG2)(a,b^2

(a,b)eV (Gi*s G2)

= E (d(Gi*QG2)(u,v) + d(Gi*QG2)(x,y))

(«,v)(x,y)eE(Gi*QG2)

^ E (d(Gi*QG2)(u,v) + d(Gi *QG2)(u,y)) «eV* vyeE2

+ E E (d(Gi*QG2)(u,v) + d(Gi*QG2)(x,v^ . veV2 «xeE(Q(Gi))

First we consider the sum in which u € V/ and vy € E2

E E (d(Gi*QG2)(u,v) + d(Gi *QG2)(u,y)) «eV* vye®2

^ ^ [(dQ(Gi)(u)+ dG2 (v)) + (dQ(Gi)(u)+ dG2 (y))] «eV* vye®2

^ ^ [2dQ(Gi)(u) + dG2 (v)+ dG2 (y)] «eV* vye®2

= ^ 2|E2|(dGi(p)+ dGi(i))+E M1(G2)=2|E2|M1(G1) + |E1|M1(G2).

e=pqeEi «eVj_*

For each edge ux € E(Q(G1)) and the vertex v € V2

Y1 Y1 (d(Gi*QG2)(u,v) + d(Gi*QG2)(x,v)) veV2 «xeE(Q(Gi))

= E E (d(Gi *QG2)(u,v)+ d(Gi*QG2)(x,v)) veV2 «xeE(Q(Gi)), «eVi,xeV*

+ E E (d(Gi*QG2)(u,v) + d(Gi*QG2)(x,v^ . veV2 «xeE(Q(Gi)), «,xeV*

Now we separately find both the sums. First,

E E (d(Gi*QG2)(u,v) + d(Gi *QG2)(x,v)) veV2 MxeE(Q(Gi)) u€Vi,x€V1*

= E E dQ(Gi)(u) + (dG2 (v) + dQ(Gi)(x)) = E E dGi (u) + dG2 (v) + dQ(Gi)(x)

veV2 wxeE(Q(Gi)) veV2 m£€E(Q(G i))

ueVi.xeV]* ueVi.xeV]*

= J] Mi(Gi) + 2|Ei|dG2(v)+2^ ^ (dGi(ui) + dGi(v*))

veV2 veV2 e=«ivieE(G i)

«¿, ViSVi

= |V2|Mi(Gi) + 4|Ei||E2| + 2| V2 |Mi(Gi). The second part of the sum is the following

E E (d(G i *QG2)(u,v) + d(G i *QG2)(x,v)) veV2 MxeE(Q(G i)),

= E E (dQ(G i)(u) + dG2 (v) + dQ(G i)(x) + dG2 (v)) veV2 «xeE(Q(G i)),

= E ( E 2dG2 (v)) + E ( E (dQ(Gi)(u) + dQ(G i)(x))) veV2 «xeE(Q(Gi)), veV2 M£€E(Q(G i)),

= E( E 2dG2 (v)) + E ( E (dG i (ui )+ dG i (uj )+ dG i (uj )+ dG i (ufc ))) v€V2 M£€E(Q(Gi)), v€V2 «¿«j «fcSEi

u.xeVj*

= 4(|E(Q(Gi ))|-2|Ei|)|E2| + |V2^2 £ C^ K)dGi (u)+ £ (dG (j - 1) £ dG (ui))

uj eVi uj eVi «¿eVi,

«¿«j eE i

*)3—dG 1 (uj)2)^ ^ iU

uj eVi «¿eVi,

«¿uj eE i

4(|E(Q(Gi))|-2|Ei|)|E2MV>| E (dGi(u)3-dGi(u)2)+ £ (dGi(u) - 1) £ dGi(u)

= 4(|E(Q(Gi))| - 2|EiQ|E2| + |V2KF(Gi) + 2M2(Gi) - 2Mi(Gi)).

Here UiUj is the edge corresponding to the vertex u and Ujuk is the edge corresponding to the vertex x.

Thus we obtain

Mi(Gi *q G2) = (|V2| + 2|E|)Mi(Gi) + |Ei|Mi(G2) + 2|V2|M2(Gi) +|V>|F (Gi) +2|E2|(2|E(Q(Gi ))| + 3|Ei|).

Similarly,

M2(Gi G2) = Y1 (d(G i*QG2)(u,v)d(Gi*QG2)(x,y))

(«,v)(x,y)€E(G i *qg2)

= E E (d(G i*QG2)(u,v)d(Gi*QG2)(u,y^^E E (d(G i*QG2)(u,v)d(Gi*QG2)(x,v)) . «eV* vy€E2 v€ V2 «x€E(Q(G i)

Now we separately find the values of each part in the sum

Y1 Y1 (d(Gi *QG2)(u,v)d(Gi *Q G2)(u,y^^ E [(dQ(Gi )(u)+dG2 (v)) (dQ(Gi )(u)+dG2 (y))] «eV* vye®2 «eV* vyeE

= E Y1 [dQ(Gi) (u)2 + dQ(Gi) (u) (dG2 (v) + dG2 (y)) + dG2 (v)dG2

: E E (dGi (uj)+ dGi (uj))2 + ^ E (dGi (uj)+ dGi (uj))(dG2 (v)+ dG2 (y)) vye®2 «¿«jeEi «¿«jeEi vyeE

+ E E dG2 (v)dG2 (y) «¿«j eEi vye®2

^ ^ (dGi (uj)2 + dGi (uj)2 + 2dGi (uj)dGi (uj)) + M2(G1 )M2(G2) + |E1 |M2(G2)

vye®2 «¿«j eEi

= |E2|F (G1) + 2|E2|M2 (G1) + M2(G1)M2 (G2 ) + |E1|M2(G2 ).

Now,

(d(Gi*QG2)(u,v)d(Gi*QG2)(x,v^^E E (d(Gi*QG2)(u,v)d(Gi*QG2)(xv)) veV2 «xeE(Q(Gi)) veV2 «xeE(Q(Gi)),

«eVi,xeV*

+ E E (d(Gi*QG2)(u,v)d(Gi*qG2)(x,v^ . veV2 «xeE(Q(Gi)), «,xeVi*

Now we find each sum separately

E E (d(Gi*QG2) (u,v)d(Gi *QG2 ) (x,v))=E E dQ(Gi )(u) (dQ(Gi)(x) + dG2(v)) veV2 «xeE(Q(Gi)), veV2 «xeE(Q(Gi)),

«eVi, xeV* «eVi,xeVi*

= E E dQ(Gi)(u)dQ(Gi)(x) + dG2 (v)dQ(Gi)(u) veV2 «xeE(Q(Gi)), «eVi,xeV*

= E E dGi (u)dQ(Gi)(x) + E E dGi (u)dG2(v) veV2 «xeE(Q(Gi)), veV2 «xeE(Q(Gi)),

«eVi.xeV-i* «eVi,xeVi*

= |V2|(F (G1) + 2M2(G1)) + 2|E2|M1(G1).

The second part is

E (dQ(Gi)(u)+ dG2 (v)) (dQ(Gi)(x)+ dG2 (v)) ) veV2 ^ «xeE(Q(Gi)), '

E E (dQ(Gi) (u)dQ(Gi) (x) + dG2 (v) (dQ(Gi)(u)+ dQ(Gi)(x)) + dG2 (v) veV2 «xeE(Q(Gi)),

= E ( Y1 (dGi (uj) + dGi (uj)) (dGi (uj) + dGi (ufc)) j veV2 ^ «¿«jeEi, '

«j «fceEi

+ E dG2(v)( £ (dGi(uj)+dGi(uj)+dGi(uj)+dGi(ufc))) +(|E(Q(G1 ))|-2|E1|)M1(G2)

veV2 «¿«j eEi,

«j «k eEi

«eV* vyeE2

«,xeV*

«,xeV *

= l^i Y1 CdGl («)dGi (uj)2+ Y1 rij(u*)dGi (ujH^ (dGi (uj) 1)dGi (ujdGi (ui)j Uj eVi «¿¡«j eVi Uj eVi «¿eVi,

«¿«j eEi

+2|E2£ G^(«.)dGi(Uj)+ £ (dGi(Uj)-1) £ dGi(ui)V(lE(Q(Gi))| - 2|E1|)M1 (G2)

^ ujeVi ujeVi «¿eVi, '

«¿«j eEi

= 1^21 Q E (dGi(t/j)4 - dGl(Uj)3) + E riidGi(t/.i)dGi(t/.J)J

«j eVi «¿,«5 eVi

+|V>|( ^ dGi (Uj )2 ^ dGi (U) - 2M2(GiA

«j eVi «¿eVi, '

«¿«j eEi

+2|E2^ £ (dGi(Uj)3-dGi(Uj)2)+ £ (dGi(Uj)-l) £ dGi(Ui)V(|E(Q(Gi))|-2|Ei|)Mi(G2)

^«j eVi «j eVi «¿eVi, '

«¿«j eEi

HV2|Qm4(Gi)-±F(Gi)+ £ ^g^/^g^O+J] ^GiK')2 E dGl(ui)-2M2(G1)]

«¿¡«j eVi «j eVi «¿eVi,

«¿«j eEi

+2|E2| (F(Gi) + 2M2(Gi) - 2Mi(Gi)) + (|E(Q(Gi))| - 2|Ei|)Mi(G2).

Here UiUj is the edge corresponding to the vertex u and Ujuk is the edge corresponding to the vertex x, rj denotes the number of common vertices adjacent to both u and Uj. Thus we obtain

M2(Gi G2) = |E2|M2(Gi) + |Ei|M2(G2) + M2(Gi)M2(G2) + 2|#2|Mi(Gi)

[|^2|M4(Gi) + (2\E2\ + |Va|F(Gi))]

+ ^ rij dGi (ui )dGi (uj)+ ^ dGi (uj )2 ^ dGi (uiH .

nV v J ~r "G^ "J )2 ^ uGi \

SVl Uj SVL UiSVi,

«¿Mj SEi

□

Theorem 4. Let Gi and G2 be two connected graphs, then

(a) Mi (Gi *t G2) = 2|E2|Mi(Gi) + |Ei|Mi(G2) + 2|V2|M2(Gi)

+ |V2|F(Gi) + 4(|E(T(Gi))| - 3|Ei|)|E2|,

(b) M2(Gi *T G2) = 5|E2|M2(Gi) + (41V21 + |Ei|)M2(G2) + M2(Gi)M2(G2) - 2|E|Mi(Gi)

+ (|S(T(Gi))| - 3|Si|)MI(G2) + i [|F2|M4(GI) + (2|F2| + |F2|)F(Gi)]

+ Y1 rij dGi (ui)dGi (uj)+ Y1 dGi (uj)2 Y1 dGi (uiH .

«¿,Mj sVl Uj SVL Ui€Vi,UiUj SEi

where rj denotes the number of common vertices adjacent to both u^uj.

Proof. We prove this theorem using Theorem 2 and Theorem 3. When u € V* and vy € E2

E E (d(Gi *TG2)(u,v)+ d(Gi*TG2)(u,y^ = E E (d(Gi*QG2)(u,v) + d(Gi*QG2) (u, y^ • USV1* vySE2 USV* vySE2

From Theorem 3

E E (d(Gi*TG2)(u,v)+ d(Gi*TG2)(u,y^ =2|E2|Mi(Gi) + |Ei|Mi(G2).

«eV* vyeE2

Also

E E (d(Gi*T G2 )(u,v) + d(Gi*TG2)(x,v)) veV2 «xeE(T(Gi))

= E E (d(Gi*T G2)(u,v)+ d(Gi*TG2)(x,v)) veV2 «xeE(T(G i)),

«evi,xev1*

+ E E (d(Gi*TG2)(u,v) + d(Gi *T G2)(x,v)) veV2 «xeE(T(G i)), u.xeVj*

+ E E (d(G i*TG2)(u,v)+ d(G i*TG2)(x,v)) . veV2 «xeE(T(G i)), Vi

Also from Theorem 2 and Theorem 3

E E (d(G i *t G2 )(u,v) + d(Gi *TG2)(x,v)) veV2 «xeE(T(G i))

= 4(|E(T(Gi))| - 3|Ei|)|E2| + |V2| (F(Gi) + 2M2(Gi) - 2Mi(Gi)) + 2|V2|Mi(Gi).

Thus,

Mi(Gi *T G2) = 2|E2|Mi(Gi) + |Ei|Mi(G2) + 2|V>|M2(Gi) +|V>|F(Gi) + 4(|E(T(Gi))| - 3|Ei|)|E2|. Similarly for M2, from Theorem 3

E E (d(G i *T G2)(u,v)d(G i *T G2)(u,y))

«ev* vyeE2

= |E2 |F (Gi) + |E2|M2 (Gi) + M2 (Gi )M2(G2) + |Ei|M2(G2). The second part of the sum is

E E (d(G i *T G2 )(u,v)d(Gi *T G2)(x,v^^ E (d(G i *TG2)(u,v)d(Gi *TG2)(x,v)) veV2 «xeE(T(G i)) veV2 «xeE(T(G i)),

«evi.xev*

+ E E (d(G i *TG2)(u,v)d(Gi *T G2)(x,v))

veV2 «xeE(T(G i)), w.xev*

+ E E (d(G i*TG2)(u,v)d(Gi*TG2)(x,v^ .

veV2 «xeE(T(G i)),«,xevi

From Theorem 2 and Theorem 3 we get

E E (d(G i *t G2) (u, v)d(G i *t G2) (x, v)) = |V2|(F(Gi) + 2M2(Gi)) +2|E2|Mi(Gi)

veV2 «xeE(T(G i))

(u •(u-) + V (u-)2 » f I /' ' I #1. * p — ,

«¿,«j eVi «j eVi «¿eVi

«¿«j eE i

+2|E2|(F(Gi) +2M2(Gi) - 2Mi(Gi)) + (|E(Q(Gi))| - 2|Ei|)Mi(G2) + 4|V2|M2(G2),

here fjj denotes the number of common vertices adjacent to both Uj,—-. Thus we obtain M2(GI *t G2) = 5|E2|M2(GI) + (4| V2| + |EI|)M2(G2) + M2(GI) M2(G2) - 2|E|Mi(Gi) + (|F(T(G1))| - 3|Si|)MI(G2) + i[|F2|M4(Gi) + (2\E2\ + \V2\)F(Gi)]

+ |V2|( Y1 rij(ui)dGi (uj)+ Y1 1 (-j)2 E 1 (Ui)J •

□

4. Applications with illustration

The above computational procedure can be used to find the respective indices for many classes of graphs very easily. As an illustration we provide the following.

Example 1. When G1 = Pn, G2 = Pm, n,m > 3, using the theorem, we easily obtain the following results

1. Mi(Pra *s P M2(P„ P

2. Mi(Pra *R P M2(P„ *r P

3. Mi(Pn *Q Pm M2(Pn Pm

4. Mi(Pn *T P

M2(Pn *T P

20mn - 22m - 14n + 14, 32mn - 40m - 24n + 38;

32mn - 40m - 14n + 14, 64mn - 48m + 24n - 80;

: 40mn - 64m - 22n + 30, 96mn - 184m + 18n + 134;

48mn - 82m - 22n + 30, 136mn - 258m - 86n + 146.

Let Tn,m denote the torus grid graph obtained from the cycle Cn and Cm. Using F* sums, we can compute the Zagreb indices of torus grid graph 72«,^ since 72«,^ = Cn Cm.

Example 2. When G1 = Cn, G2 = Cm, n, m > 3, using the theorem, we easily obtain the following results

1. Mi(Cra Cm = 20mn,

M2(Cn Pm = 32mn;

2. Mi(Cra Cm = 32mn,

M2(Cn Cm = 48mn;

3. Mi(cn Cm = 40mn,

M2(Cn Cm = 96mn;

4. Mi(Cra *T Cm = 52mn,

M2(Cn *T Cm = 136mn

We can also find the Zagreb indices of some chemical structures using the expressions of F* sums.

Example 3. Let n > 3 be an integer, then Zagreb indices of the the zigzag polyhex nanotube TUHC 6[2n, 2]

Mi(TUHC 6[2n, 2]) = 26n, M2(TUHC 6[2n, 2]) = 33n.

Since TUHC6[2n, 2] = Cra P2, then by Theorem 1.

Using F* sums, we can also find the Zagreb indices of some classes of bridge graphs. Let vi, v2,..., be vertices of graphs Gi, G2,..., Gn respectively. The bridge graph using vi, v2,..., is secured by joining the vertices vi of Gi to vi+i of Gi+i for i = 1,2,... n - 1 and it is denoted by B(Gi, G2,..., Gn; vi, v2,..., vn). If Gi = Gi+i = G and vi = vi+i = v for all i = 1,2,... n, then B (G, G, ...,G; v,v...,v) = Gn(G,v). Let Bn = Gn(P3, v) where the degree d(v) = 2 and Tn,3 = Gn(C3,v) [14] be two class of bridge graphs.

Example 4. Let n > 2 be an integer, then Mi(Bra) = 18n - 14, M2(b„) = 24n - 28;

Mi(Tra>3) = 24n - 14, M2(T„,3) = 36n - 32.

Since Bn = P2 Pn and Tn,3 = P2 Pn and by Theorem 1 and Theorem 2.

5. Summary and Conclusion

The F sum of graphs was a new sum defined by M. Eliasi, B. Taeri in [6], a lot of research has been done on this to compute various topological indices of this F sum. In this paper we have defined a similar new operation and computed the first and second Zagreb index of this sum. Computing other topological indices on these sums is an area which researchers may find helpful.

Acknowledgements

The authors are highly indebted to the anonymous referees for their valuable comments and suggestions which led to an improved presentation of the results.

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