Reduced second Zagreb index of product graphs Текст научной статьи по специальности «Математика»

Reduced second Zagreb index of product graphs

N. De

Department of Basic Sciences and Humanities (Mathematics), Calcutta Institute of Engineering and Management, Kolkata, India [email protected]

DOI 10.17586/2220-8054-2020-11-2-131-137

The reduced second Zagreb index of a graph G is defined as RM2(G) = ^^ (da(u) — 1 )(da (v) — 1), where da (v) denotes the degree

uveE(G)

of the vertex v of graph G. Recently Furtula et al. (Furtula B., Gutman I., Ediz S. Discrete Appl. Math., 2014) characterized the maximum trees with respect to reduced second Zagreb index. The aim of this paper is to compute reduced second Zagreb index of the Cartesian product of k (> 2) number of graphs and hence as a consequence the reduced second Zagreb index of some special graphs applicable in various real world problems are computed. Topological properties of different nanomaterials like nanotube, nanotorus etc. are studied here graphically in terms of the aforesaid aforementioned index.

Keywords: Reduced second Zagreb index, cartesian product of graphs, nanotube, nanotorus, Hamming graphs, Ladder graphs, Rook's graph. Received: 15 January 2020 Revised: 5 March 2020

1. Introduction

Let G be a simple connected graph with vertex set V(G) and edge set E(G). For a graph G, let dG(v) denote the degree of a vertex v in G, that is, the number of vertices adjacent with v. Throughout this article, we consider chemical graph [1,2]. By chemical graph we mean a simple connected graph where vertices and edges are supposed to be atoms and chemical bonds between them respectively.

Topological indices, also called molecular structure descriptors are used in theoretical chemistry for design of chemical compounds with given physico-chemical properties and also to model chemical biological and pharmacological properties of the molecules. A topological index is a real number and it does not depend on the labelling of a graph and must be a structural invariant. The first topological index, Wiener index, was published in 1947 [3]. Due to the importance of topological indices in chemical research, lots of topological indices are developed in the chemical graph theory. Degree based topological indices are one of them that is applicable in quantitative structure property relationship and quantitative structure activity relationship [4,5]. Among them Zagreb indices are most popular indices. First and second Zagreb indices are introduced by Gutman and Trinajestic [6], defined as follows:

Mi(G)= ^ dG(u)2,

uev(G)

M2(G)= ^ dG(u)dG(v).

uveE(G)

These graph invariants were proposed to measure the branching of carbon-atom skeleton [7]. For detail discussion on these indices, see [8-14]. Furtula et al. [15] proposed the reduced second Zagreb index of a graph G to study the difference between M1 and M2, which is defined as follows:

RM2(G)= ^ (dG(u) - 1)(dG(v) - 1).

uveE(G)

The graphs having maximum and minimum reduced second Zagreb index in the class of cyclic graphs with cut edges are studied in [16]. Mahanta et al. [17] obtained Reduced the reduced second Zagreb index of four new graph operations based on tensor product. Several useful composite graphs can be obtained by operations of different graphs. The Cartesian product is one of that the operations and which is considered in the present work.

The Cartesian product G1 ( G2 of graphs G1 and G2 has the vertex set V(G) x V(G2) and (a, x)(b, y) is an edge of G1 ( G2 if a = b and xy G E(H) or ab G E(G) and x = y. Using the Cartesian product of two graphs, the structure of several nanomaterials can be designed such as C4-nanotube, C4-nanotorus, multi-walled nanotorus etc. Carbon nanotube has diverse applications in cancer treatment, cardiac autonomic regulation, tissue regeneration etc. As topological index can predict different physico-chemical properties, it is worthy to compute that for different

structures of real world application. First and second Zagreb indices of the Cartesian product graph is presented in [18]. The PI index of the Cartesian product of bipartite graphs is computed in [19]. Klavzar et al. [20] computed the Szeged index of Cartesian product graph. In [21], the Wiener index of Cartesian product graphs are studied. The present author [22-24] studied F-index, F-coindex and reformulated first Zagreb index for Cartesian product graphs. The goal of this work is to obtain the reduced second Zagreb index of Cartesian product graphs. Using that results, RM2 index of some chemical graphs is also derived.

2. Main results

Let Gi (i = 1,2, ...,k) be a connected graph with vertex set V(Gj) and E(Gj), so that, |V(Gj)| = ni and |E(Gj)| = mi. In this section, we derive the reduced second Zagreb index of Cartesian product of k-number of connected graphs Gi, G2,...., Gk. To do this first we prove the result for two connected graphs Gi and G2.

Lemma 1. [18] Let G1 and G2 be two connected graphs, then:

(i) |V(Gi <g> G2)| = |V(Gi)|x|V(G2)|,

(ii) |E(Gi < G2)| = |E(Gi)||V(G2)| + |E(G2)||V(Gi)|,

(iii) dGl0G2 (a, b) = dGl (a) + dc2 (b).

Lemma2. [18] Let Gi, G2,....,Gk be k-number of graphs, then

k k „ T / ^ N k

Mi«g>Gj) = »£ ^+4» £ m?.

- 'H ....... 'H'v j

= i i=i j,3=i

Lemma3. [18] Let Gi, G2,....,Gk be k-number of graphs, then

k

(i) |V(® Gj)| = f»

i=i i=i

(«) |E(® Gi)| = fl»j £ m.

i=1 i=1 i=1

Theorem 1. Let G1 and G2 be two connected graphs with n1 and n2 number of vertices and, m1 and m2 number of edges respectively, then:

RM2(G1 << G2) = n1RM2(G2) + n2RM2(G1) + 3m1M1 (G2) + 3m2M1 (G1) - 8mjm2.

Proof. From definition of Cartesian product of two graphs, we have:

RM2(G1 <g> G2) = Z (dGi (a'6) - !)(dG2(c,d) - 1)

(a,b)(c,d)eE(Gi0G2)

= Z Z (dGi (u)+ dG2(b) - 1)(dGi (u) + dG2(d) - 1)

«£V(Gi) bd£E(G2)

+ Z Z (dGi (a) + dG2 (v) - 1)(dGi (c) + dG2 (v) - 1)

(G2) aceE(Gi)

= C1 + C2, (Say).

Where,

Ci = £ £ (dGi (u)+ dG2 (b) - 1)(dGi (u)+ dG2 (d) - 1)

«EV(Gi) 6d£E(G2)

= £ £ (dG2 (b) - 1)(dG2 (d) - 1)+ £ £ dGl (u)2

«£V(Gl) bdeE(G2) bdeE(G2) ueV(Gl)

+ £ £ dGl (u)(dG2 (b)+ dG2 (d) - 2)

bdeE(G2) ueV(Gl)

= niRM2(G2) + m2Mi(Gi) + 2miMi(G2) - 4mim2.

Similarly,

C = E E (dGi (a) + dG2 (v) - 1)(dG! (c)+ dG2 (v) - 1)

veV(G2) aceE(Gi)

= EE (dGi (a) - 1)(dGi (c) - 1)+ E E dGi (v)2

veV(G2) aceE(Gi) aceE(Gi) v£V(G2)

+ E (dGi (a) + dGi (c) - 2) E dGi (v)

aceE(Gi) vev (G2)

= n2RM2(G1) + m1M1(G2) + 2m2M^G^ - 4m1m2.

Therefore, combining the contributions of C1 and C2 we get the desired results. □

In the following, we calculate reduced second Zagreb index of the Cartesian product of k-number of graphs Gj for i = 1, 2,...., k.

Theorem 2. Let G1, G2,.....,Gk be k-number of connected graphs, then

RM2(® Gj) = n^ RM^ + 3e( — - nn?)M1(Gj) ^ r-f nj r-f ni n2

j=1 j=1 i=1 j

Em„m0 mr m¿ m, —-—---4n > --.

Ti..Ti. Ti.. < J Ti.Ti-

npnqnr - - . / . ninj

p,q,r=1, p=q=r ^ ^ i,,= 1, i=, ^

Where, n = |V(0 Gj)| and m = |E(0 Gj)|.

i=i i=i

k-1 k-1

Proof. Let us assume that, n' = |V(0 Gj)| and m' = |E(0 Gj)| so that, n' = — and m' = -2-. Now

j=1 i=1 by Theorem 1 and an inductive argument, we can have

nk nk

k- 1

ÄM2(0 Gi) = fiM2^ Gi ® Gk)

i=i k-i

nkÄM2(0 Gi) + |V(0 Gi)|fiM2(Gk)+3mkMi(0 Gi) i=1 i=i i=i k-i k-1 +3|E(0 Gi)|Mi(Gk) - 8|E(0 Gi)||E(Gk)|

i=i i=i

k-1 k-1 k-1

Using Lemmas 1, 2, and 3 on the above result, we obtain:

RM2(0 Gj) = nk[n'E RM^ + 3^ - n^)M1(Gj)

- 1 - 1 n i - 1 n i n j

i=1 i=1 i=1 i

+4n' E-E mPmqmr - 4n' E-E m^] + n'RM2(Gk) n^nqn^ ^^ j n j

p,q,r = 1, p=q=^ ^ j,j = 1

+3mk [n' E-E M^ + 4n' E-E —j—j] + 3n' E-E(mM1(Gk)

z—' nj z—' njnj z—' nj

j=1 j,j=1 j=1

k

mj

-8mk n'V mi n

ni i=1

After arranging the terms, we have:

k-1 n, ^ n k-1

RM2(® Gi) = [nfcn' Z RM2(Gi) + n'RM2(Gfc)] + [3nfcZ;1(m^ - ^M^G,)

z—' n, z—' n, n2

= 1 i=1 j i=1 -

+3mfcn' Z ^^ + 3n' ]T(-M1(Gfc)]

n, n,

,=1 ,=1

k-1 k-1 , x—\ mpmqmr , v-^ —,•—,•

+ [4nfcn' £ p q r + 12n'—k£ -i-^]

n^nq n^ n, n j

p,q,r=1! p=q=r i,j = 1

k-1 k

- [4n > —^ + 8mkn' > — ]

f ^ n • n • f ^ n •

After some calculations, we get:

RM2(® Gi) = [n Z nRM2(Gfe)] + [3 Z( m'nk + -n'

r ^ r) • tl 1 < J T) ■

= 1 i=1

nnkmi )M1(Gi)+3(mnk -2nmk )M1(Gk)]

n2 n2

+[4n Z mpmq — + 12n—. Z mim, ]

n^nq n^ n k ni n,

p,q,r = 1, p=q=r F 4 i,j = 1,i=j ^

- [4n y mm +8nmky — ]

n-n, nk f-f ni

i,,=1 J i=1

n ^^ RM2(Gi)+3 ( m - n- )M1(g,)

n n n2

2

ni ni ni2

i=1 i=1 j

k

—i—j

—p — q —r

+4n £ p q r - 4n £ ^ npnq n ni n j

p,q,r=1!p=q=r ^ y i,j=1 J

which is the desired result. □

k

If G1 = G2 = ....... = Gk = G then G, = Gk. Hence from the above theorem, the following corollary

i=1

follows:

Corollary 1. If G be connected graph, then

RM2(Gk) = k|V(G)|k-3[|V(G)|2RM2(G) + 3(k - 1)|V(G)||E(G)|M1(G) +4(k - 1)(k - 2)|E(G)|3 - 4(k - 1)|V(G)||E(G)|2].

3. Applications

Using various unary and binary graph operations on different elementary graphs, such as the path graph, cycle graph, complete graph etc, we can obtain several significant composite graphs having excellent usage in modern science and technology. The Cartesian product is one of that binary operations capable to construct different special structure. Using Cartesian product of two graphs, one can get ladder graph, C4-nanotube and nanotorus, rectangular grid, rook's graph, hamming graph etc. The Hamming graphs are interesting in connection with error-correcting codes and association schemes. The rook's graph represents all legal moves of rook on the chess board. So it is worth to investigate investigating topological indices for the above structures. In this section, the reduced second Zagreb index of aforesaid graphs is derived. For path, cycle, and complete graph of n vertices, the notations Pn, Cn, and Kn are used.

Example 1. The Ladder graph Ln (Fig. 1) is the Cartesian product of P2 and Pn+1, made by n sequences and (2n + 2) vertices. So, using theorem 1, the reduced second Zagreb index of Ln is given by

RM2(Ln) = 12n - 10.

Fig. 1. The ladder graph Ln

Example 2. The Cartesian product of Pn (n > 2) and Cm (m > 2) is a C4-nanotube TUC4(m, n), whose reduced second Zagreb index can be calculated using theorem 1 as follows:

RM2(Pn ( Cm) = 18mn - 25m.

Example 3. The Cartesian product of Cn (n > 3) and Cm (m > 3) is a C4-nanotorus TC4(m, n), whose reduced second Zagreb index is calculated from Theorem 1 as follows:

RM2(Cn ( Cm) = 18mn.

Example 4. The rectangular grid (Fig.2) is the Cartesian product of the Pn (n > 2) and Pm (m > 2). So, using Theorem 1, its reduced second Zagreb index is given by

RM2(Pn ( Pm) = 18mn - 25m - 25n + 28.

FIG. 2. The grid graph P5 ( P4

Example 5. The Cartesian product of Kn and Km yields the rook's graph (Fig. 3). So, using Theorem 1, its reduced second Zagreb index is given by

mn

RM2 (Km ( Kn) = — [(m + n)3 - 8m2 - 8n2 - 16mn + 21m + 21n - 14].

Fig. 3. The rook's graph K6 ( K6

N

Example 6. The graph G = Kn. is known as a Hamming graph and is denoted by H,

i=1

previous theorem to compute reduced second Zagreb index of a Hamming graph as follows:

N

RM2(H„1,„2,...,„N) = RM2(0 K„i)

ni

so, applying

1

i=1 NN

N

-nn,£(n, - 1)£(n, - 1) - 1]2.

2

i=1 i=1 i=1

If n1 = n2 =.......= nN = 2, then the graph G is known as a hypercube of dimension N and denoted by Qn (Fig. 4).

Thus from above, we directly get

RM2(Q„) = 1N2n(N - 1)2.

Fig. 4. Example of Hypercube

Example 7. The reduced second Zagreb index of the torus Cni < Cn2 <.......< Cnk is given by

RM2(C„i < C„2 <.......< C„fc) = k(2k - 1)2 n n,.

i=1

9pq

Example 8. Let T = T[p, q] be the molecular graph of a nanotorus (Fig. 5). Then |V(T)| = pq, |E(T)| = ,

M1(T) = 9pq. We consider a q-multi-walled nanotorus Gn = Pn < T. It is easy to find that RM2(Pn) = n - 3, RM2(T) = 6pq, and M1(Pn) = 4n - 6. Thus from theorem 1, we have the following result.

RM2(Gn) = 5pq(8n - 9).

We have plotted the result in Fig. 6

Fig. 5. The graph of a nanotorus

Fig. 6. Plotting of the RM2 index for Gn, n = 2,3. Cyan and green colors are used for n = 2, 3 respectively

4. Conclusion

In this article, reduced second Zagreb index of Cartesian product graph is studied. Firstly RM2 index is obtained for the product of two graphs and then the general case is considered. Applying that results, RM2 index is investigated for some special structures. As future work, some other graph operations like composition, tensor product, corona product, strong product, splice, link etc. can be discussed in terms of the reduced second Zagreb index.

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