- Research
- Open Access

# The F-coindex of some graph operations

- Nilanjan De
^{1}Email author, - Sk. Md. Abu Nayeem
^{2}and - Anita Pal
^{3}

**Received:**2 September 2015**Accepted:**16 February 2016**Published:**29 February 2016

## Abstract

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. In this paper, we introduce a new invariant which is named as F-coindex. Here, we study basic mathematical properties and the behavior of the newly introduced F-coindex under several graph operations such as union, join, Cartesian product, composition, tensor product, strong product, corona product, disjunction, symmetric difference of graphs and hence apply our results to find the F-coindex of different chemically interesting molecular graphs and nano-structures.

## Keywords

- Topological index
- Vertex degree
- First and second Zagreb indices
- F-index
- F-coindex
- Graph operations

## Mathematics Subject Classification

- Primary 05C35; Secondary 05C07
- 05C40

## Background

Topological indices are found to be very useful in chemistry, biochemistry and nanotechnology in isomer discrimination, structure–property relationship, structure-activity relationship and pharmaceutical drug design. Let *G* be a simple connected graph with vertex set *V*(*G*) and edge set *E*(*G*) respectively. Let, for any vertex \({v}\in V(G)\), \({{d}_{G}}(v)\) denotes its degree, that is the number of adjacent vertices of *v* in *G*. The complement of a graph *G* is denoted by \(\bar{G}\) and is the simple graph with the same vertex set *V*(*G*) and any two vertices \(uv\in E({\bar{G}})\) if and only if \(uv\notin E(G)\). Thus \(E(G)\cup E({\bar{G}})=E(K_n)\) and \(|E(\bar{G})|=\frac{|V(G)|(|V(G)|-1)}{2}-|E(G)|\). Also the degree of a vertex *v* in \(\bar{G}\) is given by \({{d}_{\bar{G}}}(v)=|V(G)|-1-{{d}_{G}}(v).\)

*G*is denoted by

*F*(

*G*) and is defined as the sum of cubes of the vertex degrees of the graph.

Very recently the present authors have studied the F-index of different graph operations in De et al. (2016).

*G*. Thus the Zagreb coindices of

*G*are defined as

Like Zagreb coindices, F-coindex of *G* is not the F-index of \(\bar{G}\). Here the sum runs over \(E(\bar{G})\), but the degrees are with respect to *G*.

## Motivation

According to the *International Academy of Mathematical Chemistry*, to identify whether any topological index is useful for prediction of chemical properties, the coorelation between the values of that topological index for different 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 (18) enough to make the statistical conclusion reliable. Furtula and Gutman (2015) showed that for octane isomers both \(M_1\) 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 \(({M_1}+{\lambda }F)\), where \(\lambda\) varies from −20 to 20.

*P*) and the corresponding F-coindex values of octane isomers. The dataset of octane isomers (first three columns of Table 1) are taken from www.moleculardescriptors.eu/dataset/dataset.htm and the last two columns of Table 1 are computed from the definitions of

*F*(

*G*) and \(\bar{F}(G)\). F-coindex values against \(\log P\) values are plotted in Fig. 1. Here we find that the correlation coefficient between \(\log P\) and \(\bar{F}\) is 0.966, whereas the correlation coefficient between \(\log P\) and \(M_1\) and that between \(\log P\) and

*F*are 0.077 and 0.065 respectively. Thus using this F-coindex, we can predict the \(\log P\) values with high accuracy.

Experimental values of the logarithm of the octanol–water partition coefficient and the corresponding values of different topological indices of octane isomers

Molecules | Log | \(M_1(G)\) |
| \(\bar{F}(G)\) |
---|---|---|---|---|

Octane | 3.67 | 26 | 50 | 132 |

2-Methyl-heptane | 3.61 | 28 | 62 | 134 |

3-Methyl-heptane | 3.61 | 28 | 62 | 134 |

4-Methyl-heptane | 3.61 | 28 | 62 | 134 |

3-Ethyl-hexane | 3.61 | 28 | 62 | 134 |

2,2-Dimethyl-hexane | 3.65 | 32 | 92 | 132 |

2,3-Dimethyl-hexane | 3.54 | 30 | 74 | 136 |

2,4-Dimethyl-hexane | 3.54 | 30 | 74 | 136 |

2,5-Dimethyl-hexane | 3.54 | 30 | 74 | 136 |

3,3-Dimethyl-hexane | 3.65 | 32 | 92 | 132 |

3,4-Dimethyl-hexane | 3.54 | 30 | 74 | 136 |

2-Methyl-3-ethyl-pentane | 3.54 | 30 | 74 | 136 |

3-Methyl-3-ethyl-pentane | 3.65 | 32 | 92 | 132 |

2,2,3-Trimethyl-pentane | 3.58 | 34 | 104 | 134 |

2,2,4-Trimethyl-pentane | 3.58 | 34 | 104 | 134 |

2,3,3-Trimethyl-pentane | 3.58 | 34 | 104 | 134 |

2,3,4-Trimethyl-pentane | 3.48 | 32 | 86 | 138 |

2,2,3,3-Tetramethyl-butane | 3.62 | 38 | 134 | 132 |

Graph operations play an important role in chemical graph theory. Different chemically important graphs can be obtained by applying graph operations on some general or particular graphs. For example, the linear polynomial chain (or the ladder graph \(L_n\)) is the molecular graph related to the polynomial structure obtained by the Cartesian product of \(P_2\) and \(P_{n+1}\). The \(C_4\) nanotube \(TUC_4(m, n)\) is the Cartesian product of \(P_n\) and \(P_m\) and the \(C_4\) nanotorus \(TC_4(m, n)\) is the Cartesian product of \(C_n\) and \(C_m\). For a given graph *G*, one of the hydrogen suppressed molecular graph is the bottleneck graph, which is the corona product of \(K_2\) and *G*. There are several studies on various topological indices under different graph operations available in the literature. Khalifeh et al. (2009) derived some exact formulae for computing first and second Zagreb indices under some graph operations. Das et al. (2013), derived some upper bounds for multiplicative Zagreb indices for different graph operations. Veylaki et al. (2015), computed third and hyper-Zagreb coindices of some graph operations. In De et al. (2014), the present authors computed some bounds and exact formulae of the connective eccentric index under different graph operations. Azari and Iranmanesh (2013) presented explicit formulas for computing the eccentric-distance sum of different graph operations. Interested readers are referred to Ashrafi et al. (2010), Khalifeh et al. (2008), Tavakoli et al. (2014), De et al. (2015a, b, c, d, Eskender and Vumar (2013) for other studies in this regard.

In this paper, we first derive some basic properties of F-coindex and hence present some exact expressions for the F-coindex of different graph operations such as union, join, Cartesian product, composition, tensor product, strong product, corona product, disjunction, symmetric difference of graphs. Also we apply our results to compute the F-coindex for some important classes of molecular graphs and nano-structures.

## Basic properties of F-coindex

*n*vertices can be easily obtained as follows.

- (i)
\(\bar{F}\left( {{K}_{n}}\right) =\bar{F}\left( {{\bar{K}}_{n}}\right) =0\),

- (ii)
\(\bar{F}\left( {{C}_{n}}\right) =4n(n-3)\),

- (iii)
\(\bar{F}\left( {{P}_{n}}\right) =4{{n}^{2}}-18n+20\),

- (iv)
\(\bar{F}({{K}_{m,n}})=mn(2mn-m-n).\)

Let for the graph *G* we use the notation \(|V(G)|=n\) and \({|{E}(G)|}=m\). Also let \({|{E}(\bar{G})|}=\bar{m}\). Now first we explore some basic properties of F-coindex.

###
**Proposition 1**

*Let*

*G*

*be a simple graph with*

*n*

*vertices and*

*m*

*edges, then*

###
*Proof*

###
**Proposition 2**

*Let*

*G*

*be a simple graph with*

*n*

*vertices and*

*m*

*edges, then*

###
*Proof*

An alternative expression for \(\bar{F}(G)\) can be obtained by considering sum over the edges of *G* and \(\bar{G}\) respectively as follows.

###
**Proposition 3**

*Let*

*G*

*be a simple graph with n*

*vertices and m edges, then*

###
*Proof*

###
**Proposition 4**

*Let G be a simple graph with*

*n*

*vertices and m edges, then*

###
*Proof*

## Main results

In the following, we study F-coindex of various graph operations like union, join, Cartesian product, composition, tensor product, strong product, corona product, disjunction, symmetric difference of graphs. These operations are binary and if not indicated otherwise, we use the notation \(V(G_i)\) for the vertex set, \({{E}(G_i)}\) for the edge set, \({{n}_{i}}\) for the number of vertices and \({{m}_{i}}\) for the number of edges of the graph \({{G}_{i}}\) respectively. Also let \({\bar{{m}}_{i}}\) denote the number of edges of the graph \({{\bar{G}}_{i}}.\)

### Union

The union of two graphs \({{G}_{1}}\) and \({{G}_{2}}\) is the graph denoted by \({{G}_{1}}\cup {{G}_{2}}\) with the vertex set \(V({{G}_{1}})\cup V({{G}_{2}})\) and edge set \(E({{G}_{1}})\cup E({{G}_{2}})\). In this case we assume that \(V({{G}_{1}})\) and \(V({{G}_{2}})\) are disjoint. The degree of a vertex *v* of \({{G}_{1}}\cup {{G}_{2}}\) is equal to that of the vertex in the component \({{G}_{i}}\,(i=1,2)\) which contains it. In the following preposition we calculate the F-coindex of \({{G}_{1}}\cup {{G}_{2}}.\)

###
**Proposition 5**

*Let G be a simple graph with*

*n*

*vertices and m edges, then*

###
*Proof*

### Join

###
**Proposition 6**

*Let G*

*be a simple graph with n*

*vertices and m edges, then*

###
*Proof*

###
*Example 1*

The complete bipartite graph \({{K}_{p,q}}\) can be defined as \({{K}_{p,q}}={{\bar{K}}_{p}}+{{\bar{K}}_{q}}\). So its F-coindex can be calculated from the previous proposition as \(\bar{F}({{K}_{p,q}})=pq(2pq-p-q).\)

The suspension of a graph *G* is defined as sum of *G* with a single vertex. So from the previous proposition the following corollary follows.

###
**Corollary 1**

*The F-coindex of suspension of G is given by*

###
*Example 2*

The star graph \({{S}_{n}}\) with *n* vertices is the suspension of empty graph \({{\bar{K}}_{n-1}}\). So its F-coindex can be calculated from the previous corollary as \(\bar{F}({{S}_{n}})=(n-1)(n-2).\)

###
*Example 3*

The wheel graph \({{W}_{n}}\) on \((n+1)\) vertices is the suspension of \({{C}_{n}}\). So from the previous corollary its F-coindex is given by \(\bar{F}({{W}_{n}})=9n(n-3).\)

###
*Example 4*

The fan graph \({{F}_{n}}\) on \((n+1)\) vertices is the suspension of \({{P}_{n}}\). So from the previous corollary its F-coindex is given by \(\bar{F}({{W}_{n}})=9{{n}^{2}}-37n+38.\)

We now extend the join operation to more than two graphs. Let \(G_1,\)
\(G_2,\ldots ,G_k\) be *k* graphs. Then, the degree of a vertex *v* in \({{G}_{1}}+{{G}_{2}}+\cdots +{{G}_{k}}\) is given by \({{d}_{{{G}_{1}}+{{G}_{2}}+\cdots +{G}_{k}}(v) ={{d}_{{{G}_{i}}}}(v)+{n}-{{n}_{i}}}\), where *v* is originally a vertex of the graph \(G_i\) and \(n={{n}_{1}}+{{n}_{2}}+\cdots +{{n}_{k}}\). Also let \({\bar{n}_i}=n-{n_i}\).

###
**Proposition 7**

*The F-coindex of*\({{G}_{1}}+{{G}_{2}}+\cdots +{{G}_{k}}\)

*is given by*

###
*Proof*

### Cartesian product

The Cartesian product of \(G_1\) and \(G_2\), denoted by \(G_1\times G_2\), is the graph with vertex set \(V(G_1)\times V(G_2)\) and any two vertices \(({{u}_{p}},{{v}_{r}})\) and \(({{u}_{q}},{{v}_{s}})\) are adjacent if and only if [\({{u}_{p}}={{u}_{q}}\in V(G_1)\) and \({{v}_{r}}{{v}_{s}}\in E(G_2)\)] or [\({{v}_{r}}={{v}_{s}}\in V(G_2)\) and \({{u}_{p}}{{u}_{q}}\in E(G_1)\)]. Thus we have, \({{d}_{{{G}_{1}}\times {{G}_{2}}}}(a,b)={{d}_{{{G}_{1}}}}(a)+{{d}_{{{G}_{2}}}}(b)\). In the following preposition we calculate the F-coindex of \({{G}_{1}}\times {{G}_{2}}\).

###
**Proposition 8**

*The F-coindex of*\({{G}_{1}}\times {{G}_{2}}\)

*is given by*

###
*Proof*

###
*Example 5*

###
*Example 6*

\(TUC_4(m, n)\) and \(TC_4(m,n)\) denote a \(C_4\) nanotube and nanotorus respectively. Then \(TUC_4(m, n)\cong {P_n}\times {C_m}\) and \(TC_4(m,n)\cong {C_n}\times {C_m},\) and so \(\bar{F}(TUC_4(m, n))=16{m^2}{n^2}-14{m^2}n-80mn+88m\) and \(\bar{F}(TC_4(m,n))=16{m}{n}(mn-5)\).

### Composition

The composition of two graphs \({{G}_{1}}\) and \({{G}_{2}}\) is denoted by \({{G}_{1}}[{{G}_{2}}]\) and any two vertices \(({{u}_{1}},{{u}_{2}})\) and \(({{v}_{1}},{{v}_{2}})\) are adjacent if and only if \({{u}_{1}}{{v}_{1}}\in E({{G}_{1}})\) or [\({{u}_{1}}={{v}_{1}}\) and \({{u}_{2}}{{v}_{2}}\in E({{G}_{2}})\)]. The vertex set of \({{G}_{1}}[{{G}_{2}}]\) is \(V({{G}_{1}})\times V({{G}_{2}})\) and the degree of a vertex (*a*, *b*) of \({{G}_{1}}[{{G}_{2}}]\) is given by \({{d}_{{{G}_{1}}[{{G}_{2}}]}}(a,b)={{n}_{2}}{{d}_{{{G}_{1}}}}(a)+{{d}_{{{G}_{2}}}}(b).\) In the following proposition we compute the F-coindex of the composition of two graphs.

###
**Proposition 9**

*The F-coindex of*\({{G}_{1}}[{{G}_{2}}]\) is given by

The proof of the above proposition follows from the expressions of first Zagreb index and F-index of strong product graphs given in Theorems 3 and 4 of Khalifeh et al. (2009) and De et al. respectively.

###
*Example 7*

- (i)
\(\bar{F}({{P}_{n}}[{{P}_{2}}])=100{{n}^{2}}-428n+456\),

- (ii)
\(\bar{F}({{C}_{n}}[{{P}_{2}}])=100{{n}^{2}}-300n.\)

### Tensor product

The tensor product of two graphs \({{G}_{1}}\) and \({{G}_{2}}\) is denoted by \({{G}_{1}}\otimes {{G}_{2}}\) and any two vertices \(({{u}_{1}},{{v}_{1}})\) and \(({{u}_{2}},{{v}_{2}})\) are adjacent if and only if \({{u}_{1}}{{u}_{2}}\in E({{G}_{1}})\) and \({{v}_{1}}{{v}_{2}}\in E({{G}_{2}})\). The degree of a vertex (*a*, *b*) of \({{G}_{1}}\otimes {{G}_{2}}\) is given by \({{d}_{{{G}_{1}}\otimes {{G}_{2}}}}(a,b)={{d}_{{{G}_{1}}}}(a){{d}_{{{G}_{2}}}}(b)\). In the following proposition, the F-coindex of the tensor product of two graphs is computed.

###
**Proposition 10**

*The F-coindex of*\({{G}_{1}}\otimes {{G}_{2}}\)

*is given by*

The proof follows from the expressions \({M_1}({{G}_{1}}\otimes {{G}_{2}})={{M}_{1}}({{G}_{1}}){{M}_{1}}({{G}_{2}})\) established in Theorem 2.1 of Yarahmadi (2011) and \(F({{G}_{1}}\otimes {{G}_{2}})=F({{G}_{1}})F({{G}_{2}})\) established in Theorem 7 of De et al.

###
*Example 8*

- (i)
\(\bar{F}({{P}_{n}}\otimes {{P}_{m}})=4(mn-1)(2n-3)(2m-3)-4(4n-7)(4m-7)\)

- (ii)
\(\bar{F}({{C}_{n}}\otimes {{C}_{m}})=16mn(mn-5)\)

- (iii)
\(\bar{F}({{K}_{n}}\otimes {{K}_{m}})=nm{{(n-1)}^{2}}{{(m-1)}^{2}}(m+n-1)\)

- (iv)
\(\bar{F}({{P}_{n}}\otimes {{C}_{m}})=4m(mn-1)(2n-3)(2m-3)-4(4n-7)(4m-7)\)

- (v)
\(\bar{F}({{P}_{n}}\otimes {{k}_{m}})=m(mn-1)(4n-6){{(m-1)}^{2}}-m(8n-14){(m-1)^3}\)

- (vi)
\(\bar{F}({{C}_{n}}\otimes {{K}_{m}})=4nm{{(m-1)}^{2}}(mn-2m+1)\).

### Strong product graphs

*a*,

*b*) of \({{G}_{1}}\boxtimes {{G}_{2}}\) is given by

###
**Proposition 11**

*The F-coindex of*\({{G}_{1}}\boxtimes {{G}_{2}}\)

*is given by*

The proof follows from the expressions of first Zagreb index and F-index of strong product graphs from Theorems 2.6 and 6 of Tavakoli et al. (2013) and De et al. respectively.

### Corona product

*i*th copy of \({{G}_{2}}\) to the

*i*th vertex of \({{G}_{1}}\), where \(1\le i\le {{n}_{1}}\). The corona product of \({{G}_{1}}\) and \({{G}_{2}}\) has total \(({{n}_{1}}{{n}_{2}}+{{n}_{1}})\) number of vertices and \(({{m}_{1}}+{{n}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}})\) number of edges. Different topological indices under the corona product of graphs have already been studied (Yarahmadi and Ashrafi 2012; De et al. 2015e; Pattabiraman and Kandan 2014). It is easy to see that the degree of a vertex

*v*of \({{G}_{1}}\circ {{G}_{2}}\) is given by

###
**Proposition 12**

*The F-coindex of*\({{G}_{1}}\circ {{G}_{2}}\)

*is given by*

###
*Example 9*

*G*, is the corona product of \({K}_{2}\) and

*G*, where

*G*is a given graph. F-coindex of bottleneck graph of

*G*is given by

*n*is the number of vertices of

*G*.

A *t*-thorny graph is obtained by joining *t*-number of thorns (pendent edges) to each vertex of a given graph *G*. A variety of topological indices of thorn graphs have been studied by a number of researchers (De 2012a, b; Alizadeh et al. 2014). It is well known that, the *t*-thorny graph of *G* is defined as the corona product of *G* and complement of complete graph with *t* vertices \(\bar{K_t}\). Thus from the previous theorem the following corollary follows.

###
**Corollary 2**

*The F-coindex of t-thorny graph of G is given by*

###
*Example 10*

*t*-thorny graph of \({{C}_{n}}\) and \({{P}_{n}}\) are given by

- (i)
\(\bar{F}\left( {{C}_{n}}^{t}\right) ={{n}^{2}}{{t}^{3}}-n{{t}^{3}}+6{{n}^{2}}{{t}^{2}}-7n{{t}^{2}}+9{{n}^{2}}t-18nt+4{{n}^{2}}-12n\)

- (ii)
\(\bar{F}\left( {{P}_{n}}^{t}\right) ={{n}^{2}}{{t}^{3}}-n{{t}^{3}}+6{{n}^{2}}{{t}^{2}}-11n{{t}^{2}}+9{{n}^{2}}t-28nt+4{{n}^{2}}+6{{t}^{2}}-18n+22t+20\).

### Disjunction

###
**Proposition 13**

*The F-coindex of*\({{G}_{1}}\wedge {{G}_{2}}\)

*is given by*

The proof of the above proposition follows from Proposition 3 with the relevant results from Khalifeh et al. (2009) and De et al.

### Symmetric difference

###
**Proposition 14**

*The F-coindex of*\({{G}_{1}}\oplus {{G}_{2}}\)

*is given by*

## Conclusion

In this paper, we have studied the F-coindex of different graph operations and also apply our results to find F-coindex of some special and chemically interesting graphs. However, there are still many other graph operations and special classes of graphs which are not covered here. So, for further research, F-coindex of various other graph operations and composite graphs can be considered.

## Declarations

### Authors’ contributions

All of the authors have significant contributions to this paper and the final form of this paper is approved by all of them. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Alizadeh Y, Iranmanesh A, Doslic T, Azari M (2014) The edge Wiener index of suspensions, bottlenecks, and thorny graphs. Glas Mat Ser III(49):1–12View ArticleGoogle Scholar
- Ashrafi AR, Doslic T, Hamzeh A (2010) The Zagreb coindices of graph operations. Discret Appl Math 158:1571–1578View ArticleGoogle Scholar
- Azari M, Iranmanesh A (2013) Computing the eccentric-distance sum for graph operations. Discret Appl Math 161(18):2827–2840View ArticleGoogle Scholar
- Das KC, Yurttas A, Togan M, Cevik AS, Cangul IN (2013)The multiplicative Zagreb indices of graph operations. J Inequal Appl. doi:10.1186/1029-242X-2013-90
- De N (2012a) On eccentric connectivity index and polynomial of thorn graph. Appl Math 3:931–934View ArticleGoogle Scholar
- De N (2012b) Augmented eccentric connectivity index of some thorn graphs. Int J Appl Math Res 1(4):671–680View ArticleGoogle Scholar
- De N, Pal A, Nayeem SMA (2014) On some bounds and exact formulae for connective eccentric indices of graphs under some graph operations. Int J Comb. doi:10.1155/2014/579257
- De N, Pal A, Nayeem SMA (2015a) Modified eccentric connectivity of generalized thorn graphs. Int J Comput Math. doi:10.1155/2014/436140
- De N, Pal A, Nayeem SMA (2015b) The irregularity of some composite graphs. Int J Appl Comput Math. doi:10.1007/s40819-015-0069-z
- De N, Nayeem SMA, Pal A (2015c) Reformulated First Zagreb Index of Some Graph Operations. Mathematics 3(4):945–960. doi:10.3390/math3040945 View ArticleGoogle Scholar
- De N, Nayeem SMA, Pal A (2015d) Total eccentricity index of the generalized hierarchical product of graphs. Int J Appl Comput Math. doi:10.1007/s40819-014-0016-4
- De N, Nayeem SMA, Pal A (2015e) Modified eccentric connectivity index of corona product of graphs. Int J Comput Appl 132(9):1–5. doi:10.5120/ijca2015907536 Google Scholar
- De N, Nayeem SMA, Pal A (2016) F-index of some graph operations. Discrete Math Algorithm Appl. doi:10.1142/S1793830916500257 Google Scholar
- Doslic T (2008) Vertex-weighted Wiener polynomials for composite graphs. Ars Math Contemp 1:66–80Google Scholar
- Eskender B, Vumar E (2013) Eccentric connectivity index and eccentric distance sum of some graph operations. Trans Comb 2(1):103–111Google Scholar
- Furtula B, Gutman I (2015) A forgotten topological index. J Math Chem 53(4):1184–1190View ArticleGoogle Scholar
- Gutman I, Trinajstić N (1972) Graph theory and molecular orbitals total \(\pi\)-electron energy of alternant hydrocarbons. Chem Phys Lett 17:535–538View ArticleGoogle Scholar
- Khalifeh MH, Yousefi-Azari H, Ashrafi AR (2008) The hyper-Wiener index of graph operations. Comput Math Appl 56:1402–1407View ArticleGoogle Scholar
- Khalifeh MH, Yousefi-Azari H, Ashrafi AR (2009) The first and second Zagreb indices of some graph operations. Discret Appl Math 157(4):804–811View ArticleGoogle Scholar
- Pattabiraman K, Kandan P (2014) Weighted PI index of corona product of graphs. Discret Math Algorithms Appl. doi:10.1142/S1793830914500554
- Tavakoli M, Rahbarnia F, Ashrafi AR (2013) Note on strong product of graphs. Kragujev J Math 37(1):187–193Google Scholar
- Tavakoli M, Rahbarnia F, Ashrafi AR (2014) Some new results on irregularity of graphs. J Appl Math Inform 32:675–685View ArticleGoogle Scholar
- Veylaki M, Nikmehr MJ, Tavallaee HA (2015) The third and hyper-Zagreb coindices of some graph operations. J Appl Math Comput. doi:10.1007/s12190-015-0872-z
- Yarahmadi Z (2011) Computing some topological indices of tensor product of graphs. Iran J Math Chem 2(1):109–118Google Scholar
- Yarahmadi Z, Ashrafi AR (2012) The Szeged, vertex PI, first and second Zagreb indices of corona product of graphs. Filomat 26(3):467–472View ArticleGoogle Scholar