1 Introduction and Preliminaries
Let G be a simple graph with vertex set V (G ) and edge set E (G ). We denote p = |V (G )| and q = |E (G )|, order and size of graph G respectively. The degree dν of a vertex ν in graph G is defined as the number of first neighbors of vertex ν in G . The concept of degree in graph theory is closely related to the concept of valence in chemistry. The complement of a graph G , represented through G ¯ $\begin{array}{}
\displaystyle
\overline{G}
\end{array}$ V (G ) wherein two vertices u and ν are joined by an edge uν , if and only if they are not adjacent in G . Obviously, E ( G ) ∪ E ( G ¯ ) = E ( K p ) $\begin{array}{}
\displaystyle
E(G) \cup E(\overline{G})= E(K_{p})
\end{array}$ Kp is complete graph of order p , | E ( G ¯ ) | = p ( p − 1 ) 2 − q $\begin{array}{}
\displaystyle
|E(\overline{G})| = \frac{p(p-1)}{2} -q
\end{array}$ G is alternative graph L (G ) that denotes the adjacencies among the edges of G . Line graphs are very useful in mathematical chemistry, but in recent years they were considered very little in chemical graph theory. For further facts about the applications of line graphs in chemistry, we mention the articles [6 , 25 , 32 ].
Topological indices are the arithmetical numbers which depends upon the construction of any simple graph. Topological indices are generally classified into three kinds: degree-based indices (see [2 , 5 , 7 , 8 , 28 , 29 ]), distance-based indices (see [3 , 16 ]), and spectrum-based indices (see [11 , 27 ]). There are also certain topological indices depends upon both degrees and distances (see [9 , 27 ]). The advantage of topological indices is in that they may be used directly as simple numerical descriptors in a comparison with, chemical, physical or biological parameters of molecules in Quantitative Structure Property Relationships (QSPR) and in Quantitative Structure Activity Relationships (QSAR). Wiener index is the oldest topological index and its mathematical properties and chemical applications have been extensively studied.
The Wiener index of graph G is defined as follows:
W ( G ) = 1 2 ∑ ( u , v ) d ( u , v ) . $$\begin{array}{}
\displaystyle
W(G) &=& \frac{1}{2} \sum \limits_{(u,v)} d(u,v).
\end{array}$$
where (u, ν ) is any ordered pair of vertices in G and d (u, ν ) is u − ν geodesic.
The Zagreb indices were first introduced by Gutman in [24 ], they are important molecular descriptors and have been closely correlated with many chemical properties (see [35 ]) and defined as:
M 1 ( G ) = ∑ u ∈ V ( G ) d u 2 and $$\begin{array}{}
\displaystyle
{{M_1}(G) = \mathop \sum \limits_{u \in V(G)} d_u^2}~~{{\rm{and}}}
\end{array}$$ (1)
M 2 ( G ) = ∑ u v ∈ E ( G ) d u d v . $$\begin{array}{}
\displaystyle
M_2(G) &=& \sum\limits_{uv \in E(G)} d_{u}d_{v}.\label{Z2}
\end{array}$$ (2)
In fact, one can rewrite the first Zagreb index as
M 1 ( G ) = ∑ u v ∈ E ( G ) [ d u + d v ] . $$\begin{array}{}
\displaystyle
M_1(G) &=& \sum\limits_{uv \in E(G)} \big[ d_u +d_v \big].
\end{array}$$
The third Zagreb index, introduced by Fath-Tabar in [13 ]. This index is defined as follows:
M 3 ( G ) = ∑ u v ∈ E ( G ) | d u − d v | . $$\begin{array}{}
\displaystyle
M_3(G) &=& \sum\limits_{uv \in E(G)} |d_u- d_v|.
\end{array}$$ (3)
The hyper-Zagreb index was first introduced in [34 ]. This index is defined as follows:
H M ( G ) = ∑ u v ∈ E ( G ) ( d u + d v ) 2 . $$\begin{array}{}
\displaystyle
HM(G) &=& \sum\limits_{uv \in E(G)} (d_u+d_v)^{2}.
\end{array}$$ (4)
In 2008, Doslic put forward the first Zagreb coindex, defined as (see [1 ]):
M 1 ¯ = M 1 ¯ ( G ) ∑ u v ∉ E ( G ) [ d u + d v ] . $$\begin{array}{}
\displaystyle
\overline{M_1}=\overline{M_{1}}(G) = \sum\limits_{uv \not\in E(G)} \big[
d_u +d_v \big].
\end{array}$$
The second Zagreb coindex is defined as (see [1 ]):
M 2 ¯ = M 2 ¯ ( G ) = ∑ u v ∉ E ( G ) d u d v . $$\begin{array}{}
\displaystyle
\overline{M_2}= \overline{M_{2}}(G) &=& \sum\limits_{uv \not\in E(G)}
d_{u}d_{v}.
\end{array}$$
The degree distance index for graphs developed by Dobrynin and Kochetova in [10 ] and Gutman in [17 ] as
a weighted version of the Wiener index. The degree distance of G , denoted by DD (G ), is defined as follows:
D D ( G ) = ∑ { u , v } ⊆ V ( G ) d ( u , v ) [ d u + d v ] . $$\begin{array}{}
\displaystyle
DD\left( G \right) = \sum\limits_{\{ u,v\} \subseteq V\left( G \right)} {d\left( {u,v} \right)} [{d_u} + {d_v}].
\end{array}$$
Randić index introduced by Milan Randić in 1975 (see [30 ]). This index is defined as follows:
R ( G ) = ∑ u v ∈ E ( G ) 1 d u d v . $$\begin{array}{}
\displaystyle
R(G)=\sum\limits_{uv \in E(G)} \frac{1}{\sqrt{d_ud_v}}.
\end{array}$$ (5)
Later, this index was generalized by Bollobás and Erdös (see [4 ]) to the following form for any real number α , and named the general Randić index:
R ( G ) = ∑ u v ∈ E ( G ) [ d u d v ] α . $$\begin{array}{}
\displaystyle
R(G)=\sum\limits_{uv \in E(G)} [{d_ud_v}]^\alpha .
\end{array}$$
The Atom-Bond Connectivity index (ABC), introduced by Estrda et al. in [12 ] which has been applied up until now to study the stability of alkanes and the strain energy of cycloalkanes. The ABC index of G is defined as:
A B C ( G ) = ∑ u v ∈ E ( G ) d u + d v − 2 d u d v . $$\begin{array}{}
\displaystyle
ABC(G)=\sum\limits_{uv \in E(G)} \sqrt{\frac{d_u+d_v-2}{d_ud_v}}.
\end{array}$$ (6)
In 2010, the general sum-connectivity index χ (G ) has been introduced in [37 ]. This index is defined as follows:
χ ( G ) = ∑ u v ∈ E ( G ) 1 d u + d v . $$\begin{array}{}
\displaystyle
\chi(G)=\sum\limits_{uv \in E(G)} \frac{1}{\sqrt{d_u+d_v}}.
\end{array}$$ (7)
D. Vukicevic and B. Furtula introduced the geometric arithmetic (GA) index in [36 ]. The GA index for G is defined by
G A ( G ) = ∑ u v ∈ E ( G ) 2 d u d v d u + d v . $$\begin{array}{}
\displaystyle
GA(G)=\sum\limits_{uv \in E(G)} \frac{2\sqrt{d_ud_v}}{d_u+d_v}.
\end{array}$$ (8)
Inspired by the work on the ABC index, Furtula et al. proposed the following modified version of the ABC index and called it as Zagreb index (AZI) in [14 ]. This index is defined as follows:
A Z I ( G ) = ∑ u v ∈ E ( G ) ( d u d v d u + d v − 2 ) 3 . $$\begin{array}{}
\displaystyle
AZI(G)=\sum\limits_{uv \in E(G)} (\frac{d_ud_v}{d_u+d_v-2})^3.
\end{array}$$ (9)
The fourth member of the class of ABC index was introduced by M. Ghorbani et al. in [20 , 22 , 23 ] as:
A B C 4 ( G ) = ∑ u v ∈ E ( G ) S u + S v − 2 S u S v . $$\begin{array}{}
\displaystyle
ABC_4(G)=\sum\limits_{uv \in E(G)} \sqrt{\frac{S_u+S_v-2}{S_uS_v}}.
\end{array}$$ (10)
where Su is the summation of degrees of all neighbors of vertex u in G . In other words, Su = Σuν∈E (G )dν . Similarly for Sν .
The 5th GA index was introduced by Graovac et al. in [21 ] as:
G A 5 ( G ) = ∑ u v ∈ E ( G ) 2 S u S v S u + S v . $$\begin{array}{}
\displaystyle
GA_5(G)=\sum\limits_{uv \in E(G)} \frac{2\sqrt{S_uS_v}}{S_u+S_v}.
\end{array}$$ (11)
The Sanskruti index S (G ) of a graph G is defined in [33 ] as follows:
S ( G ) = ∑ u v ∈ E ( G ) ( S u S v S u + S v − 2 ) 3 . $$\begin{array}{}
\displaystyle
S(G)=\sum\limits_{uv \in E(G)} (\frac{S_uS_v}{S_u+S_v-2})^3.
\end{array}$$ (12)
Theorem 1
[17 ] Let G be a graph of order p and size q. Then
M 1 ( G ¯ ) = M 1 ( G ) + p ( p − 1 ) 2 − 4 q ( p − 1 ) ; $$\begin{array}{}
\displaystyle
M_{1}(\overline{G}) &=& M_{1}(G)+ p(p-1)^{2}- 4q(p-1);\\
\end{array}$$ (13)
M 1 ¯ ( G ) = 2 q ( p − 1 ) − M 1 ( G ) ; $$\begin{array}{}
\displaystyle
\overline{M_{1}}(G) &=& 2q(p-1)- M_{1}(G); \\
\end{array}$$ (14)
M 1 ¯ ( G ¯ ) = 2 q ( p − 1 ) − M 1 ( G ) . $$\begin{array}{}
\displaystyle
\overline{M_{1}}(\overline{G}) &=& 2q(p-1)- M_{1}(G).
\end{array}$$ (15)
Theorem 2
[19 ] Let G be a graph of order p and size q. Then
M 2 ( G ¯ ) = 1 2 p ( p − 1 ) 3 − 3 q ( p − 1 ) 2 + 2 q 2 + 2 p − 3 2 M 1 ( G ) − M 2 ( G ) ; $$\begin{array}{}
\displaystyle
M_{2}(\overline{G}) &=& \frac{1}{2}p(p-1)^{3}- 3q(p-1)^{2} + 2q^{2} +\frac{2p-3}{2}M_{1}(G) -M_{2}(G);\\
\end{array}$$ (16)
M 2 ¯ ( G ) = 2 q 2 − 1 2 M 1 ( G ) − M 2 ( G ) ; $$\begin{array}{}
\displaystyle
\overline{M_{2}}(G) &=& 2q^{2}-\frac{1}{2} M_{1}(G) - M_{2}(G); \\
\end{array}$$ (17)
M 2 ¯ ( G ¯ ) = q ( p − 1 ) 2 − ( p − 1 ) M 1 ( G ) + M 2 ( G ) . $$\begin{array}{}
\displaystyle
\overline{M_{2}}(\overline{G}) = q(p-1)^{2}- (p-1)M_{1}(G)
+M_{2}(G).
\end{array}$$ (18)
Theorem 3
[26 ] Let G be a graph of order p and size q. Then
M 1 ¯ ( G ) ≥ 2 W ( G ) − 2 M 1 ( G ) + 6 q ( p − 1 ) − p 3 + p 2 . $$\begin{array}{}
\displaystyle
\overline{M_{1}}(G) &\geq& 2W(G) -2M_{1}(G) +6q(p-1) -p^{3} +p^{2}.
\end{array}$$ (19)
Theorem 4
[26 ] Let G be a nontrivial graph of diameter d ≥ 2. Then
M 1 ¯ ( G ) ≤ D D ( G ) − M 1 ( G ) 2 $$\begin{array}{}
\displaystyle
\overline{M_{1}}(G) &\leq& \frac{DD(G) - M_{1}(G)}{2}
\end{array}$$ (20)
with equality if and only if d = 2.
The following lemma is helpful for computing the degree of a vertex of line graph.
Lemma 5
Let G be a graph with u, ν ∈ V (G ) and e = uν ∈ E (G ). Then:
d e = d u + d v − 2. $$\begin{array}{}
\displaystyle
d_e=d_u+d_v-2.
\end{array}$$
Lemma 6
[18 ] Let G be a graph of order p and size q, then the line graph L (G ) of G is a graph of order p and size 1 2 M 1 ( G ) − q $\begin{array}{}
\displaystyle
\frac{1}{2}M_{1}(G) -q
\end{array}$
In this paper we discuss the topological indices and co-indices of the line graphs of Banana tree graph and Firecracker graph and their complement graphs.
2 Topological indices of line graph of Banana tree graph
In this section we computed the topological indices of the line graph of Banana tree graph. The Banana tree graph Bn,k is the graph obtained by connecting one leaf of each of n copies of an k −star graph with a single root vertex that is distinct from all the stars. The Bn,k has order nk + 1 and size nk . B 3,5 is shown in the Fig. 1 .
Theorem 7
Let G be the line graph of the Banana tree graph Bn,k . Then
M 1 (G ) = nk 3 − 5nl 2 + 10nk + n 3 − 7n ;
M 2 ( G ) = 1 2 n k 4 − 7 2 n k 3 + n 2 k + 10 n k 2 − 14 n k + 1 2 n 4 − 1 2 n 3 − n 2 + 8 n ; $\begin{array}{}
\displaystyle
{M_2}(G) = \frac{1}{2}n{k^4} - \frac{7}{2}n{k^3} + {n^2}k + 10n{k^2} - 14nk + \frac{1}{2}{n^4} - \frac{1}{2}{n^3} - {n^2} + 8n;
\end{array}$ Fig. 1 The Banana tree graph B 3,5 .
Fig. 2 The line graph of Banana tree graph B 3,5 .
M 3 (G ) = −n 2 + 2nk − 3n ;
HM (G ) = 2n 4 + 2nk 4 − 14nk 3 + 2n 2 k + 41nk 2 − 57nk − n 3 − 2n 2 + 31n ;
R ( G ) = 1 2 n 2 − n n 2 + n ( k − 1 ) n + k n − 2 n ( k − 1 ) ( k − 2 + 1 2 n k 2 + 6 n − 5 n k ( k − 2 ) 2 ; $\begin{array}{}
\displaystyle
R(G) = \frac{1}{2}\frac{{{n^2} - n}}{{\sqrt {{n^2}} }} + \frac{n}{{\sqrt {(k - 1)n} }} + \frac{{kn - 2n}}{{\sqrt {(k - 1)(k - 2} }} + \frac{1}{2}\frac{{n{k^2} + 6n - 5nk}}{{\sqrt {{{(k - 2)}^2}} }};
\end{array}$
A B C ( G ) = 1 2 n 3 + 2 3 n + 1 6 n 15 + 1 4 n 6 ; $\begin{array}{}
\displaystyle
ABC(G) = \frac{1}{2}n\sqrt 3 + \frac{2}{3}n + \frac{1}{6}n\sqrt {15} + \frac{1}{4}n\sqrt 6 ;
\end{array}$
G A ( G ) = 2 n ( k − 1 ) n k − 1 + n + 2 n ( k − 2 ) ( k − 1 ) ( k − 2 ) 2 k − 3 + 1 2 ( n 2 + n k 2 − 5 n k + 5 n ) ; $\begin{array}{}
\displaystyle
GA(G) = \frac{{2n\sqrt {(k - 1)n} }}{{k - 1 + n}} + \frac{{2n(k - 2)\sqrt {(k - 1)(k - 2)} }}{{2k - 3}} + \frac{1}{2}({n^2} + n{k^2} - 5nk + 5n);
\end{array}$
χ ( G ) = ( n 2 − n ) 2 4 n + n k − 1 + n + k n − 2 n 2 k − 3 + n k 2 + 6 n − 5 n k 2 k − 4 ; $\begin{array}{}
\displaystyle
\chi (G) = \frac{{({n^2} - n)\sqrt 2 }}{{4\sqrt n }} + \frac{n}{{\sqrt {k - 1 + n} }} + \frac{{kn - 2n}}{{2k - 3}} + \frac{{n{k^2} + 6n - 5nk}}{{\sqrt {2k - 4} }};
\end{array}$
A Z I ( G ) = ( n 2 − n ) n 6 ( 2 n − 2 ) 3 + n 4 ( k − 1 ) 3 ( k − 3 + n ) 3 + ( k n − 2 n ) ( k − 1 ) 3 ( k − 2 ) 3 ( 2 k − 5 ) 3 + ( n k 2 + 6 n − 5 n k ) ( k − 2 ) 6 ( 2 k − 6 ) 3 . $\begin{array}{}
\displaystyle
AZI(G) = \frac{{({n^2} - n){n^6}}}{{{{(2n - 2)}^3}}} + \frac{{{n^4}{{(k - 1)}^3}}}{{{{(k - 3 + n)}^3}}} + \frac{{(kn - 2n){{(k - 1)}^3}{{(k - 2)}^3}}}{{{{(2k - 5)}^3}}} + \frac{{(n{k^2} + 6n - 5nk){{(k - 2)}^6}}}{{{{(2k - 6)}^3}}}.
\end{array}$
Proof. The graph G for n = 3 and k = 5 is shown in Fig. 2 . By using Lemma 5 , it is easy to see that the order of G is nk out of which (k − 2)n vertices are of degree k − 2, n vertices are of degree k − 1 and n vertices are of degree n . Therefore by using Lemma 6 , G has size n 2 + 3 n + n k 2 − 3 n k 2 $\begin{array}{}
\displaystyle
\frac{{{n^2} + 3n + n{k^2} - 3nk}}{2}
\end{array}$ G into edges of the type E (du,dν ) where uν is an edge. In G , we get edges of the type E (n,n ) , E (k −1,n ) , E (k −1,k −2) and E (k −2,k −2) . The number of edges of these types are given in the Table 1 . By using Formulas (1 )-(9 ) and Table 1 , we can obtain the required results.
Table 1 The size partition of G .
(du ,dν ) where uν ∈ E (G ) (n,n ) (k − 1,n ) (k − 1,k − 2) (k − 2,k − 2) Number of edges n ( n − 1 ) 2 $\begin{array}{}
\displaystyle
\frac{{n(n - 1)}}{2}
\end{array}$ n (k − 2)n n k 2 + 6 n − 5 n k 2 $\begin{array}{}
\displaystyle
\frac{{n{k^2} + 6n - 5nk}}{2}
\end{array}$
Theorem 8
Let G be the Web graph Wn . Then
M 1 ( G ¯ ) = − 2 n 3 k − 6 n 2 k + n k 2 + 5 n k + n 3 + 2 n 2 − n ; $\begin{array}{}
\displaystyle
{M_1}(\bar G) = - 2{n^3}k - 6{n^2}k + n{k^2} + 5nk + {n^3} + 2{n^2} - n;
\end{array}$
M 1 ¯ ( G ) = M 1 ¯ ( G ¯ ) = − n k 3 + 3 n k 2 − 7 n k − n 3 + n 2 + 10 n ; $\begin{array}{}
\displaystyle
\overline {{M_1}} (G) = \overline {{M_1}} (\bar G) = - n{k^3} + 3n{k^2} - 7nk - {n^3} + {n^2} + 10n;
\end{array}$
M 2 ( G ¯ ) = n 4 k − 1 2 n 3 k 2 − 8 n 2 k − 3 2 n 2 k 2 + 3 n k 2 + 3 n k + 2 n 3 + 4 n 2 − 2 n ; $\begin{array}{}
\displaystyle
{M_2}(\bar G) = {n^4}k - \frac{1}{2}{n^3}{k^2} - 8{n^2}k - \frac{3}{2}{n^2}{k^2} + 3n{k^2} + 3nk + 2{n^3} + 4{n^2} - 2n;
\end{array}$
M 2 ¯ ( G ) = − n k 3 + 3 n k 2 − 7 n k − n 3 + n 2 + 10 n ; $\begin{array}{}
\displaystyle
\overline {{M_2}} (G) = - n{k^3} + 3n{k^2} - 7nk - {n^3} + {n^2} + 10n;
\end{array}$ Fig. 3 The Firecracker graph F 4,7 .
Fig. 4 The line graph of Firecracker graph F 4,7 .
M 2 ¯ ( G ¯ ) = − n 4 k + 1 2 n 3 k 2 − n 3 k + 3 2 n 2 k 2 + 5 n 2 k − 3 2 n k 2 − 11 2 n k + 1 2 n 4 + 1 2 n 3 − 1 2 n 2 + 5 2 n ; $\begin{array}{}
\displaystyle
\overline {{M_2}} (\bar G) = - {n^4}k + \frac{1}{2}{n^3}{k^2} - {n^3}k + \frac{3}{2}{n^2}{k^2} + 5{n^2}k - \frac{3}{2}n{k^2} - \frac{{11}}{2}nk + \frac{1}{2}{n^4} + \frac{1}{2}{n^3} - \frac{1}{2}{n^2} + \frac{5}{2}n;
\end{array}$
W ( G ) ≤ ( 1 − 3 k ) n 3 + ( − 3 k 3 + 6 k 2 − 9 k ) n 2 + ( 4 k 2 + 4 k + 12 ) n + ( n k − 1 ) 3 − ( n k − 1 ) 2 2 ; $\begin{array}{}
\displaystyle
W(G) \le \frac{{(1 - 3k){n^3} + ( - 3{k^3} + 6{k^2} - 9k){n^2} + (4{k^2} + 4k + 12)n + {{(nk - 1)}^3} - {{(nk - 1)}^2}}}{2};
\end{array}$
DD (G ) ≥ −n 3 + 2(−3k 2 + 1)n 2 + (−k 3 + 7k 2 − 4k + 13)n .
Table 2 The size partition of G .
(Su ,Sv ) where uv ∈ E (G ) (n 2 − n + k − 1,n 2 − n + k − 1) (k 2 − 4k + n + 4,n 2 − n + k − 1) Number of edges n ( n − 1 ) 2 $\begin{array}{}
\displaystyle
\frac{{n(n - 1)}}{2}
\end{array}$ n (Su ,Sv ) where uv ∈ E (G ) (k 2 − 4k + n + 4,k 2 − 4k + 5) (k 2 − 4k + 5,k 2 − 4k + 5) Number of edges (k − 2)n n k 2 + 6 n − 5 n k 2 $\begin{array}{}
\displaystyle
\frac{{n{k^2} + 6n - 5nk}}{2}
\end{array}$
Proof. By using Theorems (1 )-(4 ) and (7 ), we can obtain the required results.
Theorem 9
Let G be the Banana tree graph Bn,k . Then
A B C 4 ( G ) = 1 2 ( n 2 − n ) 2 n 2 + 2 k − 2 n − 4 ( n 2 + k − n − 1 ) 2 + n k 2 + n 2 − 3 k + 1 ( k 2 − 4 k + n + 4 ) ( n 2 + k − n − 1 ) + ( k n − 2 n ) 2 k 2 − 8 k + n + 7 ( k 2 − 4 k + n + 4 ) ( k 2 − 4 k + 5 ) + 1 2 ( n k 2 + 6 n − 5 n k ) 2 k 2 − 8 k + 8 ( k 2 − 4 k + 5 ) 2 ; $\begin{array}{}
\displaystyle
AB{C_4}(G) = \frac{1}{2}({n^2} - n)\sqrt {\frac{{2{n^2} + 2k - 2n - 4}}{{{{({n^2} + k - n - 1)}^2}}}} + n\sqrt {\frac{{{k^2} + {n^2} - 3k + 1}}{{({k^2} - 4k + n + 4)({n^2} + k - n - 1)}}} + (kn - 2n)\sqrt {\frac{{2{k^2} - 8k + n + 7}}{{({k^2} - 4k + n + 4)({k^2} - 4k + 5)}}} + \frac{1}{2}(n{k^2} + 6n - 5nk)\sqrt {\frac{{2{k^2} - 8k + 8}}{{{{({k^2} - 4k + 5)}^2}}}} ;
\end{array}$
G A 5 ( G ) = ( n 2 − n ) ( n 2 + k − n − 1 ) 2 2 n 2 + 2 k − 2 n − 2 + 2 n ( k 2 − 4 k + n + 4 ) k 2 + n 2 − 3 k + 3 + 2 ( k n − 2 n ) ( k 2 − 4 k + n + 4 ) ( k 2 − 4 k + 5 ) 2 k 2 − 8 k + n + 9 + ( n k 2 + 6 n − 5 n k ) ( k 2 − 4 k + 5 ) 2 2 k 2 − 8 k + 10 ; $\begin{array}{}
\displaystyle
G{A_5}(G) = ({n^2} - n)\frac{{\sqrt {{{({n^2} + k - n - 1)}^2}} }}{{2{n^2} + 2k - 2n - 2}} + 2n\frac{{\sqrt {({k^2} - 4k + n + 4)} }}{{{k^2} + {n^2} - 3k + 3}} + \frac{{2(kn - 2n)\sqrt {({k^2} - 4k + n + 4)({k^2} - 4k + 5)} }}{{2{k^2} - 8k + n + 9}} + \frac{{(n{k^2} + 6n - 5nk)\sqrt {{{({k^2} - 4k + 5)}^2}} }}{{2{k^2} - 8k + 10}};
\end{array}$
S ( G ) = ( n 2 − n ) ( n 2 + k − n − 1 ) 6 ( 2 n 2 + 2 k − 2 n − 4 ) 3 + n ( k 2 − 4 k + n + 4 ) 3 ( n 2 + k − n − 1 ) 3 ( k 2 + n 2 − 3 k + 1 ) 3 + ( k n − 2 n ) ( k 2 − 4 k + n + 4 ) 3 ( k 2 − 4 k + 5 ) 3 ( 2 k 2 − 8 k + n + 7 ) 3 + ( n k 2 + 6 n − 5 n k ) ( k 2 − 4 k + 5 ) 6 2 ( 2 k 2 − 8 k + 8 ) 3 . $\begin{array}{}
\displaystyle
S(G) = \frac{{({n^2} - n){{({n^2} + k - n - 1)}^6}}}{{{{(2{n^2} + 2k - 2n - 4)}^3}}} + \frac{{n{{({k^2} - 4k + n + 4)}^3}{{({n^2} + k - n - 1)}^3}}}{{{{({k^2} + {n^2} - 3k + 1)}^3}}} + \frac{{(kn - 2n){{({k^2} - 4k + n + 4)}^3}{{({k^2} - 4k + 5)}^3}}}{{{{(2{k^2} - 8k + n + 7)}^3}}} + \frac{{(n{k^2} + 6n - 5nk){{({k^2} - 4k + 5)}^6}}}{{2{{(2{k^2} - 8k + 8)}^3}}}.
\end{array}$
Proof.
We partition the size of G into edges of the type (Su ,Sv ) where uv ∈ E (G ) as shown in Table 2 . Hence we get the required results by using Table 2 and Formulas (10 )-(12 ).
3 Topological indices of line graph of Firecracker graph
In this section we computed the topological indices of the line graph of Firecracker graph. The Firecracker graph Fn,k is the graph obtained by the concatenation of nk −stars by linking one leaf from each. The Fn,k has order nk and size nk − 1. F 4,7 is shown in the Fig. 3 .
Theorem 10
Let G be the line graph of the Firecracker graph Fn,k . Then
M 1 (G ) = nk 3 − 5nk 2 + 12nk − 4k + 8n − 28;
M 2 ( G ) = 1 2 n k 4 − 7 2 n k 3 + 11 n k 2 − 10 n k − 2 k 2 + 28 n − 4 k − 54 $\begin{array}{}
\displaystyle
{M_2}(G) = \frac{1}{2}n{k^4} - \frac{7}{2}n{k^3} + 11n{k^2} - 10nk - 2{k^2} + 28n - 4k - 54;
\end{array}$
M 3 (G ) = 4nk − 4k − 12n + 16;
HM (G ) = 2nk 4 + 4nk 2 − 14nk 3 − 52nk − 10k 2 − 2k + 136n − 248;
R ( G ) = 1 3 3 + 2 3 3 k + 2 3 ( k − 1 ) + 1 4 n − 1 + n − 3 k + 2 k − 2 k − 1 + n k 2 + 6 n − 5 n k k − 2 + ( n − 2 ) k − 2 k $\begin{array}{}
\displaystyle
R(G) = \frac{1}{3}\sqrt 3 + \frac{2}{3}\sqrt {\frac{3}{k}} + \frac{2}{{\sqrt {3(k - 1)} }} + \frac{1}{4}n - 1 + \frac{{n - 3}}{{\sqrt k }} + 2\sqrt {\frac{{k - 2}}{{k - 1}}} + \frac{{n{k^2} + 6n - 5nk}}{{k - 2}} + (n - 2)\sqrt {\frac{{k - 2}}{k}} ;
\end{array}$
A B C ( G ) = 1 3 15 + 2 3 3 k + 1 k + 2 3 3 k k − 1 + 1 4 ( n − 4 ) 6 + 1 2 ( 2 n − 6 ) 2 + k k + ( 2 k − 4 ) 2 k − 5 ( k − 2 ) ( k − 1 ) + 1 2 ( n k 2 + 6 n − 5 n k ) 2 k − 6 ( k − 2 ) 2 + ( n k − 2 n − 2 k + 4 ) 2 k − 4 ( k − 2 ) k $\begin{array}{}
\displaystyle
ABC(G) = \frac{1}{3}\sqrt {15} + \frac{2}{3}\sqrt 3 \sqrt {\frac{{k + 1}}{k}} + \frac{2}{3}\sqrt 3 \sqrt {\frac{k}{{k - 1}}} + \frac{1}{4}(n - 4)\sqrt 6 + \frac{1}{2}(2n - 6)\sqrt {\frac{{2 + k}}{k}} + (2k - 4)\sqrt {\frac{{2k - 5}}{{(k - 2)(k - 1)}}} + \frac{1}{2}(n{k^2} + 6n - 5nk)\sqrt {\frac{{2k - 6}}{{{{(k - 2)}^2}}}} + (nk - 2n - 2k + 4)\sqrt {\frac{{2k - 4}}{{(k - 2)k}}} ;
\end{array}$
G A ( G ) = 8 7 3 + 4 3 k 3 + k + 4 3 ( k − 3 ) 2 + k + 8 ( n − 3 ) k 4 + k + 4 ( k − 2 ) ( k − 1 ) ( k − 2 ) 2 k − 3 + 1 2 ( n k 2 + 8 n − 5 n k − 8 ) + ( n − 2 ) k ( k − 2 ) $\begin{array}{}
\displaystyle
GA(G) = \frac{8}{7}\sqrt 3 + \frac{{4\sqrt {3k} }}{{3 + k}} + \frac{{4\sqrt {3(k - 3)} }}{{2 + k}} + \frac{{8(n - 3)\sqrt k }}{{4 + k}} + \frac{{4(k - 2)\sqrt {(k - 1)(k - 2)} }}{{2k - 3}} + \frac{1}{2}(n{k^2} + 8n - 5nk - 8) + (n - 2)\sqrt {k(k - 2)} ;
\end{array}$
χ ( G ) = 2 7 + 2 3 + k + 2 2 + k + 2 ( n − 3 ) 4 + k + 2 ( k − 2 ) 2 k − 3 + 1 2 n k 2 + 6 n − 5 n k 2 ( k − 2 ) + ( n − 2 ) 1 2 ( k − 2 ) + 1 4 ( n − 4 ) 2 $\begin{array}{}
\displaystyle
\chi (G) = \frac{2}{{\sqrt 7 }} + \frac{2}{{\sqrt {3 + k} }} + \frac{2}{{\sqrt {2 + k} }} + \frac{{2(n - 3)}}{{\sqrt {4 + k} }} + \frac{{2(k - 2)}}{{\sqrt {2k - 3} }} + \frac{1}{2}\frac{{n{k^2} + 6n - 5nk}}{{\sqrt {2(k - 2)} }} + (n - 2)\sqrt {\frac{1}{2}(k - 2)} + \frac{1}{4}(n - 4)\sqrt 2 ;
\end{array}$
A Z I ( G ) = − 162668 3375 + 54 k 3 ( 1 + k ) 3 + 128 ( n − 3 ) k 3 ( k + 2 ) 3 + 2 ( k − 2 ) ( k − 1 ) 3 ( k − 2 ) 3 ( 2 k − 5 ) 3 + ( n k 2 − 6 n − 5 n k ) ( k − 2 ) 6 16 ( k − 3 ) 3 + 1 8 ( n k − 2 n − 2 k + 4 ) k 3 $\begin{array}{}
\displaystyle
AZI(G) = - \frac{{162668}}{{3375}} + \frac{{54{k^3}}}{{{{(1 + k)}^3}}} + \frac{{128(n - 3){k^3}}}{{{{(k + 2)}^3}}} + \frac{{2(k - 2){{(k - 1)}^3}{{(k - 2)}^3}}}{{{{(2k - 5)}^3}}} + \frac{{(n{k^2} - 6n - 5nk){{(k - 2)}^6}}}{{16{{(k - 3)}^3}}} + \frac{1}{8}(nk - 2n - 2k + 4){k^3}.
\end{array}$
Proof.
The graph G for n = 4 and k = 7 is shown in Fig. 4 . By using Lemma 5 , it is easy to see that the order of G is nk − 1 out of which 2 vertices are of degree 3, 2 vertices are of degree k − 1, n − 3 vertices are of degree 4, n (k − 2) vertices are of degree k − 2, and n − 2 vertices are of degree k . Therefore by using Lemma 6 , G has size n k 2 − 3 n k + 8 n − 8 2 $\begin{array}{}
\displaystyle
\frac{{n{k^2} - 3nk + 8n - 8}}{2}
\end{array}$ G into edges of the type E (du,dv ) where uv is an edge. In G , we get edges of the type E (3,4) , E (3,k) , E (3,k −1) , E (4,4) , E (4,k ) , E (k −1,k −2) , E (k −2,k −2) , and E (k,k −2) . The number of edges of these types are given in the Table 3 . By using Formulas (1 )-(9 ) and Table 3 , we can obtain the required results.
Table 3 The size partition of G .
(du ,dv ) : uv ∈ E (G ) (3,4) (3,k ) (3,k − 1) (4,4) (4,k ) (k − 1,k − 2) (k − 2,k − 2) (k,k − 2) Number of edges 2 2 2 n − 42n − 6 2(k − 2) n k 2 − 5 n k + 6 n 2 $\begin{array}{}
\displaystyle
\frac{{n{k^2} - 5nk + 6n}}{2}
\end{array}$ (n − 2)(k − 2)
Theorem 11
Let G be the line graph of the Firecracker graph Fn,k . Then
M 1 ( G ¯ ) = 2 n k 2 − 16 n 2 k + 19 n k − 4 k + 40 n − 61 ; $\begin{array}{}
\displaystyle
{M_1}(\bar G) = 2n{k^2} - 16{n^2}k + 19nk - 4k + 40n - 61;
\end{array}$
M 1 ¯ ( G ) = M 1 ¯ ( G ¯ ) = 8 n 2 k − 14 n k + 4 k − 24 n + 44 ; $\begin{array}{}
\displaystyle
\overline {{M_1}} (G) = \overline {{M_1}} (\bar G) = 8{n^2}k - 14nk + 4k - 24n + 44;
\end{array}$
M 2 ( G ¯ ) = 3 2 n k 3 + 8 n 2 k 2 − 15 n k 2 + 32 n 2 k − 56 n k + 2 k 2 + 14 k − 160 n + 409 2 ; $\begin{array}{}
\displaystyle
{M_2}(\bar G) = \frac{3}{2}n{k^3} + 8{n^2}{k^2} - 15n{k^2} + 32{n^2}k - 56nk + 2{k^2} + 14k - 160n + \frac{{409}}{2};
\end{array}$
M 2 ¯ ( G ) = 8 n 2 k 2 − 24 n 2 k − 12 n k 2 + 2 k 2 + 32 n 2 + 28 n k − 96 n + 6 k + 100 ; $\begin{array}{}
\displaystyle
\overline {{M_2}} (G) = 8{n^2}{k^2} - 24{n^2}k - 12n{k^2} + 2{k^2} + 32{n^2} + 28nk - 96n + 6k + 100;
\end{array}$
M 2 ¯ ( G ¯ ) = − 8 n 2 k 2 − 24 n 2 k + 9 n k 2 + 52 n k − 2 k 2 − 12 k + 60 n − 126 ; $\begin{array}{}
\displaystyle
\overline {{M_2}} (\bar G) = - 8{n^2}{k^2} - 24{n^2}k + 9n{k^2} + 52nk - 2{k^2} - 12k + 60n - 126;
\end{array}$
W ( G ) ≤ − 8 n 2 k + 1 2 n k 2 + 21 2 n k − 2 k + 20 n − 31 ; $\begin{array}{}
\displaystyle
W(G) \le - 8{n^2}k + \frac{1}{2}n{k^2} + \frac{{21}}{2}nk - 2k + 20n - 31;
\end{array}$
DD (G ) ≥ nk 3 − 5nk 2 + 16n 2 k − 16nk + 2k − 40n + 60.
Proof.
By using Theorems (1 )-(4 ) and (10 ), we can obtain the required results.
Theorem 12
Let G be the line graph of the Firecracker graph Fn,k . Then
A B C 4 ( G ) = 1 2 ( k − 2 ) ( k − 3 ) ( n − 2 ) 2 k 2 − 8 k + 11 ( k 2 − 4 k + 6 ) ( k 2 − 4 k + 7 ) + ( n − 4 ) ( k − 2 ) \ 2 k 2 − 8 k + 16 ( k 2 − 4 k + 12 ) ( k 2 − 4 k + 6 ) + ( k 2 − 3 k + 2 ) 2 k 2 − 4 k + 7 + 2 k − 4 2 k 2 − 8 k + 15 ( k 2 − 4 k + 11 ) ( k 2 − 4 k + 6 ) + 2 k 2 − 2 k + 12 2 k 3 − 5 k 2 + 10 k + 33 + 2 k 2 − 2 k + 16 2 k 3 − k 2 − 6 k + 77 + 2 k 2 − 2 k + 17 2 k 3 − k 2 − 4 k + 84 + 2 k 2 − 2 k + 8 2 k 3 − 5 k 2 + 2 k + 21 + ( n − 5 ) 2 k 2 − 2 k + 18 ( k 2 − 4 k + 12 ) ( 4 + k ) + 2 4 k + 13 4 k 2 + 30 k + 56 + 4 k + 2 4 k 2 + 20 k + 21 + 1 2 ( n − 6 ) 2 k + 7 k + 4 ; $\begin{array}{}
\displaystyle
AB{C_4}(G) = \frac{1}{2}(k - 2)(k - 3)(n - 2)\sqrt {\frac{{2{k^2} - 8k + 11}}{{({k^2} - 4k + 6)({k^2} - 4k + 7)}}} + (n - 4)(k - 2){\rm{\backslash }}\sqrt {\frac{{2{k^2} - 8k + 16}}{{({k^2} - 4k + 12)({k^2} - 4k + 6)}}} + ({k^2} - 3k + 2)\sqrt {\frac{2}{{{k^2} - 4k + 7}}} + 2k - 4\sqrt {\frac{{2{k^2} - 8k + 15}}{{({k^2} - 4k + 11)({k^2} - 4k + 6)}}} + 2\sqrt {\frac{{{k^2} - 2k + 12}}{{2{k^3} - 5{k^2} + 10k + 33}}} + 2\sqrt {\frac{{{k^2} - 2k + 16}}{{2{k^3} - {k^2} - 6k + 77}}} + 2\sqrt {\frac{{{k^2} - 2k + 17}}{{2{k^3} - {k^2} - 4k + 84}}} + 2\sqrt {\frac{{{k^2} - 2k + 8}}{{2{k^3} - 5{k^2} + 2k + 21}}} + (n - 5)\sqrt 2 \sqrt {\frac{{{k^2} - 2k + 18}}{{({k^2} - 4k + 12)(4 + k)}}} + 2\sqrt {\frac{{4k + 13}}{{4{k^2} + 30k + 56}}} + 4\sqrt {\frac{{k + 2}}{{4{k^2} + 20k + 21}}} + \frac{1}{{\sqrt 2 }}(n - 6)\frac{{\sqrt {2k + 7} }}{{k + 4}};
\end{array}$
G A 5 = 4 ( 2 k + 3 ) ( k 2 − 4 k + 7 ) k 2 − 2 k + 10 + 2 ( 2 k + 3 ) ( 2 k + 7 ) 2 k + 5 + 4 ( 2 k + 3 ) ( k 2 − 4 k + 11 ) k 2 − 2 k + 14 + 4 ( 2 k + 7 ) ( k 2 − 4 k + 11 ) k 2 − 2 k + 18 + 4 ( 2 k + 7 ) ( k 2 − 4 k + 12 ) k 2 − 2 k + 19 + 4 ( 2 k + 7 ) ( 2 k + 8 ) 4 k + 15 + 2 ( k − 2 ) ( k 2 − 4 k + 5 ) ( k 2 − 4 k + 7 ) k 2 − 4 k + 6 + ( n − 2 ) ( k 2 − 5 k + 6 ) ( k 2 − 4 k + 6 ) ( k 2 − 4 k + 7 ) 2 k 2 − 8 k + 13 + 4 ( k − 2 ) ( k 2 − 4 k + 11 ) ( k 2 − 4 k + 6 ) 2 k 2 − 8 k + 17 + ( n − 4 ) ( k − 2 ) ( k 2 − 4 k + 12 ) ( k 2 − 4 k + 6 ) k 2 − 4 k + 9 + 4 ( n − 5 ) ( k 2 − 4 k + 12 ) ( 2 k + 8 ) k 2 − 2 k + 20 + ( k 2 − 5 k + 6 ) ( k 2 − 4 k + 5 ) ( k 2 − 4 k + 7 ) k 2 − 4 k + 6 + n − 6 ; $\begin{array}{}
\displaystyle
G{A_5} = 4\frac{{\sqrt {(2k + 3)({k^2} - 4k + 7)} }}{{{k^2} - 2k + 10}} + \frac{{2\sqrt {(2k + 3)(2k + 7)} }}{{2k + 5}} + \frac{{4\sqrt {(2k + 3)({k^2} - 4k + 11)} }}{{{k^2} - 2k + 14}} + \frac{{4\sqrt {(2k + 7)({k^2} - 4k + 11)} }}{{{k^2} - 2k + 18}} + \frac{{4\sqrt {(2k + 7)({k^2} - 4k + 12)} }}{{{k^2} - 2k + 19}} + \frac{{4\sqrt {(2k + 7)(2k + 8)} }}{{4k + 15}} + \frac{{2(k - 2)\sqrt {({k^2} - 4k + 5)({k^2} - 4k + 7)} }}{{{k^2} - 4k + 6}} + \frac{{(n - 2)({k^2} - 5k + 6)\sqrt {({k^2} - 4k + 6)({k^2} - 4k + 7)} }}{{2{k^2} - 8k + 13}} + \frac{{4(k - 2)\sqrt {({k^2} - 4k + 11)({k^2} - 4k + 6)} }}{{2{k^2} - 8k + 17}} + \frac{{(n - 4)(k - 2)\sqrt {({k^2} - 4k + 12)({k^2} - 4k + 6)} }}{{{k^2} - 4k + 9}} + \frac{{4(n - 5)\sqrt {({k^2} - 4k + 12)(2k + 8)} }}{{{k^2} - 2k + 20}} + \frac{{({k^2} - 5k + 6)\sqrt {({k^2} - 4k + 5)({k^2} - 4k + 7)} }}{{{k^2} - 4k + 6}} + n - 6{\rm{;}}
\end{array}$
S ( G ) = 2 ( 2 k + 3 ) 3 ( k 2 − 4 k + 7 ) 3 ( k 2 − 2 k + 8 ) 3 + ( 2 k + 3 ) 3 ( 2 k + 7 ) 3 32 ( 2 + k ) 3 + 2 ( 2 k + 3 ) 3 ( k 2 − 4 k + 11 ) 3 ( k 2 − 2 k + 12 ) 3 + 2 ( 2 k + 7 ) 3 ( k 2 − 4 k + 11 ) 3 ( k 2 − 2 k + 16 ) 3 + 2 ( 2 k + 7 ) 3 ( k 2 − 4 k + 12 ) 3 ( k 2 − 2 k + 17 ) 3 + 2 ( 2 k + 7 ) 3 ( 2 k + 8 ) 3 ( 4 k + 13 ) 3 + 8 ( n − 6 ) ( 4 + k ) 6 ( 2 k + 7 ) 3 + 1 4 ( k − 2 ) ( k 2 − 4 k + 7 ) 3 + ( n − 2 ) ( k − 2 ) ( k − 3 ) ( k 2 − 4 k + 6 ) 3 ( k 2 − 4 k + 7 ) 3 2 ( 2 k 2 − 8 k + 11 ) 3 + 2 ( k − 2 ) ( k 2 − 4 k + 11 ) 3 ( k 2 − 4 k + 6 ) 3 ( 2 k 2 − 8 k + 15 ) 3 + ( n − 4 ) ( k − 2 ) ( k 2 − 4 k + 12 ) 3 ( k 2 − 4 k + 6 ) 3 8 ( k 2 − 4 k + 8 ) 3 + 16 ( n − 5 ) ( k 2 − 4 k + 12 ) 3 ( 4 + k ) 3 ( k 2 − 2 k + 18 ) 3 + 1 8 ( k − 2 ) ( k − 3 ) ( k 2 − 4 k + 7 ) 3 . $\begin{array}{}
\displaystyle
S(G) = \frac{{2{{(2k + 3)}^3}{{({k^2} - 4k + 7)}^3}}}{{{{({k^2} - 2k + 8)}^3}}} + \frac{{{{(2k + 3)}^3}{{(2k + 7)}^3}}}{{32{{(2 + k)}^3}}} + \frac{{2{{(2k + 3)}^3}{{({k^2} - 4k + 11)}^3}}}{{{{({k^2} - 2k + 12)}^3}}} + \frac{{2{{(2k + 7)}^3}{{({k^2} - 4k + 11)}^3}}}{{{{({k^2} - 2k + 16)}^3}}} + \frac{{2{{(2k + 7)}^3}{{({k^2} - 4k + 12)}^3}}}{{{{({k^2} - 2k + 17)}^3}}} + \frac{{2{{(2k + 7)}^3}{{(2k + 8)}^3}}}{{{{(4k + 13)}^3}}} + \frac{{8(n - 6){{(4 + k)}^6}}}{{{{(2k + 7)}^3}}} + \frac{1}{4}(k - 2){({k^2} - 4k + 7)^3} + \frac{{(n - 2)(k - 2)(k - 3){{({k^2} - 4k + 6)}^3}{{({k^2} - 4k + 7)}^3}}}{{2{{(2{k^2} - 8k + 11)}^3}}} + \frac{{2(k - 2){{({k^2} - 4k + 11)}^3}{{({k^2} - 4k + 6)}^3}}}{{{{(2{k^2} - 8k + 15)}^3}}} + \frac{{(n - 4)(k - 2){{({k^2} - 4k + 12)}^3}{{({k^2} - 4k + 6)}^3}}}{{8{{({k^2} - 4k + 8)}^3}}} + \frac{{16(n - 5){{({k^2} - 4k + 12)}^3}{{(4 + k)}^3}}}{{{{({k^2} - 2k + 18)}^3}}} + \frac{1}{8}(k - 2)(k - 3){({k^2} - 4k + 7)^3}.
\end{array}$
Proof.
We partition the size of G into edges of the type (Su ,Sv ) where uv ∈ E (G ) as shown in Table 4 . Hence we get the required results by using Table 4 and Formulas (10 )-(12 ).
Table 4 The size partition of G .
(Su ,Sv ) : uv ∈ E (G ) (2k + 3,k 2 − 4k + 7) (2k + 3,2k + 7) (2k + 3,k 2 − 4k + 11) (2k + 7,k 2 − 4k + 11) Number of edges 2 2 2 2 (Su ,Sv ) : uv ∈ E (G ) (2k + 7,k 2 − 4k + 12) (2k + 7,2k + 8) (2k + 8,2k + 8) (k 2 − 4k + 5,k 2 − 4k + 7) Number of edges 2 2 n − 62(k − 2) (Su ,Sv ) : uv ∈ E (G ) (k 2 − 4k + 6,k 2 − 4k + 7) (k 2 − 4k + 11,k 2 − 4k + 6) (k 2 − 4k + 12,k 2 − 4k + 6) (k 2 − 4k + 12,2k + 8) Number of edges ( n − 2 ) ( k 2 − 5 k + 6 ) 2 $\begin{array}{}
\displaystyle
\frac{{(n - 2)({k^2} - 5k + 6)}}{2}
\end{array}$ 2(k − 2) (n − 4)(k − 2) 2(n − 5) (Su ,Sv ) : uv ∈ E (G ) (k 2 − 4k + 5,k 2 − 4k + 7) Number of edges k 2 − 5k + 6
