### Leap Zagreb coindex and Wiener polarity index of

### Jahangir graphs

Raad Sehen Haoera,∗

a_{Open Educational College, Ministry of Education, Department of Mathematics, Al-Qadisiya Centre, Iraq.}

Abstract

The Leap Zagreb coindices and the Wiener polarity index are examples of less frequently studied topological indices that are based on the number of vertices lying at second and third distances from a given vertex. These indices have applications in QSAR and QSPR studies. In this paper, we obtain leap Zagreb coindices and the Wiener polarity index of a rotationally symmetric subfamily of circulant graphs known as Jahangir graphs.

Keywords: Topological indices, leap Zagreb index, Jahangir graphs.

1. Introduction

Let Γ denote a simple connected graph with vertex set V and edge set E. The distance dΓ(v, w) between

two vertices v, w ∈ V (Γ) is the length of a path of minimal length joining v with w in Γ. Let u, x, y ∈ V (Γ) and e = xy ∈ E(Γ). The number of vertices lying at distance k from a vertex v ∈ V (Γ) is called the k-th degree d(v|k) of the vertex v.

The Zagreb indices were first introduced in [1] where the authors examined the dependence of total π-electron energy of molecular structures. For a molecular graph Γ, the first Zagreb index M1(Γ) and the

second Zagreb index M2(Γ) are defined, respectively, as follows.

M1(Γ) = X v∈V (Γ) dΓ(v)2= X uv∈E(Γ) (dΓ(u) + dΓ(v)) M2(Γ) = X uv∈E(Γ) dΓ(u)dΓ(v). ∗ Corresponding author

Email address: raadsehen@gmail.com (Raad Sehen Haoer)

The Zagreb coindices were defined as the modification of the Zagreb index by Ashrafi et al. [2] as follows. M1(Γ) = X uv /∈E(Γ) (dΓ(u) + dΓ(v)) M2(Γ) = X uv /∈E(Γ) dΓ(u)dΓ(v).

Modifications of the Zagreb indices based on the second neighbors of the vertices of a graph were defined by Gutman et al. [3]. These indices are called the first leap Zagreb indices, second leap Zagreb indices and third leap Zagreb indices respectively, are defined as follows.

LM1(Γ) =
X
v∈V (Γ)
d2_{Γ}(v | 2)
LM2(Γ) =
X
uv∈E(Γ)
dΓ(u | 2)dΓ(v | 2)
LM3(Γ) =
X
v∈V (Γ)
dΓ(v | 1)dΓ(v | 2) =
X
uv∈E(Γ)
(dΓ(u | 2) + dΓ(v | 2)).

It can be seen that the coindices of the second and third version of leap Zagreb indices can be established as follows. LM2(Γ) = X uv /∈E(Γ) dΓ(u | 2)dΓ(v | 2) LM3(Γ) = X uv /∈E(Γ) (dΓ(u | 2) + dΓ(v | 2)).

Wiener [4] introduced another topological invariant of a graph Γ known as Wiener polarity index Wp(Γ).

It is defined as the number of unordered pairs of vertices {u, v} ⊆ V (Γ) which are at distance 3 in Γ. Mathematically, Wp(Γ) is defined in the following equation along with an equivalent and comparatively easy

representation. Wp(Γ) = {{u, v} ⊆ V (Γ) | dΓ(u, v) = 3} = 1 2 X v∈V (Γ) dΓ(v | 3). (1.1)

Wiener polarity index has attracted the attention of a remarkably large number of mathematicians. Hosoya [5] found a physical-chemical interpretation of Wp(Γ). The Wiener polarity index of fullerenes and hexagonal

systems was studied in [6]. The Wiener polarity index of square, hexagonal and triangular lattices were studied in [7]. Recently, Arockiaraj et al. [8] studied the hyper-Wiener and Wiener polarity indices of silicate and oxide networks. Authors in [9, 10] and [11] studied some degree and distance based indices of product graphs and nanotubes. In [12, 13, 14, 15], authors studied some newly defined degree based topological indices of some important families of graphs. Also, Liu and Liu [16] studied the Wiener polar-ity index of dendrimers. For a survey of results on Wiener index, Wiener polarpolar-ity index and some other variations of Wiener index, see [17]. Topological indices have been extensively studied for their applications in QSAP and QSPR studies, see [18, 19, 20, 21]. More on the applications of topological indices can be found in [22, 23, 24, 25, 26, 27, 28]. The study of topological indices for several types of graphs have been carried out in the literature, for example see [2, 29]. Several degree and distance-based topological indices of Jahangir graphs are studied in [30, 31, 32].

In this paper we study leap Zagreb coindices and the Wiener polarity index of Jahangir graphs Jn,m.

2. Jahangir graphs

The Jahangir graph Jn,m is a graph with nm + 1 number of vertices and m(n + 1) number of edges for

all n ≥ 2 and m ≥ 3. The Jahangir graph Jn,m consists of a cycle Cmn and a single vertex that is adjacent

to m vertices of the cycle that are at distance n from each other. The Jahangir graphs J6,4 is shown in

*v*

*u*

*u*

*u*

_{u}

_{u}

*w*

*w*

*w*

*w*

*w*

*w*

*w*

*w*

*x''*

*x'*

*x'*

*x''*

*x''*

*x'*

*x'*

*x''*

*x*

*x*

*x*

*x*

Figure 1: The Jahangir graph J6,4.

3. First, second and third degrees of vertices in Jahangir graphs Jn,m

In the following lemma, we present some structural properties of the Jahangir graphs Jn,m.

Lemma 3.1. Let Jn,m denote the Jahangir graphs (see Figure 1). Then the first, second and third degrees

of vertices in Jn,m are given by the following expressions.

(a) dΓ(y|1) = m, if y = v, 3, if y = u, 2, otherwise, where n ≥ 3, m ≥ 4. (b) dΓ(y|2) = 2m, if y = v, m + 1, if y = u, 3, if y = w, 2, otherwise, where n ≥ 4, m ≥ 4. (c) dΓ(y|3) = 2m, if y = v, 2m − 2, if y = u, m + 1, if y = w, 3, if y = x0, 2, otherwise, where n ≥ 5, m ≥ 4.

The computation of coindices of a graph requires information about the edge set of complement of the given graph. Let E and E respectively denote the edge set of the Jahangir graph and edge set of the complement of the Jahangir graph Jn,m, that is, E = E(Jn,m) and E = E(Jn,m). Then the edge partition

of the edge set of Jahangir graph can be presented by the Table 1.

In Table 2, we give the edge partition of the complement of Jahangir graph with respect to the type of its vertices.

4. Leap Zagreb coindices of Jahangir graphs Jn,m

Now we proceed towards the main calculations of the indices of Jahangir graphs Jn,m. First we calculate

Table 1: The edge partition of the graph Jn,m. E u v w x u 0 m 2m 0 v m 0 0 0 w 2m 0 0 2m x 0 0 2m m(n − 4)

Table 2: The edge partition of the graph Jn,m.

E u v w x
u m(m−1)_{2} 0 m(2m − 2) m2_{(n − 3)}
v 0 0 2m m(n − 3)
w m(2m − 2) 2m 2m(2m−1)_{2} 2m[m(n − 3) − 1]
x m2(n − 3) m(n − 3) 2m[m(n − 3) − 1]
m(n−3)[m(n−3)−1]
2 − m(n − 4)

Theorem 4.1. Then second leap Zagreb coindex of the Jahangir graph Jn,m is given by

LM2(Jn,m) = (1/2)m4+ (1/2)m3− (13/2)m2− (35/2)m + 2m3n + 6m2n + 2m2n2− 3mn.

Proof. We use Lemma 3.1, in the definition of second leap Zagreb coindex to simplify the results as follows.
LM2(Jn,m) =
X
xy /∈E(Jn,m)
dJn,m(x|2) dJn,m(y|2)
= d(u|2)d(u|2) m(m − 1)
2
+ d(u|2)d(w|2)m(2m − 2)
+d(u|2)d(x|2)m2(n − 3) + d(w|2)d(v|2)2m + d(w|2)d(w|2) 2m(2m − 1)
2
+d(w|2)d(x|2)2m[m(n − 3) − 1] + d(x|2)d(v|2)m(n − 3)
+d(x|2)d(x|2) m(n − 3)(m(n − 3) − 1)
2 − m(n − 4)
= (m + 1)(m + 1)m(m − 1)
2 + (m + 1)(3)m(2m − 2)
+(m + 1)(2)m2(n − 3) + (3)(2m)2m
+(3)(3)2m(2m − 1)
2 + (3)(2)2m[m(n − 3) − 1]
+(2)(2m)m(n − 3) + (2)(2) m(n − 3)(m(n − 3) − 1)
2 − mn − 4m
= m(m − 1)(m + 1)
2
2 + 3m(m + 1)(2m − 2) + 2m
2_{(m + 1)(n − 3) + 12m}2
+9m(2m − 1) + 12m[m(n − 3) − 1] + 4m2(n − 3)
+2m(n − 3)[m(n − 3) − 1] − mn + 4m
= (1/2)m4+ (1/2)m3− (13/2)m2− (35/2)m + 2m3n + 6m2n + 2m2n2− 3mn.
This completes the proof.

Theorem 4.2. Then third leap Zagreb coindex of the Jahangir graph Jn,m is given by

Proof. We use Lemma 3.1, in the definition of third leap Zagreb coindex to simplify the results as follows. LM3(Jn,m) = X xy /∈E(Jn,m) (dJn,m(x|2) + dJn,m(y|2)) = (d(u|2) + d(u|2)) m(m − 1) 2 + (d(u|2) + d(w|2))m(2m − 2) +(d(u|2) + d(x|2))m2(n − 3) + (d(w|2) + d(v|2))2m + (d(w|2) +d(w|2)) 2m(2m − 1) 2 +(d(w|2) + d(x|2))2m[m(n − 3) − 1] + (d(x|2) + d(v|2))m(n − 3) +(d(x|2) + d(x|2)) m(n − 3)(m(n − 3) − 1) 2 − m(n − 4) = ((m + 1) + (m + 1))m(m − 1) 2 + ((m + 1) + (3))m(2m − 2) +((m + 1) + (2))m2(n − 3) + ((3) + (2m))2m +((3) + (3))2m(2m − 1) + ((3) + (2))2m[m(n − 3) − 1] + ((2) + (2m))m(n − 3) +((2) + (2)) m(n − 3)(m(n − 3) − 1) 2 − mn + 4m = m(m + 1)(m − 1) + 2m(m + 4)(m − 1) +m2(m + 3)(n − 3) + 2m(2m + 3) +12m(2m − 1) + 10m[m(n − 3) − 1] +2m(m + 1)(n − 3) + 2m(n − 3)[m(n − 3) − 1] − mn + 4m = m3n + 2m2n2+ 3m2n − 5m2− 4mn − 3m.

This completes the proof.

Theorem 4.3. Then Wiener polarity index of the Jahangir graph Jn,m is given by

Wp(Jn,m) =

1 2(4m

2_{+ 3nm − 7m).}

Proof. We use Lemma 3.1, in the definition of Wiener polarity index to simplify the results as follows.
Wp(Jn,m) =
X
x∈V (Γ)
1
2dΓ(x|3)
= 1
2 2m + (2m − 2)m + (m + 1)2m + (3)(n − 3)m
= 1
2(4m
2_{+ 3nm − 7m).}

This completes the proof.

5. Conclusion and discussion

The pairwise comparisons of the 3D surfaces of the second and third kind of leap Zagreb coindices LM1

and LM2 and the Wiener polarity index of the Jahangir graphs Jn,m is provided in Figure 2.

The Jahangir graphs Jn,m is an infinite family of cycle-based graphs that form a subclass of circulant

Figure 2: The comparison between the graphs of LM1verses LM2 (Left), LM2 verses Wp(Middle) and LM1 verses Wp.

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