On the number of geodesics of Petersen graph $GP(n,2)$

TL;DR

This paper investigates the number of geodesics and betweenness centrality in the Petersen graph $GP(n,2)$, providing explicit formulas and enhancing understanding of its network structure.

Contribution

It derives explicit expressions for the number of geodesics and betweenness centrality specifically for the Petersen graph $GP(n,2)$, a novel analysis for this class of graphs.

Findings

Formulas for the number of geodesics in $GP(n,2)$

Expressions for betweenness centrality in $GP(n,2)$

Enhanced understanding of geodesic structure in Petersen graphs

Abstract

In any network, the interconnection of nodes by means of geodesics and the number of geodesics existing between nodes are important. There exists a class of centrality measures based on the number of geodesics passing through a vertex. Betweenness centrality indicates the betweenness of a vertex or how often a vertex appears on geodesics between other vertices. It has wide applications in the analysis of networks. Consider $GP(n,k)$. For each $n$ and $k \,(n > 2k)$, the generalized Petersen graph $GP ( n , k )$ is a trivalent graph with vertex set $\{ u_ i ,\, v_ i \,|\, 0 \leq i \leq n - 1 \}$ and edge set $\{ u_ i u_ {i + 1} , u_ i v _i , v_ i v_{ i + k}\, |\, 0\leq i \leq n - 1, \hbox{ subscripts reduced modulo } n \}$. There are three kinds of edges namely outer edges, spokes and inner edges. The outer vertices generate an $n$-cycle called outer cycle and inner vertices generate one or more inner cycles. In this paper, we consider $GP(n,2)$ and find expressions for the number of geodesics and betweenness centrality.

Equations

$$\sigma (u_i,u_j)=\{\begin{array}{ll}1& forr=1,2,3\\ 2& forr=4\\ 3& forr=5;r<k\\ 4& forr=5;r=k\\ 1& forr=6,8,10,\dots \\ 2& forr=7,9,11,\dots ;r<k\\ 3& forr=7,9,11,\dots ;r=k\end{array}$$

$$\{u_i,v_i,v_{i+2},v_{i+4},\dots ,v_{i+r},u_{i+r}\}$$

$$\{u_i,v_i,v_{i-2},v_{i-4},\dots ,v_{i-r-1},u_{i-r-1}(=u_{i+r})\}$$

$$\sigma (v_i,v_j)=\{\begin{array}{ll}1& for\; evenr\\ 1& for\; oddr,r>k-2\\ \frac{r+1}{2}& for\; oddr,r<k-2\\ \frac{r+3}{2}& for\; oddr,r=k-2\end{array}$$

$$\sigma (u_i,v_j)=\{\begin{array}{ll}1& forr<k\\ 1& for\; evenr,r=k\\ 2& for\; oddr,r=k\end{array}$$

$$\sigma (u_i,u_{i+r})=\{\begin{array}{ll}1& forr=1,2,3;r<k\\ 2& forr=3;r=k\\ 2& forr=4;r<k\\ 4& forr=4;r=k\\ 3& forr=5;r<k\\ 6& forr=5;r=k\\ 1& forr=6,8,10,\dots ;r<k\\ 2& forr=6,8,10,\dots ;r=k\\ 2& forr=7,9,11,\dots ;r<k\\ 4& forr=7,9,11,\dots ;r=k\end{array}$$

$$\sigma (v_i,v_j)=\{\begin{array}{ll}1& for\; evenr,r<k\\ 2& for\; evenr,r=k\\ \frac{r+1}{2}& for\; oddr,r<k\\ r+1& for\; oddr,r=k\end{array}$$

$$\sigma (u_i,v_j)=\{\begin{array}{ll}1& r<k\\ 2& r=k\end{array}$$

$$d(u_i,u_j)=\{\begin{array}{ll}r& forr\le 5\\ \frac{r+4}{2}& for\; evenr,r>5\\ \frac{r+5}{2}& for\; oddr,r>5\end{array}$$

$$d(v_i,v_j)=\{\begin{array}{ll}\frac{r}{2}& for\; evenr\\ \frac{r+5}{2}& for\; oddr\end{array}$$

$$d(u_i,v_j)=\{\begin{array}{ll}\frac{r+2}{2}& for\; evenr\\ \frac{r+3}{2}& for\; oddr\end{array}$$

$$diamGP(n,2)=\lceil \frac{n-1}{4}\rceil +2$$

$$B_G(x)=\sum _{s,t\in V(G)\setminus \{x\}}\frac{{\sigma }_{st}(x)}{{\sigma }_{st}}$$

$$B_H(x)=\sum _{s,t\in V(H)\setminus \{x\}}\frac{{\sigma }_{st}^H(x)}{{\sigma }_{st}^H}$$

$$B(x,H)=\sum _{s,t(\ne x)\in V(H)}\frac{{\sigma }_{st}(x)}{{\sigma }_{st}}$$

$$B(x,S)=\sum _{s,t(\ne x)\in S}\frac{{\sigma }_{st}(x)}{{\sigma }_{st}}$$

$$B_G(x,x_0)=\sum _{t\in V(G)\setminus x}\frac{{\sigma }_{x_{0}t}(x)}{{\sigma }_{x_{0}t}}$$

$$B(x_i,x_j)=\{\begin{array}{ll}i-1& ifi<j\\ n-i& ifj<i\end{array}$$

$$B(x_i,x_0)=\{\begin{array}{ll}\frac{n-1-2i}{2}& if\; 1\le i<n/2\\ 0& ifi=n/2\end{array}$$

$$B(x_i,x_0)=\frac{n-1-2i}{2}if\; 1\le i\le \frac{n-1}{2}$$

$$B(x_i,x_0)=B(x_{n-1},x_0)$$

$$B(x_i,x_0)=0B(x_0,x_i)=n-2$$

$$B(x_i,x_j)=0\; fori,j\ne 0$$

$$B(x_i,x_0)=0,B(x_0,x_i)=n-5$$

$$B(x_i,x_{i\pm 1})=1/2,B(x_i,x_{i\pm j})=0\; forj\ge 2$$

$$B_G(x_i)=\frac{1}{2}\sum _{j\ne i}{B}_G(x_i,x_j)$$

$$B(x,S,T)=B(x,S)+B(x,T)$$

$$B(x,S|T)=\sum _{s\in S,t\in T}\frac{{\sigma }_{st}(x)}{{\sigma }_{st}}$$

$$B(u)=\{\begin{array}{ll}\frac{1}{4}(5n+1)& forn=13,17,21,\dots \\ \frac{15n^2+32n-79}{12(n+1)}& forn=15,19,23,\dots \\ \frac{1}{4}(5n+14)& for\; evenn,n\ge 12\end{array}$$

$$B(u_0,U)=\{\begin{array}{ll}\frac{1}{4}(n+13)& forn=13,17,21,\dots \\ \frac{1}{12}(3n+41)& forn=15,19,23,\dots \\ \frac{1}{4}(n+14)& for\; evenn,n\ge 12\end{array}$$

Full text

Keywords: Petersen graph, geodesics, wicket, Möbius strip, betweenness centrality, induced betweenness centrality.

1 Introduction

Generalized Petersen graphs were first defined by Watkins[1] who was interested in trivalent graphs without proper three edge-colorings. For integers $\displaystyle n$ and $\displaystyle k$ with $\displaystyle 1\leq k<n/2$, the generalized Petersen graph $\displaystyle GP(n,k)$ has been defined as an undirected graph with vertex-set $\displaystyle V=\{u_{0},u_{1},\ldots,u_{n-1},v_{0},v_{1},\ldots,v_{n-1}\}$ and edge set $\displaystyle E$ consisting of all pairs of the three forms $\displaystyle(u_{i},u_{i+1}),(u_{i},v_{i})$ and $\displaystyle(v_{i},v_{i+k})$ where $\displaystyle i$ is an integer and all subscripts are read modulo $\displaystyle n$. The above three forms of edges are called outer edges, spokes, and inner edges respectively. In this original definition, $\displaystyle GP(n,k)$ is a trivalent graph of order $\displaystyle 2n$ and size $\displaystyle 3n$. It can be seen that when $\displaystyle n$ is even and $\displaystyle k=n/2$ the resulting graph is not cubic. And because of the obvious isomorphism $\displaystyle GP(n,k)\cong GP(n,n-k)$, $\displaystyle k<n/2$. In $\displaystyle GP(n,k)$, there exist one outer cycle and one or more inner cycles. In this paper we may refer even (odd) subscripted vertices by even (odd) vertices.

Theorem 1.1**.**

[2]** If $\displaystyle d$ denotes the greatest common divisor of $\displaystyle n$ and $\displaystyle k$, then the set of inner edges generates a subgraph which is the union of $\displaystyle d$ pairwise-disjoint $\displaystyle(n/d$)-cycles.

A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set.

Theorem 1.2**.**

[2]** $\displaystyle G(n,k)$ is vertex-transitive if and only if $\displaystyle k^{2}\equiv\pm 1(\pmod{n})$ or $\displaystyle n=10$ and $\displaystyle k=2$

The well known Petersen graph $\displaystyle GP(5,2)$ is the smallest vertex-transitive graph which is not a Cayley graph[3]. It has many interesting properties and can be taken as counter examples for many conjectures[4, 5]. The generalized Petersen graphs $\displaystyle GP(n,1)$ are prisms, isomorphic to the Cartesian product $\displaystyle C_{n}\square K_{2}$. It can be easily seen that for even values of $\displaystyle n$ i.e, $\displaystyle n=2k$, $\displaystyle GP(2k,2)$ is planar for each $\displaystyle k$. Again Robertson[6] has shown that $\displaystyle GP(n,2)$ is Hamiltonian unless $\displaystyle n\equiv 5\pmod{6}$. A set of vertices of the form $\displaystyle\{v_{i},u_{i},u_{i+1},\ldots,u_{i+n},v_{i+n}\}$ in a generalized Petersen graph is called an n-wicket(or simply a wicket) [7].

The number of shortest paths or geodesics between two vertices $\displaystyle u$ and $\displaystyle v$ in a graph will be denoted by $\displaystyle\sigma(u,v)$.

2 Generalized Petersen graph $\displaystyle GP(n,2)$

The graph $\displaystyle GP(n,2)$ is defined for $\displaystyle n\geq 5$. It contains either one or two inner cycles according as $\displaystyle n$ is odd or even (See Fig.1). If $\displaystyle n$ is odd, the inner cycle contains even vertices followed by odd vertices and when $\displaystyle n$ is even there are two inner cycles - the cycle of even vertices and the cycle of odd vertices each having $\displaystyle n/2$ vertices. (See Fig3). Two inner vertices $\displaystyle v_{i}$ and $\displaystyle v_{j}$ are consecutive if $\displaystyle|i-j|=1$ or $\displaystyle|i-j|=-1\mod(n)$ and adjacent if $\displaystyle|i-j|=2$ or $\displaystyle|i-j|=-2\mod(n)$. If $\displaystyle n$ is odd, say $\displaystyle n=2k+1$, then for each vertex $\displaystyle u_{i}$ there are two eccentric vertices $\displaystyle u_{i\pm k}$ in the outer cycle as diametric vertices. The vertices on either side of $\displaystyle u_{i}$ are identical with respect to the metric. When $\displaystyle k$ is increased by one, the eccentric pair advances one more distance away from $\displaystyle u_{i}$. If $\displaystyle n$ is even, say $\displaystyle n=2k$, then these diametric vertices coincide to a single vertex.

$\displaystyle GP(n,2)$ when $\displaystyle n$ is odd can be viewed as a Möbius strip with outer vertices lying in the middle and inner vertices lying on the border. If $\displaystyle n=2k+1$, it is easy to see that there are $\displaystyle k$ odd vertices lying on the upper ends and $\displaystyle k+1$ even vertices lying on the lower ends of the strip (See Fig2). Moving along the middle of the Möbius strip, the outer vertices comes in a regular manner as $\displaystyle u_{0},u_{1},u_{2},\ldots,u_{2k}$ and along the border of the strip, odd vertices follow the even vertices. One movement along the edge in the border of the strip is equivalent to two movements along the edges in the middle. Hence shortest paths prefer edges along the border when the number of vertices increases.

3 Geodesics in $\displaystyle GP(n,2)$

3.1 Geodesics in $\displaystyle GP(n,2)$ when $\displaystyle n$ is odd

3.1.1 Number of geodesics between a pair of vertices in the outer cycle of $\displaystyle GP(n,2)$ when $\displaystyle n$ is odd

In $\displaystyle GP(n,2)$ where $\displaystyle n=2k+1$, by symmetry we consider the vertices $\displaystyle u_{0}$ and $\displaystyle u_{r}$ where $\displaystyle 1\leq r\leq k,\;k\geq 2$ and any shortest path joining them may be denoted by $\displaystyle P(u_{0},u_{r})$.

Lemma 3.1**.**

In $\displaystyle GP(2k+1,2),\,k\geq 2$ for the vertices $\displaystyle u_{0}$ and $\displaystyle u_{r}$ in the outer cycle, $\displaystyle 1\leq r\leq k$, there is always a geodesic joining them contained in the outer cycle for $\displaystyle r\leq 5$.

Proof.

By symmetry, we consider $\displaystyle r$ for $\displaystyle 1\leq r\leq k$. It is obvious for $\displaystyle r=1$ since $\displaystyle(u_{0},u_{1})$ makes an outer edge. For $\displaystyle r=2$, the only geodesic is the one joining $\displaystyle\{u_{0},u_{1},u_{2}\}$ since any path intersecting the inner cycle contains two spokes and atleast one inner edge. For $\displaystyle r=3$, the only geodesic is the one joining $\displaystyle\{u_{0},u_{1},u_{2},u_{3}\}$ since the inner vertices $\displaystyle v_{0}$ and $\displaystyle v_{3}$ are non-adjacent, they make a path of minimum length 4 joining either $\displaystyle\{u_{0},u_{1},v_{1},v_{3},u_{3}\}$, $\displaystyle\{u_{0},v_{0},v_{2},u_{2},u_{3}\}$ or $\displaystyle\{u_{0},v_{0},v_{5},v_{3},u_{3}\}$ in the case $\displaystyle k=3$. For $\displaystyle r=4$ there are two geodesics of length 4, one joining $\displaystyle\{u_{0},u_{1},u_{2},u_{3},u_{4}\}$ and the other joining $\displaystyle\{u_{0},v_{0},v_{2},v_{4},u_{4}\}$. For $\displaystyle r=5(<k)$, there are three geodesics joining $\displaystyle\{u_{0},u_{1},u_{2},u_{3},u_{4},u_{5}\}$, $\displaystyle\{u_{0},u_{1},v_{1},v_{3},v_{5},u_{5}\}$ and $\displaystyle\{u_{0},v_{0},v_{2},v_{4},u_{4},u_{5}\}$ each of length 5. When $\displaystyle r=5(=k)$, $\displaystyle u_{0}$ and $\displaystyle u_{5}$ become the extreme vertices and hence there are four geodesics between $\displaystyle u_{0}$ and $\displaystyle u_{5}$ including the one passing through the inner cycle in the reverse direction. ∎

Lemma 3.2**.**

In $\displaystyle GP(2k+1,2)$, for even $\displaystyle r\leq k$, there is a unique geodesic joining $\displaystyle u_{0}$ and $\displaystyle u_{r}$ for $\displaystyle r>5$ and it passes through the spokes at $\displaystyle u_{0}$ and $\displaystyle u_{r}$ and the inner vertices lying between $\displaystyle v_{0}$ and $\displaystyle v_{r}$.

Proof.

Consider $\displaystyle u_{r}$ where $\displaystyle r$ is even and $\displaystyle r>5$. Let $\displaystyle P(u_{0},u_{r})$ be any geodesic joining $\displaystyle u_{0}$ and $\displaystyle u_{r}$ contained in the outer cycle. If $\displaystyle P(u_{0},u_{r})$ is contained in the outer cycle, its length becomes $\displaystyle r$. Since $\displaystyle r$ is even, $\displaystyle v_{0}$ and $\displaystyle v_{r}$ are even vertices and they lie on a unique shortest path of length $\displaystyle r/2$ contained in the inner cycle. Considering the spokes at $\displaystyle v_{0}$ and $\displaystyle v_{r}$, the length of the path $\displaystyle P(u_{0},u_{r})$ becomes $\displaystyle r/2+2<r$ for $\displaystyle r>5$. Thus $\displaystyle P(u_{0},u_{r})$ passes through the inner vertices and the spokes at $\displaystyle u_{0}$ and $\displaystyle u_{r}$ for $\displaystyle r>5$. ∎

Lemma 3.3**.**

In $\displaystyle GP(2k+1,2)$, for odd $\displaystyle r$, there are two geodesics between $\displaystyle u_{0}$ and $\displaystyle u_{r}$ for $\displaystyle 5<r<k$ and three for $\displaystyle r=k$ all passing through the inner vertices.

Proof.

When $\displaystyle r$ is odd and $\displaystyle 5<r<k$, it can be easily seen that $\displaystyle P(u_{0},u_{r})$ contains exactly two spokes either at $\displaystyle u_{0}$ and $\displaystyle u_{r-1}$ or at $\displaystyle u_{1}$ and $\displaystyle u_{r}$ having length $\displaystyle\frac{r-1}{2}+3$. Otherwise $\displaystyle r<\frac{r-1}{2}+3$, a contradiction. When $\displaystyle r$ is odd, the distance between $\displaystyle v_{0}$ and $\displaystyle v_{r}$ in the inner cycle is $\displaystyle k-\frac{r-1}{2}$. When $\displaystyle r=k$, including the spokes at $\displaystyle u_{0}$ and $\displaystyle u_{r}$, the distance becomes $\displaystyle\frac{r-1}{2}+3$. Thus there are three geodesics between $\displaystyle u_{0}$ and $\displaystyle u_{r}$ when $\displaystyle r=k$. ∎

Proposition 3.1**.**

In $\displaystyle GP(2k+1,2)$, $\displaystyle k\geq 2$, for the vertices $\displaystyle u_{0}$ and $\displaystyle u_{r}$ in the outer cycle, $\displaystyle 1\leq r\leq k$, there is no geodesic joining them contained in the outer cycle for $\displaystyle r>5$.

Theorem 3.1**.**

*If $\displaystyle u_{i}$ and $\displaystyle u_{j}$ are any two distinct vertices in the outer cycle of $\displaystyle GP(2k+1,2)$ where $\displaystyle|i-j|=r\leq k$, then the number of geodesics $\displaystyle\sigma(u_{i},u_{j})$ between $\displaystyle u_{i}$ and $\displaystyle u_{j}$ is given by

[TABLE]

Proof.

By symmetry, we consider $\displaystyle r\leq k$. When $\displaystyle n$ is odd, there exists only one inner cycle in $\displaystyle GP(n,2)$. There is a unique geodesic between $\displaystyle u_{i}$ and $\displaystyle u_{i+r}$ for $\displaystyle r=1,2,3$ lying on the outer cycle (Lemma 6.2) and two geodesics joining $\displaystyle u_{i}$ and $\displaystyle u_{i+4}$ namely,

$\displaystyle\{u_{i},u_{i+1},u_{i+2},u_{i+3},u_{i+4}\}$ and $\displaystyle\{u_{i},v_{i},v_{i+2},v_{i+4},u_{i+4}\}$

When $\displaystyle r=5(=k)$, the vertices $\displaystyle u_{i}$ and $\displaystyle u_{i+5}$ becomes diametric pair on the outer cycle and hence there are four geodesics joining them namely, $\displaystyle\{u_{i},u_{i+1},u_{i+2},u_{i+3},u_{i+4},u_{i+5}\}$ lying on the outer cycle and $\displaystyle\{u_{i},u_{i+1},v_{i+1},v_{i+3},v_{i+5},u_{i+5}\}$, $\displaystyle\{u_{i},v_{i},v_{i+2}$, $\displaystyle v_{i+4},u_{i+4},u_{i+5}\}$, through inner cycle in the forward direction and $\displaystyle\{u_{i},v_{i},v_{i-2},v_{i-4},v_{i-6},u_{i-6}(=u_{i+5})\}$ in the reverse direction.

But when $\displaystyle r<k$, the path in the reverse direction does not become a geodesic. Therefore there are only three geodesics between $\displaystyle u_{i}$ and $\displaystyle u_{i+5}$ when $\displaystyle r<k$.

When $\displaystyle r>5$, all geodesics pass through the inner cycle and no geodesic lies entirely on the outer cycle.

When $\displaystyle r(>5)$ is even, $\displaystyle u_{i}$ and $\displaystyle u_{i+r}$ have the same parity and hence there exists only one geodesic joining $\displaystyle u_{i}$ and $\displaystyle u_{i+r}$ namely,

[TABLE]

When $\displaystyle r$ is odd and $\displaystyle 5<r<k$, there are two geodesics namely,

$\displaystyle\{u_{i},u_{i+1},v_{i+1},v_{i+3},\ldots,v_{i+r},u_{i+r}\}$ and $\displaystyle\{u_{i},v_{i},v_{i+2},v_{i+4},\ldots,v_{i+r-1},u_{i+r-1},u_{i+r}\}$.

When $\displaystyle r=k$, there is one more geodesics in the reverse direction i.e,

[TABLE]

∎

3.1.2 Number of geodesics between a pair of vertices in the inner cycle of $\displaystyle GP(n,2)$ when $\displaystyle n$ is odd

Theorem 3.2**.**

*If $\displaystyle v_{i}$ and $\displaystyle v_{j}$ are any two distinct vertices in the inner cycle of $\displaystyle GP(2k+1,2)$ where $\displaystyle|i-j|=r\leq k$, then number of geodesics $\displaystyle\sigma(v_{i},v_{j})$ between $\displaystyle v_{i}$ and $\displaystyle v_{j}$ is given by

[TABLE]

Proof.

When $\displaystyle n$ is odd, there exists only one inner cycle in $\displaystyle GP(n,2)$ containing all even vertices followed by all odd vertices.

When $\displaystyle n=2k+1$, there are $\displaystyle k+1$ even vertices and $\displaystyle k$ odd vertices in the inner cycle. Consider two consecutive inner vertices $\displaystyle v_{i}$ and $\displaystyle v_{i+1}$. Since $\displaystyle v_{i}$ and $\displaystyle v_{i+1}$ are non adjacent inner vertices, there is a unique geodesic between $\displaystyle v_{i}$ and $\displaystyle v_{i+1}$ passing through the 1-wicket $\displaystyle\{v_{i},u_{i},u_{i+1},v_{i+1}\}$ containing two spokes and an outer edge.

When $\displaystyle r$ is even, both $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ have the same parity and therefore there is a unique geodesic $\displaystyle P(v_{i},v_{i+r})$ of length $\displaystyle r/2$ namely $\displaystyle\{v_{i},v_{i+2},v_{i+4},\ldots,v_{i+r}\}$.

When $\displaystyle r$ is odd, $\displaystyle r<k-2$; $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ have opposite parity and a geodesic from $\displaystyle v_{i}$ to $\displaystyle v_{i+r}$ passes through the 1-wicket at any one of the consecutive pairs

$\displaystyle(v_{i},v_{i+1}),(v_{i+2},v_{i+3}),\ldots,(v_{i+r-1},v_{i+r})$ and thus there exist $\displaystyle\frac{r+1}{2}$ geodesics joining $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ having length $\displaystyle\frac{r+1}{2}+3$. When $\displaystyle r=k-2$, the pair $\displaystyle(v_{i},v_{i+r})$ lies sufficiently apart so that there is one more geodesic in the reverse direction. When $\displaystyle r>k-2$, the geodesic in the reverse direction alone exists. ∎

3.1.3 Number of geodesics between a pair of vertices in the outer and inner cycle of $\displaystyle GP(n,2)$ when $\displaystyle n$ is odd

Theorem 3.3**.**

*If $\displaystyle u_{i}$ and $\displaystyle v_{j}$ are any two vertices in the outer and inner cycles respectively of $\displaystyle GP(2k+1,2)$ where $\displaystyle|i-j|=r\leq k$, then the number of geodesics $\displaystyle\sigma(u_{i},v_{j})$ between $\displaystyle u_{i}$ and $\displaystyle v_{j}$ is given by

[TABLE]

Proof.

When $\displaystyle n$ is odd, there exists only one inner cycle in $\displaystyle GP(n,2)$ containing all even vertices followed by all odd vertices. Consider $\displaystyle u_{i}$ and $\displaystyle v_{i+r}$. If both vertices are either even or odd, $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ are also the same and there is a unique geodesic of length $\displaystyle r/2+1$ passing through the spoke $\displaystyle(u_{i},v_{i})$ joining $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ along the inner cycle . Otherwise the geodesic passes through the outer edge $\displaystyle(u_{i},u_{i+1})$, the spoke $\displaystyle(u_{i+1},v_{i+1})$ and joins $\displaystyle v_{i+1}$ to $\displaystyle v_{i+r}$ along the inner cycle. It is of length $\displaystyle\frac{r-1}{2}+2$. In the extreme case $\displaystyle r=k$, there is one more geodesic in the reverse direction passing through the spoke $\displaystyle(u_{i},v_{i})$. ∎

3.2 Geodesics in $\displaystyle GP(n,2)$ when $\displaystyle n$ is even

3.2.1 Number of geodesics between a pair of vertices in the outer cycle of $\displaystyle GP(n,2)$ when $\displaystyle n$ is even

Theorem 3.4**.**

*If $\displaystyle u_{i}$ and $\displaystyle u_{j}$ are any two distinct vertices in the outer cycle of $\displaystyle GP(2k,2)$, $\displaystyle k\geq 3$ where $\displaystyle|i-j|=r\leq k$, then the number of geodesics $\displaystyle\sigma(u_{i},u_{j})$ between $\displaystyle u_{i}$ and $\displaystyle u_{j}$ is given by

[TABLE]

Proof.

Since $\displaystyle n$ is even, there are two inner cycles - the cycle of even vertices and the cycle of odd vertices. When $\displaystyle r=1,2,3;\;r<k$, there is a unique geodesic between $\displaystyle u_{i}$ and $\displaystyle u_{i+r}$ lying on the outer cycle and in the extreme case, i.e, when $\displaystyle r=3$, $\displaystyle r=k$, the outer cycle itself makes two geodesics on either sides.

When $\displaystyle r=4$, $\displaystyle r<k$, there are two geodesics of length $\displaystyle 4$, one over the outer cycle and the other over the inner cycle of even or odd vertices according as $\displaystyle i$ is even or odd by means of the two spokes at the given vertices. In its extreme case there are two more geodesics passing over the outer cycle and the inner cycle in the reverse direction.

When $\displaystyle r=5,\;r<k$, there are $\displaystyle 3$ geodesics of length $\displaystyle 5$. One lying on the outer cycle and the others passing through the spokes either at $\displaystyle u_{i+1}$ and $\displaystyle u_{i+5}$ or at $\displaystyle u_{i}$ and $\displaystyle u_{i+4}$. In its extreme case when $\displaystyle r=k$, there are $\displaystyle 3$ more geodesics passing over the outer and inner cycles in the reverse direction.

When $\displaystyle r=6,8,\ldots;\;r<k$, there is no geodesic passing over the outer cycle. Since $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ lie on the same inner cycle, there is a geodesic joining them of length $\displaystyle r/2$. Therefore there is a unique geodesic of length $\displaystyle r/2+2$ joining $\displaystyle u_{i}$ and $\displaystyle u_{i+r}$ when $\displaystyle r=6,8,\ldots;\;r<k$. In the extreme case there is one more geodesic passing over the inner cycle in the reverse direction.

When $\displaystyle r=7,9,\ldots;\;r<k$, $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ lie on different inner cycles and therefore there are two geodesics one joining the vertices $\displaystyle\{u_{i},u_{i+1},v_{i+1},v_{i+3},\ldots,v_{i+r},u_{i+r}\}$ and the other joining the vertices $\displaystyle\{u_{i},v_{i},v_{i+2},v_{i+4},\ldots,v_{i+r-1},u_{i+r-1},u_{i+r}\}$. In the extreme case, reversing the direction over the outer and inner cycles, there lie two more geodesics. ∎

3.2.2 Number of geodesics between a pair of vertices in the inner cycle of $\displaystyle GP(n,2)$ when $\displaystyle n$ is even

Theorem 3.5**.**

If $\displaystyle v_{i}$ and $\displaystyle v_{j}$ are any two distinct vertices in the inner cycle of $\displaystyle GP(2k,2)$, $\displaystyle k\geq 3$ where $\displaystyle|i-j|=r\leq k$, then the number of geodesics $\displaystyle\sigma(v_{i},v_{j})$ between $\displaystyle v_{i}$ and $\displaystyle v_{j}$ is given by

[TABLE]

Proof.

When $\displaystyle r$ is even and $\displaystyle r<k$ both the vertices $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ lie on the same inner cycle. Therefore, there exists a unique geodesic lying on the same inner cycle. In the extreme case, there is one more geodesic on the reverse side of the inner cycle.

When $\displaystyle r$ is odd and $\displaystyle r<k$ the vertices lie on different inner cycles and hence joined by a wicket containing two spokes and an outer edge at any vertex $\displaystyle v_{i},v_{i+2},\ldots,v_{i+r-1}$. Hence there are $\displaystyle\frac{r+1}{2}$ geodesics. In the extreme case the above method can be repeated along the reverse side of the inner cycle. ∎

3.2.3 Number of geodesics between a pair of vertices in the outer and inner cycle of $\displaystyle GP(n,2)$ when $\displaystyle n$ is even

Theorem 3.6**.**

If $\displaystyle u_{i}$ and $\displaystyle v_{j}$ are any two vertices in the outer and inner cycles respectively of $\displaystyle GP(2k,2)$, $\displaystyle k\geq 3$ where $\displaystyle|i-j|=r\leq k$, then the number of geodesics $\displaystyle\sigma(u_{i},v_{j})$ between $\displaystyle u_{i}$ and $\displaystyle v_{j}$ is given by

[TABLE]

Proof.

When $\displaystyle r$ is even and $\displaystyle r<k$, both $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ are either even or odd and hence there is a unique geodesic joining the spoke $\displaystyle(u_{i},v_{i})$ to $\displaystyle v_{i+r}$. In the extreme case there is one more geodesic along the opposite side of the same inner cycle.

When $\displaystyle r$ is odd and $\displaystyle r<k$, $\displaystyle u_{i}$ and $\displaystyle v_{i+r}$ lie on different inner cycles. So there is a unique geodesic passing through $\displaystyle\{u_{i},u_{i+1},v_{i+1}\}$. In the extreme case there is one more geodesic lying in the opposite direction. ∎

It can be seen that in $\displaystyle GP(n,2)$ when $\displaystyle n$ is even, the outer and inner cycles are even and hence the number of geodesics in each of the extreme cases doubles.

4 Distance between two vertices in $\displaystyle GP(n,2)$

Theorem 4.1**.**

The distance between a pair of vertices $\displaystyle(u_{i},u_{j})$, $\displaystyle(v_{i},v_{j})$ and $\displaystyle(u_{i},v_{j})$ in $\displaystyle GP(n,2)$ is given by

[TABLE]

[TABLE]

[TABLE]

where $\displaystyle|i-j|=r\leq\left\lceil\frac{n-1}{2}\right\rceil$

Proof.

For $\displaystyle r\leq\left\lceil\frac{n-1}{2}\right\rceil$, from any vertex $\displaystyle u_{i}$ to $\displaystyle u_{i+r}$ (or $\displaystyle u_{i-r}$) for $\displaystyle r\leq 5$ there is a geodesic of length $\displaystyle r$ lying on the outer cycle. When $\displaystyle r$ is even, $\displaystyle r>5$ there is a geodesic of length $\displaystyle\frac{r}{2}+2$ joining $\displaystyle u_{i}$ and $\displaystyle u_{i+r}$ passing through $\displaystyle\{v_{i},v_{i+2},\ldots,v_{i+r}\}$ and the two spokes at $\displaystyle u_{i}$ and $\displaystyle u_{i+r}$.

When $\displaystyle r$ is odd, there is a geodesic of length $\displaystyle\frac{r-1}{2}+3$ joining $\displaystyle u_{i}$ and $\displaystyle u_{i+r}$ passing through $\displaystyle\{u_{i+1},v_{i+1},v_{i+3},\ldots,v_{i+r-1},u_{i+r-1}\}$.

If both $\displaystyle v_{i}$ and $\displaystyle v_{i+r}$ are either odd or even there is a unique geodesic of length $\displaystyle r/2$ joining them along the inner cycle, if not so, there is a geodesic $\displaystyle\{v_{i},u_{i},u_{i+1},v_{i+1},v_{i+3},\ldots,v_{i+r}\}$ of length $\displaystyle\frac{r-1}{2}+3$.

If $\displaystyle r$ is even, $\displaystyle u_{i}$ and $\displaystyle v_{i}$ both are either odd or even and there is a geodesic of length $\displaystyle\frac{r}{2}+1$, otherwise there is a geodesic of length $\displaystyle\frac{r-1}{2}+2$ including a spoke and an outer edge. ∎

Corollary 4.1**.**

*The diameter of $\displaystyle GP(n,2)$ when $\displaystyle n\geq 8,$ is given by

[TABLE]

5 Betweenness centrality

Betweenness centrality[8, 9, 10, 11] measures the relative importance of vertex in a graph. A vertex is said to be central if it can effectively monitor the communication between vertices. It describes how a vertex acts as a bridge among all the pairs of vertices. Betweenness centrality of a vertex $\displaystyle x$ is the sum of the fraction of all-pairs shortest paths that pass through $\displaystyle x$. It has wide applications in the analysis of networks[12, 13, 14, 15, 16].

Betweenness centrality of a vertex in a graph

Definition 1**.**

[17]**. Let $\displaystyle G$ be a graph and $\displaystyle x\in V(G)$, then the betweenness centrality of $\displaystyle x$ in $\displaystyle G$ denoted by $\displaystyle B_{G}(x)$ or simply $\displaystyle B(x)$ may be defined as

[TABLE]

where $\displaystyle\sigma_{st}(x)$ denotes the number of shortest $\displaystyle s$-$\displaystyle t$ paths in $\displaystyle G$ passing through $\displaystyle x$ and $\displaystyle\sigma_{st}$, the number of shortest $\displaystyle s$-$\displaystyle t$ paths in $\displaystyle G$. The ratio $\displaystyle\frac{\sigma_{st}(x)}{\sigma_{st}}$ is called pair dependency of $\displaystyle\{s,t\}$ on $\displaystyle x$, denoted by $\displaystyle\delta_{G}(s,t,x)$.

We may now define the following terms related to betweenness centrality.

Definition 2**.**

*Let $\displaystyle G$ be a graph and $\displaystyle H$ a subgraph of $\displaystyle G$. Let $\displaystyle x\in V(H)$, then the betweenness centrality of $\displaystyle x$ in $\displaystyle H$ denoted by $\displaystyle B_{H}(x)$ may be defined as

[TABLE]

where $\displaystyle\sigma^{H}_{st}(x)$ and $\displaystyle\sigma^{H}_{st}$ denotes the number of shortest $\displaystyle s$-$\displaystyle t$ paths passing through $\displaystyle x$ and the number of shortest $\displaystyle s$-$\displaystyle t$ paths respectively, being their vertices in $\displaystyle H$.

Definition 3**.**

*Let $\displaystyle G$ be a graph and $\displaystyle H$ a subgraph of $\displaystyle G$. Let $\displaystyle x\in V(G)$, then the betweenness centrality of $\displaystyle x$ induced by $\displaystyle H$ denoted by $\displaystyle B(x,H)$ may be defined as

[TABLE]

Definition 4**.**

*Let $\displaystyle G$ be a graph and $\displaystyle S$ a subset of $\displaystyle V(G)$. Let $\displaystyle x\in V(G)$, then the betweenness centrality of $\displaystyle x$ induced by $\displaystyle S$ denoted by $\displaystyle B(x,S)$ may be defined as

[TABLE]

Definition 5**.**

Let $\displaystyle G$ be a graph and $\displaystyle x,x_{0}\in V(G)$, then the betweenness centrality of $\displaystyle x$ induced by $\displaystyle x_{0}$ in $\displaystyle G$, denoted by $\displaystyle B_{G}(x,x_{0})$ or simply $\displaystyle B(x,x_{0})$ is defined by

[TABLE]

It can be easily seen that in any graph $\displaystyle G$, the betweenness centrality induced by a vertex on its extreme vertex or an end vertex is zero. $\displaystyle B(x_{i},x_{j})=0$ for complete graph $\displaystyle K_{n}$. Let $\displaystyle P_{n}$ be a path on $\displaystyle n$ vertices $\displaystyle\{x_{1},\ldots,x_{n}\}$, then

[TABLE]

If $\displaystyle C_{n}$ is a cycle on $\displaystyle n$ vertices $\displaystyle\{x_{0},\ldots,x_{n-1}\}$, then

if $\displaystyle n$ is even,

[TABLE]

if $\displaystyle n$ is odd,

[TABLE]

By symmetry,

[TABLE]

For a star $\displaystyle S_{n}$ with central vertex $\displaystyle x_{0}$,

[TABLE]

[TABLE]

For a wheel $\displaystyle W_{n},n>5$ with central vertex $\displaystyle x_{0}$,

[TABLE]

[TABLE]

Theorem 5.1**.**

*Let $\displaystyle G$ be a graph and $\displaystyle x_{i}\in V(G)$, then

[TABLE]

Definition 6**.**

*Let $\displaystyle G$ be a graph and $\displaystyle x\in V(G)$. Let $\displaystyle S$, $\displaystyle T$ be two disjoint subsets of $\displaystyle V(G)$, then the betweenness centrality of $\displaystyle x$ induced by $\displaystyle S$ and $\displaystyle T$ denoted by $\displaystyle B(x,S,T)$ may be defined as

[TABLE]

Definition 7**.**

*Let $\displaystyle G$ be a graph and $\displaystyle x\in V(G)$. Let $\displaystyle S$, $\displaystyle T$ be two disjoint subsets of $\displaystyle V(G)$ where $\displaystyle s(\neq x)\in S$ and $\displaystyle t(\neq x)\in T$, then the betweenness centrality of $\displaystyle x$ induced by $\displaystyle S$ and $\displaystyle T$ one against the other denoted by $\displaystyle B(x,S|T)$ may be defined as

[TABLE]

6 Betweenness centrality of a vertex in GP(n,2)

Let us consider the betweenness centrality of $\displaystyle GP(n,2)$, where the vertices lie on two vertex transitive subgraphs namely the outer cycle generated by $\displaystyle U=\{u_{0},u_{1},\ldots,u_{n-1}\}$ and the inner cycle generated by $\displaystyle V=\{v_{0},v_{1},\ldots v_{n-1}\}$.

6.1 Betweenness centrality of an outer vertex in GP(n,2)

Theorem 6.1**.**

*The betweenness centrality of an outer vertex $\displaystyle u$ in $\displaystyle GP(n,2)$ is given by

[TABLE]

Proof.

The betweenness centrality of a vertex in $\displaystyle G$ is the sum of the betweenness centralities induced by $\displaystyle U$, $\displaystyle V$ and $\displaystyle U$ $\displaystyle Vs$ $\displaystyle V$ determined in lemma 6.2 -6.4. ∎

Lemma 6.1**.**

For any vertex $\displaystyle u_{0}$ in the outer cycle of $\displaystyle GP(n,2)$, $\displaystyle n\geq 6$, there are 10 pairs of outer vertices, and for each pair there is a geodesic lying on the outer cycle with $\displaystyle u_{0}$ as an internal vertex. More over, these pairs contribute the value 6.5 for its betweenness centrality.

Proof.

Consider $\displaystyle GP(n,2)$, $\displaystyle n\geq 6$. Now for any vertex $\displaystyle u_{0}\in U$, followed by lemma 6.2, it can be easily seen that there exists a geodesic joining $\displaystyle u_{-1}$ to $\displaystyle u_{r}$ and $\displaystyle u_{-2}$ to $\displaystyle u_{s}$ where $\displaystyle r=1,2,3,4$ and $\displaystyle s=2,3$ lying entirely on the outer cycle, contributing 1, 1, 1/2, 2/3 and 1/2, 1/3 respectively to the betweenness centrality of $\displaystyle u_{0}$. Hence by the symmetry of metric, there are 10 pairs of vertices with total contribution 13/2. ∎

Lemma 6.2**.**

In $\displaystyle GP(n,2)$, the betweenness centrality of an outer vertex $\displaystyle u_{0}$ induced by the outer cycle is given by

[TABLE]

Proof.

Consider an outer vertex $\displaystyle u_{0}\in U$ in $\displaystyle GP(n,2)$ for $\displaystyle n\geq 12$. First we take all possible $\displaystyle U$-$\displaystyle U$ pairs of outer vertices and find their contributions to $\displaystyle B(u_{0})$. By lemma 6.1 the outer cycle contains 10 geodesics passing through $\displaystyle u_{0}$ contributing 13/2. When $\displaystyle n=2k+1$, for even $\displaystyle k$, $\displaystyle k\geq 8$, the pair $\displaystyle(u_{-1},u_{r})$, $\displaystyle r=6,8,\ldots,k-2$ contributes 1/2 for each $\displaystyle r$ and when $\displaystyle k$ is incremented, there is one more pair $\displaystyle(u_{-1},u_{k-1})$ with contribution 1/3. Therefore, by symmetry, the outer pairs contribute the sum $\displaystyle\frac{1}{2}(k+7)$ when $\displaystyle k$ is even and $\displaystyle\frac{1}{6}(3k+22)$ when $\displaystyle k$ is odd. When $\displaystyle n=2k$, for even $\displaystyle k$, $\displaystyle k\geq 8$, the pair $\displaystyle(u_{-1},u_{r})$, $\displaystyle r=6,8,\ldots,k-2$ contributes 1/2 for each $\displaystyle r$ and when $\displaystyle k$ is incremented, there is one more pair $\displaystyle(u_{-1},u_{k-1})$ with contribution 1/4. Hence by symmetry, it can be seen that outer pairs contribute $\displaystyle\frac{1}{2}(k+7)$ for $\displaystyle k$, even or odd. ∎

Lemma 6.3**.**

In $\displaystyle GP(n,2)$, the betweenness centrality of an outer vertex $\displaystyle u_{0}$ induced by the inner cycle is given by

[TABLE]

Proof.

Consider the possible $\displaystyle V$-$\displaystyle V$ pairs of inner vertices. When $\displaystyle n=2k+1$ for even $\displaystyle k$, $\displaystyle k\geq 6$, the pair $\displaystyle(v_{0},v_{r})$ for $\displaystyle r=1,3,\ldots,k-3$ contributes the betweenness centrality $\displaystyle 1/t$ where $\displaystyle t=(r+1)/2$. By the symmetry of the metric, there are 2$\displaystyle t$ similar pairs with a total contribution 2 for each $\displaystyle r$. Since there are $\displaystyle(k-2)/2$ of such $\displaystyle(v_{0},v_{r})$ pairs, the total contribution of $\displaystyle V$-$\displaystyle V$ pairs is $\displaystyle k-2$. When $\displaystyle k$ is incremented, there is one new pair $\displaystyle(v_{0},v_{r})$ where $\displaystyle r={k-2}$ with contribution $\displaystyle 1/t$ where $\displaystyle t=(k+1)/2$ and there are $\displaystyle k-1$ similar pairs. Therefore, by symmetry, inner pairs contribute the sum $\displaystyle\frac{k^{2}-5}{k+1}$. Consider the case $\displaystyle n=2k$, for even $\displaystyle k,\;k\geq 6$ the inner pairs $\displaystyle(v_{0},v_{r})$ for $\displaystyle r=1,3,\ldots,k-1$ contributes $\displaystyle 1/t$ where $\displaystyle t=(r+1)/2$. Because of symmetry, each pair $\displaystyle(v_{0},v_{r})$ belongs to a set of $\displaystyle 2t$ similar pairs giving 2 and these sets contributes a total $\displaystyle k$ to the centrality of $\displaystyle u_{0}$. When $\displaystyle k$ is incremented, the leading pair $\displaystyle(v_{0},v_{r})$ for $\displaystyle r=k$ gives $\displaystyle 1/t$ where $\displaystyle t=(k+1)/2$ and so get the total contribution as $\displaystyle k$. ∎

Lemma 6.4**.**

In $\displaystyle GP(n,2)$, the betweenness centrality of an outer vertex $\displaystyle u_{0}$ induced by the vertices of outer Vs inner cycle is given by

[TABLE]

Proof.

Consider the possible $\displaystyle U$-$\displaystyle V$ pairs. When $\displaystyle n=2k+1$, for even $\displaystyle k$, $\displaystyle k\geq 6$, the pair $\displaystyle(u_{1},v_{-r})$ for $\displaystyle r=0,2,4,\dots,k-2$ contributes 1, When $\displaystyle k$ is incremented, the leading pair $\displaystyle(u_{1},v_{-r})$ makes the contribution 1/2 for $\displaystyle r=k-1$. Therefore by symmetry, in either case the total contribution can be found as $\displaystyle k$. Similar argument is there for $\displaystyle n=2k$. ∎

6.2 Betweenness centrality of an inner vertex in GP(n,2)

Theorem 6.2**.**

*The betweenness centrality of an inner vertex $\displaystyle v$ in $\displaystyle GP(n,2)$ is given by

[TABLE]

Proof.

The betweenness centrality of $\displaystyle v\in V$ is the sum of its betweenness centralities induced by the subsets $\displaystyle U$, $\displaystyle V$ and $\displaystyle U$ $\displaystyle Vs$ $\displaystyle V$ in $\displaystyle G$ determined in lemma 6.5 -6.7. ∎

Lemma 6.5**.**

In $\displaystyle GP(n,2)$, the betweenness centrality of an inner vertex $\displaystyle v_{0}$ induced by the outer cycle is given by

[TABLE]

Proof.

Consider all outer pairs $\displaystyle(u_{i},u_{j})$ such that $\displaystyle v_{0}$ lies on atleast one geodesic joining them. Let $\displaystyle d=d(u_{i},u_{j})$, then clearly $\displaystyle 4\leq d\leq D$, where $\displaystyle D=diam(G)$. Let $\displaystyle B_{d}(v_{0})$ denotes the total contribution of those pairs at a distance $\displaystyle d$ towards the betweenness centrality $\displaystyle B(v_{0})$ of $\displaystyle v_{0}$. Consider $\displaystyle n=2k+1$ where $\displaystyle k=2l,\,l\geq 3$. Now for $\displaystyle d=4$, there exists 3 pairs of 1/2 and for $\displaystyle d=5$, there exists 6 pairs of 1/3 and 4 pairs of 1. For $\displaystyle 6\leq d\leq D$, there exist $\displaystyle 2d-4$ pairs of 1/2 and $\displaystyle d-1$ pairs of 1 giving the sum $\displaystyle 2d-3$. Therefore,

[TABLE]

In the case $\displaystyle n=2k+1$, when $\displaystyle k$ is incremented from odd to even, the leading diametric pairs $\displaystyle(u_{i},u_{j})$ i.e, vertices at a distance $\displaystyle D=(n+9)/4$, contains 2 pairs of 2/3 and $\displaystyle 3(n-3)/4$ pairs of 1/3 and hence there is a contribution of an extra sum $\displaystyle(3n+7)/12$. Therefore,

[TABLE]

Consider the case $\displaystyle n=2k$ where $\displaystyle k=2l,\,l\geq 3$, then $\displaystyle D=(n+8)/4$ and when $\displaystyle d=D$, there are $\displaystyle(3n-4)/4$ pairs of 1/2 and a single pair of 1 giving the sum $\displaystyle(3n+4)/8$. Therefore,

[TABLE]

Consider the case $\displaystyle n=2k$ where $\displaystyle k=2l+1,\,l\geq 3$, then $\displaystyle D=(n+10)/4$ and when $\displaystyle d=D$, there are $\displaystyle(n-2)/2$ pairs of 1/4 and one diametric pair of 2/4 giving the sum $\displaystyle(n+2)/8$. Therefore,

[TABLE]

∎

Lemma 6.6**.**

In $\displaystyle GP(n,2)$, the betweenness centrality of an inner vertex $\displaystyle v_{0}$ induced by the inner cycle is given by

[TABLE]

Proof.

We consider all inner pairs $\displaystyle(v_{i},v_{j})$ such that $\displaystyle v_{0}$ lies on their shortest paths. Let $\displaystyle d=d(v_{i},v_{j})$ and $\displaystyle D=diam(G)$, then clearly $\displaystyle 2\leq d\leq D$. Let $\displaystyle B_{d}(v_{0})$ denotes the total contribution of pairs at a distance $\displaystyle d$ towards the betweenness centrality $\displaystyle B(v_{0})$ of $\displaystyle v_{0}$. Let $\displaystyle n=2k+1$ where $\displaystyle k=2l,\,l\geq 3$, then $\displaystyle D=(n+7)/4$. For $\displaystyle d=2$, there exists a pair contributing 1 and for $\displaystyle 2\leq d\leq D-1$ we have $\displaystyle B_{d}(v_{0})=2d-4$. Therefore,

[TABLE]

In the case $\displaystyle n=2k+1$ where $\displaystyle k=2l+1,Gplay\,l\geq 3$, we have $\displaystyle D=(n+9)/4$ and

[TABLE]

Consider $\displaystyle n=2k$ where $\displaystyle k=2l,\,l\geq 3$. There are two inner $\displaystyle k$-cycles. Since $\displaystyle v_{0}$ lies on a $\displaystyle k$-cycle, subscripted with even numbers, we need not consider the pair $\displaystyle(v_{i},v_{j})$ with odd subscripts $\displaystyle i$ and $\displaystyle j$. The even subscripted pairs $\displaystyle(v_{i},v_{j})$ give the betweenness centrality $\displaystyle(k-2)^{2}/8$. Now for $\displaystyle j=2,4,6,$ etc, the pair $\displaystyle(v_{-1},v_{j})$ gives $\displaystyle 1/2,1/3,\ldots,1/l$; $\displaystyle(v_{-3},v_{j})$ gives $\displaystyle 2/3,2/4,\ldots,2/l$ and finally $\displaystyle(v_{-(k-3)},v_{2})$ gives $\displaystyle(l-1)/l$. Considering vertices of these two inner $\displaystyle k$-cycles. $\displaystyle B_{d}(v_{0})=d-3$ for $\displaystyle 4\leq d\leq D$ where $\displaystyle D=k/2+2$. Therefore,

[TABLE]

In the case $\displaystyle n=2k$ where $\displaystyle k=2l+1,\,l\geq 3$, the vertices of the same cycle contribute $\displaystyle(k-1)(k-3)/8$ to $\displaystyle v_{0}$ and for the vertices of the different cycles $\displaystyle B_{d}(v_{0})=d-3$ for $\displaystyle 4\leq d\leq D-1$ and $\displaystyle B_{D}(v_{0})=(k-1)/4$ where $\displaystyle D=(k+5)/2$. Now

[TABLE]

∎

Lemma 6.7**.**

In $\displaystyle GP(n,2)$, the betweenness centrality of an inner vertex $\displaystyle v_{0}$ induced by the vertices of outer Vs inner cycle is given by

[TABLE]

Proof.

In $\displaystyle GP(n,2)$ where $\displaystyle n=2k+1$, $\displaystyle k=2l$, $\displaystyle l\geq 3$, consider the possible $\displaystyle u$-$\displaystyle v$ pairs determining the value of $\displaystyle B(v_{0})$. From $\displaystyle u_{0}$ there are $\displaystyle k/2$ geodesics passing through $\displaystyle v_{0}$ to either sides. For an even index $\displaystyle i\leq k-2$, from $\displaystyle u_{i}$ and $\displaystyle u_{i-1}$ there is an equal number of geodesics, i.e, $\displaystyle(k-i)/2$ passing through $\displaystyle v_{0}$ and therefore, by symmetry, there exists $\displaystyle k^{2}/2$, $\displaystyle u$-$\displaystyle v$ geodesics through $\displaystyle v_{0}$. Hence

[TABLE]

In the case of odd $\displaystyle k$, i.e, $\displaystyle k=2l+1$, $\displaystyle l\geq 3$ there is one more geodesic of 1/2 from each outer vertex and the expression $\displaystyle k^{2}/2$ turns to be $\displaystyle(k-1)^{2}/2+k$ Hence

[TABLE]

Consider the case $\displaystyle n=2k,\,k\geq 7$. From $\displaystyle u_{0}$ to any inner even vertex $\displaystyle v_{j}$, $\displaystyle(j\neq 0)$ there is a unique geodesic passing through $\displaystyle v_{0}$, and from $\displaystyle u_{i}$ to $\displaystyle v_{j}$ there are two if $\displaystyle v_{i}$ and $\displaystyle v_{j}$ are diametric pairs of the inner even cycle. Thus $\displaystyle k(k-1)/2$ stands for $\displaystyle B(v_{0})$. Hence

[TABLE]

∎

7 Conclusion

Here we found the number of geodesics between two vertices and the betweenness centrality of $\displaystyle GP(n,k)$ for $\displaystyle k=2$. The study of geodesics is extremely important in the context of interconnection networks and it has wide applications in routing , fault-tolerance, time delays and in the calculation of many centrality measures. This work may be extended for any $\displaystyle k<n/2$.