The Reliability of Cayley Graphs Generated by Transposition Trees Based on Edge Failures

Abstract

Extra edge connectivity is an important parameter for measuring the reliability of interconnection networks. Given a graph G and a non-negative integer h, the h-extra edge connectivity of G, denoted by ${\lambda }_h\left(G\right)$, is the minimum cardinality of a set of edges in G (if it exists) whose deletion disconnects G such that each remaining component contains at least $h+1$ vertices. In this paper, we obtain the h-extra edge connectivity of Cayley graphs generated by transposition trees for $h\le 5$. As byproducts, we derive the h-extra edge connectivity of the star graph $S_n$ and the bubble-sort graph $B_n$ for $h\le 5$.

1. Introduction

The interconnection network plays a crucial role in multiprocessor systems. It is rarely possible to make sure that a multiprocessor system is fault-free at all times, so reliability assessment is particularly important in the process of designing and maintaining multiprocessor systems. To analyze multiprocessor system properties, the interconnection network can be modeled by a graph $G=(V,E)$, where V is the set of processors and E is the set of communication links in the network.

The edge connectivity of a connected graph G, denoted by $\lambda \left(G\right)$, is the minimum number of edges whose removal disconnects G. In general, a higher $\lambda \left(G\right)$ indicates greater network reliability, making it a traditional measurement for reliability evaluation. However, this measurement relies on the assumption that any subset of system components is likely to be faulty simultaneously, which is impossible in real applications.

To compensate for this shortcoming, many researchers have made improvements. By imposing additional constraints on disconnected components, Harary [1] introduced the concept of conditional connectivity. Following this idea, Esfahanian [2] proposed restricted edge connectivity, which is defined as the minimum size of the edges whose removal disconnects the interconnection network while restricting the candidate edge sets to some subsets of the system. This concept was further generalized by Latifi et al. [3] to h-edge connectivity, which requires that each vertex of every remaining component has at least h fault-free neighbors. For a connected graph G, an edge set F is called an edge-cut of G if $G-F$ is disconnected. An h-edge-cut of G is an edge-cutF of G for which every component of $G-F$ has a minimum degree of at least h. The h-edge connectivity of G, denoted by ${\lambda }^{\left(h\right)}\left(G\right)$, is the minimum cardinality of an h-edge-cut (when such a cut exists). Analogously, we can define h-connectivity when F is a vertex cut set. These refined connectivity parameters provide more accurate measures for the fault tolerance of the networks, as they avoid the unrealistic assumption that arbitrary subsets of components may fail simultaneously.

Meanwhile, Fábrega and Fiol [4] proposed extra edge connectivity in 1996. For a non-negative integer h, an h-extra edge-cut of a connected graph G is defined as an edge set F of graph G whose deletion disconnects G, with each resulting component of $G-F$ containing at least $h+1$ vertices. The h-extra edge connectivity of a connected graph G, denoted by ${\lambda }_h\left(G\right)$, is the minimum cardinality of such edge cuts of G. For a graph G, it is obvious that ${\lambda }_0\left(G\right)=\lambda \left(G\right)$, ${\lambda }_1\left(G\right)={\lambda }^{\left(1\right)}\left(G\right)$, and ${\lambda }_h\left(G\right)\le {\lambda }^{\left(h\right)}\left(G\right)$ for $h\ge 2$. As a generalization of traditional edge connectivity, extra edge connectivity can better measure the fault tolerance of interconnection networks. Consequently, this parameter has attracted significant research attention, with numerous studies investigating the extra edge connectivity for various interconnection network topologies [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].

Given a finite group $\Gamma$ and a nonempty subset S of $\Gamma$, the Cayley graph $Cay(\Gamma ,S)$ of $\Gamma$ relative to S is defined as the graph with vertex set $\Gamma$ and an edge set consisting of all ordered pairs $(x,y)$ such that $xs=y$ for some $s\in S$. If the set S is inverse-closed (i.e., $s\in S$ leads to $s^{-1}\in S$), $Cay(\Gamma ,S)$ becomes an undirected graph. Cayley graphs form an important class of interconnection networks, encompassing some well-known networks including hypercubes $Q_n$, k-ary n-cubes $Q_n^k$, star graphs $S_n$, and bubble-sort graphs $B_n$. Due to their symmetric properties and applications in parallel computing, the Cayley graphs of groups have been extensively studied (see [21]). For example, the Cayley graph $Q_n^k$ has been utilized to build the architecture of a known supercomputer, the Fujitsu Tofu interconnect D [22].

For a given integer $n\ge 2$, let $〈n〉$ denote the set $\left\{1,2,\dots ,n\right\}$ and $Sym\left(n\right)$ denote the group of permutations on $〈n〉$. Let $\mathcal{T}$ be a subset of transpositions in $Sym\left(n\right)$ and $G\left(\mathcal{T}\right)$ be a graph on $〈n〉$ with edge $(i,j)\in G\left(\mathcal{T}\right)$ if and only if transposition $\left(ij\right)\in \mathcal{T}$. If $G\left(\mathcal{T}\right)$ is a tree, then $G\left(\mathcal{T}\right)$ is called a transposition-generating tree and $Cay\left(Sym\right(n),\mathcal{T})$ is a Cayley graph generated by a transposition tree. If $G\left(\mathcal{T}\right)$ is isomorphic to the star $K_{1,n-1}$, $Cay\left(Sym\right(n),\mathcal{T})$ is the star graph $S_n$; if not, it will be denoted by $G_n$. Note that the graph $K_{1,n-1}$ is called a star, instead of a star graph. For $n\ge 4$, the star graph $S_n$ has a girth of 6, while $G_n$ has a girth of 4, where the girth is the length of the shortest cycle (see [23]). Various fault tolerances and reliability in this class of Cayley graph have received much attention, such as Hamiltonian laceability [24], linearly many faults [25], component connectivity [26], error detecting [27], two-edge connectivity [28], and so on. Yang et al. [28] showed that ${\lambda }_2\left(S_n\right)={\lambda }_2\left(G_n\right)=3n-7$ for $n\ge 5$.

In this paper, we extend these results and determine the exact values of the h-extra edge connectivity of Cayley graphs generated by transposition trees for an h less than 6. For an integer $n\ge 4$, if $h\le 4$ and $n\ne h$, then ${\lambda }_h\left(S_n\right)=(h+1)n-3h-1$; if $h=5$ or $(n,h)=(4,4)$, then ${\lambda }_h\left(S_n\right)=6n-18$. For an integer $n\ge 5$, if $h\le 2$, then ${\lambda }_h\left(G_n\right)=(h+1)n-3h-1$; ${\lambda }_3\left(G_n\right)=4n-12$; ${\lambda }_4\left(G_n\right)=5n-15$; and ${\lambda }_5\left(G_n\right)=6n-20$.

For a vertex subset X and a vertex v in a graph G, $d_X\left(v\right)$ denotes the number of neighbors of v in X, $G\left[X\right]$ denotes the subgraph of G induced by X, $\delta \left(X\right)$ denotes the minimum degree of $G\left[X\right]$, and $E_G\left(X\right)$ denotes the set of edges between X and $V\left(G\right)\setminus Y$. A path with h vertices is called an h-path, denoted by $P_h$. A cycle with h vertices is called an h-cycle, denoted by $C_h$. When it is clear from the context, we do not distinguish between a vertex set and its induced subgraph. Undefined graph terminology follows Xu [29].

The remainder of this paper is organized as follows: Section 2 determines the h-extra edge connectivity of the star graph $S_n$ for $h\le 5$. Section 3 establishes the h-extra edge connectivity of $G_n$ for $h\le 5$. Conclusions are given in Section 4.

2. The Extra Edge Connectivity of Star Graphs

Using the group theoretical model, Akers and Krishnamurthy [21] proposed the star graph $S_n$, which is one of the most promising interconnection networks. Compared to alternative networks, it has a superior degree and diameter while maintaining excellent hierarchical and symmetric characteristics. The n-dimensional star graph $S_n$ is a Cayley graph $Cay\left(Sym\right(n),\mathcal{T})$, where the set of transpositions is $\mathcal{T}=\left\{\right(12),(13),\dots ,(1n\left)\right\}$. That is, $G\left(\mathcal{T}\right)$ is isomorphic to a star $K_{1,n-1}$ (see Figure 1). The following definition gives an easy way to understand the star graph $S_n$:

Definition 1. 

An n-dimensional star graph $S_n$ is a graph with $n!$ vertices labeled with permutations in $〈n〉$. There is an edge between two vertices in $S_n$ if and only if one vertex can be obtained from another one by swapping the symbol of the first position and the i-th position where $2\le i\le n$.

Moreover, the star graph $S_n$ is a vertex- and edge-transitive graph with regular degree $(n-1)$ (see Figure 1).

Lemma 1 

([30]).  If $n\ge 3$, then the girth of $S_n$ is 6.

Lemma 2 

([28,31]). If $n\ge 4$, then ${\lambda }^{\left(2\right)}\left(S_n\right)=6n-18$.

Lemma 3. 

For two non-negative integers $n\ge 4$ and $h\le 5$, the h-extra edge connectivity of n-dimensional star graph $S_n$ satisfies that

$$\begin{array}{c}\hfill {\lambda }_h\left(S_n\right)\le \left\{\begin{array}{cc}(h+1)n-3h-1\hfill & \mathrm{if}h\le 4,n\ne h,\hfill \\ 6n-18\hfill & otherwise.\hfill \end{array}\right.\end{array}$$

Proof. 

If $h\le 4$ and $n\ne h$, let X be an $(h+1)$-path in $S_n$. By Lemma 1, the induced subgraph $S_n\left[X\right]$ is also an $(h+1)$-path. Assume that F is the set of edges with exactly one end in X. Then, we have

$$\begin{array}{cc}\hfill \left|F\right|& =(n-1)\left|X\right|-2\left|E\right(S_n\left[X\right]\left)\right|\hfill \\ \hfill & =(n-1)(h+1)-2h\hfill \\ \hfill & =(h+1)n-3h-1.\hfill \end{array}$$

It is easy to see that $S_n-F$ has exactly two components, one of which is the $(h+1)$-path X. Since $|S_n|-|X|=n!-(h+1)>h+1$ for $n\ge 4$, F is an h-extra edge-cut of $S_n$. Therefore,

$${\lambda }_h\left(S_n\right)\le \left|F\right|=(h+1)n-3h-1.$$

(1)

If $h=5$ or $(n,h)=(4,4)$, let X be a 6-cycle in $S_n$, and F be the set of edges with exactly one end in X. Then, we have $\left|F\right|=6(n-3)=6n-18$.

Clearly, $S_n-F$ has two components, one of which is the 6-cycle X. Since $|S_n|-|X|=n!-6>h+1$ for $n\ge 4$, F is an h-extra edge-cut of $S_n$. Hence,

$${\lambda }_h\left(S_n\right)\le \left|F\right|=6n-18.$$

(2)

By (1) and (2), the lemma follows.  □

Theorem 1. 

For two non-negative integers $n\ge 4$ and $h\le 5$, the h-extra edge connectivity of n-dimensional star graph $S_n$ is

$$\begin{array}{c}\hfill {\lambda }_h\left(S_n\right)=\left\{\begin{array}{cc}(h+1)n-3h-1\hfill & \mathrm{if}h\le 4,n\ne h,\hfill \\ 6n-18\hfill & otherwise.\hfill \end{array}\right.\end{array}$$

Proof. 

Let F be a minimum h-extra edge-cut of $S_n$. Then, ${\lambda }_h\left(S_n\right)=\left|F\right|$. Clearly, there are exactly two components in $S_n-F$, denoted by X and Y, respectively. We assume $\left|X\right|\le \left|Y\right|$ without loss of generality.

By Lemma 3, we only need to show that

$$\begin{array}{c}\hfill \left|F\right|\ge \left\{\begin{array}{cc}(h+1)n-3h-1\hfill & \mathrm{if}h\le 4\mathrm{and}n\ne h,\hfill \\ 6n-18\hfill & otherwise.\hfill \end{array}\right.\end{array}$$

(3)

Based on the minimum degree of X and Y, we consider the following two cases:

Case 1. $\delta \left(X\right)\ge 2$ and $\delta \left(Y\right)\ge 2$.

Notice that F is also a 2-edge-cut of $S_n$. By Lemma 2, we know that the size of a minimum 2-edge-cut of $S_n$ is ${\lambda }^{\left(2\right)}\left(S_n\right)=6n-18$. Then, we have $\left|F\right|\ge 6n-18$.

If $h\le 4$ and $n\ne h,$ the inequality $6n-18-\left[\right(h+1)n-3h-1]=(5-h)n+3h-17\ge 0$ holds. Therefore,

$$\left|F\right|\ge 6n-18\ge (h+1)n-3h-1.$$

(4)

Case 2. $\delta \left(X\right)=1$ or $\delta \left(Y\right)=1$.

Without loss of generality, suppose that a vertex v in X satisfies $d_X\left(v\right)=1$.

Since F is an h-extra edge-cut of $S_n$, $\left|X\right|\ge h+1$ and $\left|Y\right|\ge h+1$. If $\left|X\right|\ge h+2,$ then $\left|V\right(X\setminus \left\{v\right\}\left)\right|\ge h+1$. Let $X^{\prime }=X\setminus \left\{v\right\}$, $Y^{\prime }=Y\cup \left\{v\right\}$. Clearly, $d_{Y^{\prime }}\left(v\right)=n-2$ since $d_{S_n}\left(v\right)=n-1$. Let $F^{\prime }$ be the set of edges between $X^{\prime }$ and $Y^{\prime }$. Then, we have $\left|F\right|=|F^{\prime }|+n-3$. One can see that $\left|F\right|>|F^{\prime }|$ for $n\ge 4$ (see Figure 2). Therefore, $F^{\prime }$ is a smaller set of the h-extra edge-cut than F, a contradiction. It follows that $\left|X\right|=h+1\le 6$. By Lemma 1 and $d_X\left(v\right)=1$, the subgraph of $S_n$ induced by X is a tree. Therefore, $\left|F\right|=(n-1)\left|X\right|-2\left(\right|X|-1)=(n-3)\left|X\right|+2=(h+1)n-3h-1$.

If $h\le 4$ and $n\ne h,$ we have

$$\left|F\right|=(h+1)n-3h-1;$$

(5)

If $h=5$ or $(n,h)=(4,4)$, we have that the inequality $(h+1)n-3h-1-(6n-18)=(h-5)n-3h+17>0$ holds. Hence,

$$\left|F\right|=(h+1)n-3h-1>6n-18.$$

(6)

By (4)–(6), the inequality (3) is satisfied. Hence, the theorem follows.  □

Corollary 1 

([28]).  If $n\ge 4$, then ${\lambda }_1\left(S_n\right)=2n-4$, ${\lambda }_2\left(S_n\right)=3n-7$.

Corollary 2. 

If $n\ge 4$, then ${\lambda }_3\left(S_n\right)=4n-10$, ${\lambda }_5\left(S_n\right)=6n-18$. If $n\ge 5$, then ${\lambda }_4\left(S_n\right)=5n-13$. Moreover, ${\lambda }_4\left(S_4\right)=6$.

3. Extra Edge Connectivity of ${\mathit{G}}_{\mathit{n}}$

In this section, we focus on the h-extra edge connectivity of Cayley graphs $G_n$ generated by transposition trees other than the star.

Lemma 4 

([23]). If $n\ge 4$, then the girth of $G_n$ is 4, and $\lambda \left(G_n\right)=n-1$.

Lemma 5 

([28]).  If $n\ge 4$, then ${\lambda }^{\left(2\right)}\left(G_n\right)=4n-12$.

Lemma 6. 

For two non-negative integers $n\ge 4$ and $h\le 3$, the h-extra edge connectivity of $G_n$ satisfies

$$\begin{array}{c}\hfill {\lambda }_h\left(G_n\right)\le \left\{\begin{array}{cc}(h+1)n-3h-1\hfill & \mathrm{if}h\le 2,(n,h)\ne (4,2),\hfill \\ 4n-12\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

Proof. 

If $h\le 2$ and $(n,h)\ne (4,2)$, let X be an $(h+1)$-path in $G_n$. Let F be the set of edges with exactly one end in X. Then, we have $\left|F\right|=(h+1)n-3h-1$.

Clearly, $G_n-F$ has two components, one of which is the $(h+1)$-path X. Since $|V\left(G_n\right)|-|X|=n!-(h+1)\ge h+1$ for $n\ge 4$, F is an h-extra edge-cut of $G_n$ for $h\le 2$. Thus, we have

$${\lambda }_h\left(G_n\right)\le \left|F\right|=(h+1)n-3h-1.$$

(7)

If $h=3$ or $(n,h)=(4,2)$, suppose that X is a 4-cycle of $G_n$. Let F be the set of edges with exactly one end in X. Then, we have $\left|F\right|=4(n-3)=4n-12$. Clearly, $G_n-F$ has exactly two components, including the 4-cycle X and a large component $G_n-X$. Since $\left|V\right(G_n\left)\right|-\left|X\right|=n!-4\ge h+1$ for $n\ge 4$, F is an h-extra edge-cut of $G_n$. Thus, we have

$${\lambda }_h\left(G_n\right)\le \left|F\right|=4n-12.$$

(8)

By (7) and (8), the lemma holds.  □

Theorem 2. 

For two non-negative integers $n\ge 4$ and $h\le 3$, the h-extra edge connectivity of $G_n$ is

$$\begin{array}{c}\hfill {\lambda }_h\left(G_n\right)=\left\{\begin{array}{cc}(h+1)n-3h-1\hfill & \mathrm{if}h\le 2,(n,h)\ne (4,2),\hfill \\ 4n-12\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

Proof. 

Let F be a minimum h-extra edge-cut of $G_n$. Then, ${\lambda }_h\left(G_n\right)=\left|F\right|$. By Lemma 6, we only need to show that

$$\begin{array}{c}\hfill \left|F\right|\ge \left\{\begin{array}{cc}(h+1)n-3h-1\hfill & \mathrm{if}h\le 2,(n,h)\ne (4,2),\hfill \\ 4n-12\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(9)

Since F is a minimum h-extra edge-cut, there are exactly two components in $G_n-F$, denoted by X and Y, respectively. And assume $\left|X\right|\le \left|Y\right|$ without loss of generality. According to the minimum degree of the graphs X and Y, we discuss the following two cases:

Case 1. $\delta \left(X\right)\ge 2$ and $\delta \left(Y\right)\ge 2$.

Notice that F is also a 2-extra edge-cut of $G_n$. By Lemma 5, we know that the size of a minimum 2-extra edge-cut in $G_n$ is ${\lambda }^{\left(2\right)}\left(G_n\right)=4n-12$. Thus, $\left|F\right|\ge 4n-12$.

If $h\le 2$ and $(n,h)\ne (4,2)$, then the inequality $4n-12-\left[\right(h+1)n-3h-1]=(3-h)n+(3h-11)\ge 0$ holds. Hence,

$$\left|F\right|\ge 4n-12\ge (h+1)n-3h-1.$$

(10)

If $h=3$ or $(n,h)=(4,2)$, we have

$$\left|F\right|\ge 4n-12.$$

(11)

Case 2. $\delta \left(X\right)=1$ or $\delta \left(Y\right)=1$.

Without loss of generality, suppose that $\delta \left(X\right)=1$ and there exists a vertex v in X such that $d_X\left(v\right)=1$. An analysis similar to that in the proof of Theorem 1 shows that $\left|X\right|=h+1\le 4$. By Lemma 4 and the fact that $d_X\left(v\right)=1$, we know that the subgraph induced by X is a tree. Thus, we have

$$\left|F\right|=(h+1)n-3h-1\mathrm{for}h\le 3.$$

(12)

Hence, if $h=3$ or $(n,h)=(4,2)$, we have

$$\left|F\right|=(h+1)n-3h-1\ge 4n-12.$$

(13)

By (10)–(13), the inequality (9) holds. Hence, the theorem holds.  □

Corollary 3 

([28]).  If $n\ge 5$, then ${\lambda }_2\left(G_n\right)=3n-7$. Moreover, ${\lambda }_2\left(G_4\right)=4$.

The n-dimensional bubble-sort graph $B_n$ was first introduced by Akers et al. [21]. It has high symmetry and a hierarchical structure [32]. In addition, $B_n$ is a Cayley graph $Cay\left(Sym\right(n),\mathcal{T})$, where $G\left(\mathcal{T}\right)$ is an n-path; thus, $B_n$ belongs to $G_n$. In other words, an n-dimensional bubble-sort graph $B_n$ is a graph with $n!$ vertices labeled with permutations on $〈n〉$ where two vertices are adjacent in $B_n$ if and only if there exists $i\in 〈n-1〉$ such that one vertex can be obtained from another by swapping the symbol of the i-th position and the $(i+1)$-th position (see Figure 3). Many properties of $B_n$ have been explored, such as the h-connectivity [33], embedded connectivity [34,35], and conditional diagnosability [36].

For $i\in 〈n〉$, let $H_i$ be the subgraph of $G_n$ induced by the vertices with i in the n-th position. Then, $H_i$ is also a Cayley graph generated by a transposition tree. The collection $\{H_1,H_2,\dots ,H_n$} forms a decomposition of $G_n$ with dimension n. For a vertex $u\in V\left(H_i\right)$, we call the neighbors of u in $G-H_i$ external neighbors of u. Moreover, for $i\ne j$, we call the edges between $H_i$ and $H_j$ crossing edges.

Lemma 7 

([25]).  For two distinct integers $i,j$ in $〈n〉$, the following is true:

(1) Each vertex in $H_i$ has exactly one external neighbor.

(2) There are $(n-2)!$ independent edges between $H_i$ and $H_j$.

Let F be an h-extra edge-cut of $G_n$ with minimum cardinality. Let X be a connected component in $G_n-F$ and Y be the subgraph $G_n-F-X$. For any $i\in 〈n〉$, let

$$X_i=X\cap H_i,Y_i=Y\cap H_i,\mathrm{and}{F}_i=F\cap E\left(H_i\right).$$

Let $J_X=\{i\in 〈n〉:X_i\ne \varnothing \}$, $J_Y=\{i\in 〈n〉:Y_i\ne \varnothing \}$, and $J_0=J_X\cap J_Y$.

Let $a=|J_X\setminus J_0|$, $b=|J_Y\setminus J_0|$, and $c=|J_0|$. Then, $c=n-a-b$. By the symmetry of X and Y, we assume that $a\le b$. Let F be a minimum h-extra edge-cut of $G_n$ such that $\left|X\right|\le \left|Y\right|$ and $\left|X\right|$ is as large as possible. Let $E_C(X,Y)$ denote all the crossing edges between X and Y. For $i_1\in J_X\setminus J_0$ and $i_2\in J_Y\setminus J_0$, by Lemma 7, there are $(n-2)!$ independent edges between $H_{i_1}$ and $H_{i_2}$. Therefore,

$$|E_C(X,Y)|\ge ab(n-2)!.$$

(14)

Because $i\in J_0$, $H_i$ is isomorphic to $G_{n-1}$ and $H_i-F_i$ is not connected, by Lemma 4, we have

$$|F_i|\ge (n-2)\mathrm{for}i\in J_0.$$

(15)

Theorem 3. 

${\lambda }_4\left(B_4\right)=5$, ${\lambda }_5\left(B_4\right)=6$.

Proof. 

For $h=4$, let X be the subgraph in $B_4$ induced by $1234,2134,2314,1243,2143$ and F be the edges between X and $B_4-X$ (see the dotted edges in Figure 3). It is easy to see that F is a 4-extra edge-cut in $B_4$, that is, ${\lambda }_4\left(B_4\right)\le \left|F\right|=5$.

For $h=5$, let X be the subgraph induced by $3412,3142,1342,1432,4132,4312$. Let F be the edges between X and $B_4-X$ (see the dashed edges in Figure 3). It is easy to see that F is a 5-extra edge-cut in $B_4$, that is, ${\lambda }_5\left(B_4\right)\le \left|F\right|=6$.

Now we only need to prove that ${\lambda }_4\left(B_4\right)\ge 5$, ${\lambda }_5\left(B_4\right)\ge 6$. For $h=4,5$, let F be a minimum h-extra egde-cut in $B_4$. It suffices to show that $\left|F\right|\ge h+1$. We consider two cases as follows:

Case 1. $a\ne 0$.

In this case, note that $a\le b$; hence, $ab\ge 1$. Note that $E_C(X,Y)\subset F$. Recalling that $n=4$, by (14) and (15), we have,

$$\begin{array}{cc}\hfill \left|F\right|& \ge |E_C(X,Y)|+{\sum }_{i\in J_0}\left|F_i\right|\hfill \\ & \ge ab(n-2)!+c(n-2)\hfill \\ & \ge (n-a-b+ab)(n-2)\hfill \\ & =(n-1+(a-1\left)\right(b-1\left)\right)(n-2)\hfill \\ & \ge h+1.\hfill \end{array}$$

Case 2. $a=0$.

In this case, $|J_X|=|J_0|=c$.

If $c=1$, without loss of generality, we assume that X is contained in $H_1$. Then, $|E_C(X,Y)|=|X|\ge h+1$. Therefore, $\left|F\right|\ge |F_1|+|E_C(X,Y)|\ge 2+|X|\ge h+3$.

If $c=2$, assume that $J_X=\{1,2\}$. Then, $|X_1|+|X_2|\ge h+1$. Thus, $|Y_1|+|Y_2|=2|H_1|-|X_1|-|X_2|\le 12-(h+1)=11-h$. That is to say, there are at most $11-h$ edges between $Y_1\cup Y_2$ and $H_3\cup H_4$. By Lemma 7, there are eight independent edges between $H_1\cup H_2$ and $H_3\cup H_4$. Therefore, there are at least $8-(11-h)=h-3$ edges between $X_1\cup X_2$ and $H_3\cup H_4$. Then, $|E_C(X,Y)|\ge h-3$. Hence, $\left|F\right|\ge |F_1|+|F_2|+|E_C(X,Y)|\ge 2+2+(h-3)=h+1$.

If $c\ge 3$, by (15), we have $\left|F\right|\ge {\sum }_{i\in J_0}\left|F_i\right|\ge c(n-2)\ge h+1$.

That is to say, $\left|F\right|\ge h+1$, and the theorem follows.  □

Lemma 8. 

If $n\ge 5$, then ${\lambda }_4\left(G_n\right)\le 5n-15$ and ${\lambda }_5\left(G_n\right)\le 6n-20$.

Proof. 

Since $G\left(\mathcal{T}\right)$ is not $K_{1,n-1}$, $G\left(\mathcal{T}\right)$ contains two disjoint subgraphs isomorphic to $K_2$ and $K_{1,2}$, respectively. Without loss of generality, assume that $(1,2)$ is the edge in $K_2$, and $(3,4),(4,5)$ are the two edges in $K_{1,2}$.

For $h=4$, let X be the subgraph in $G_n$ induced by the five vertices, which have the first five positions with 12,345, 12,435, 21,435, 21,345, 12,453 and the last $(n-5)$ positions with $6\dots n$. Note that $(1,2)$, $(3,4)$, and $(4,5)$ are the edges in $G\left(\mathcal{T}\right)$. Let F be the edges between X and $G_n-X$. Note that $G_n$ has no odd cycle, and there are no edges linking 21,345 and 12,453 since we cannot obtain the vertex 21,345 by swapping two symbols in the vertex 12,453. Thus, there are exactly five edges in X (see Figure 4). Therefore, $\left|F\right|=5(n-1)-2\times 5=5n-15$. We can also obtain that $G_n-F$ has two components, and both of them contain at least five vertices. Hence, F is a 4-extra edge-cut in $G_n$, that is, ${\lambda }_4\left(G_n\right)\le \left|F\right|=5n-15$.

For $h=5$, let $X^{\prime }$ be the subgraph induced by the six vertices which have the first five positions with 12,345, 12,435, 12,453, 21,453, 21,435, 21,345 and the last $(n-5)$ positions with $6\dots n$ (see Figure 4). Let F be the edges between $X^{\prime }$ and $G_n-X^{\prime }$. Note that $(1,2)$, $(3,4)$, and $(4,5)$ are the edges in $G\left(\mathcal{T}\right)$, and there are exactly seven edges in $X^{\prime }$ (see Figure 4). Therefore, $\left|F\right|=6(n-1)-2\times 7=6n-20$. Therefore, F is a 5-extra edge-cut in $G_n$, that is, ${\lambda }_5\left(G_n\right)\le \left|F\right|=6n-20$.  □

Lemma 9. 

If $n\ge 5$, $h=4,5$, F is a minimum h-extra edge-cut in $G_n$, then $|J_X|=|J_0|$.

Proof. 

If $|J_X|\ne |J_0|$, then $|J_X|>|J_0|$, that is to say, $a=|J_X\setminus J_0|\ge 1$.

Note that $a\le b$; hence, $ab\ge 1$. Note that $E_C(X,Y)\subset F$. By (14) and (15), we have,

$$\begin{array}{cc}\hfill \left|F\right|& \ge |E_C(X,Y)|+{\sum }_{i\in J_0}\left|F_i\right|\hfill \\ & \ge ab(n-2)+c(n-2)\hfill \\ & \ge (n-a-b+ab)(n-2)\hfill \\ & =(n-1+(a-1\left)\right(b-1\left)\right)(n-2)\hfill \\ & \ge (n-1)(n-2).\hfill \end{array}$$

Together with Lemma 8, we have $5n-15\ge (n-1)(n-2)$ and $6n-20\ge (n-1)(n-2)$, a contradiction.  □

Theorem 4. 

If $n\ge 5$, then ${\lambda }_4\left(G_n\right)=5n-15$ and ${\lambda }_5\left(G_n\right)=6n-20$.

Proof. 

By Lemma 8, it suffices to show that

$${\lambda }_4\left(G_n\right)\ge 5n-15\mathrm{and}{\lambda }_5\left(G_n\right)\ge 6n-20\mathrm{for}n\ge 4.$$

(16)

We prove (16) by induction on n. When $n=4$, note that $G\left(\mathcal{T}\right)$ is not $K_{1,n-1}$. Then, $G\left(\mathcal{T}\right)$ is isomorphic to a path $P_4$, and thus $G_n$ is four-dimensional bubble-sort graph. By Theorem 3, the conclusion (16) is correct. Assume (16) holds for $n-1$ with $n\ge 5$.

Let F be a minimum h-extra edge-cut in $G_n$. It suffices to show that $\left|F\right|\ge 5n-15$ for $h=4$ and $\left|F\right|\ge 6n-20$ for $h=5$.

By Lemma 9, we have $|J_X|=|J_0|=c$. For each $i\in J_0$, let $Y_i^{\prime }$ be a connected component in $H_i-X_i$ with minimum cardinality. We first show that

$$|Y_i^{\prime }|\ge h+1\mathrm{for}\mathrm{each}i\in J_0.$$

(17)

For each $i\in J_0$, if there exists a component $Y_i^{\prime }$ in $H_i-X_i$ that has at most h vertices, we contract the component $Y_i^{\prime }$ to a new vertex $v_{Y_i^{\prime }}$. Eventually, we obtain a new graph $Y^{\prime }$ from Y, and assume that $v_{Y_1^{\prime }}$ has the largest distance to the old vertices in $Y^{\prime }$. Then, $Y-Y_1^{\prime }$ is connected.

Let $|Y_1^{\prime }|=s$. If $s\le 3$ and $(n,s)\ne (5,3)$, by Theorem 2, we have $|E_{H_1}\left(Y_1^{\prime }\right)|\ge {\lambda }_{s-1}\left(G_{n-1}\right)=s(n-1)-3s+2\ge s$. If $s=4$ or $(n,s)=(5,3)$, by Theorem 2, we have $|E_{H_1}\left(Y_1^{\prime }\right)|\ge {\lambda }_{s-1}\left(G_{n-1}\right)=4(n-1)-12\ge s$. If $s=5$, by the inductive hypothesis, we have $|E_{H_1}\left(Y_1^{\prime }\right)|\ge {\lambda }_4\left(G_{n-1}\right)\ge 5(n-1)-15\ge s$. Note that $|E_C(Y-Y_1,Y_1^{\prime })|\le s$, and we always have $|E_C(Y-Y_1,Y_1^{\prime })|\le |E_{H_1}\left(Y_1^{\prime }\right)|$. Then, $F^{\prime }=F\cup E_{G_n}\left(Y_1^{\prime }\right)\setminus E_{H_1}\left(Y_1^{\prime }\right)$ is also an h-extra edge-cut in $G_n$, and $|F^{\prime }|\le |F|$, but the component $X\cup Y_1^{\prime }$ has a larger size, a contradiction. Thus, (17) always holds. Therefore, for $i\in J_0$, if $X_i$ is a connected component in $H_i$ and $|X_i|\le h+1$, then $F_i$ is an $\left(\right|X_i|-1)$-extra edge-cut in $H_i$.

In the following, we consider the case of $h=4$. The proof of the case of $h=5$ is similar.

If $c=1$, assume that $X_1=X$. Then, $|X_1|\ge 5$. By (17), we have that every component in $H_1-X_1$ has at least five vertices. Then, $F_1$ is a 4-extra edge-cut in $H_1$. We have $|F_1|\ge 5(n-1)-15=5n-20$ by an inductive hypothesis. Note that each vertex in X has an external neighbor in $G_n-H_1$. Then, $|E_C(X,Y)|\ge 5$. Hence, $\left|F\right|\ge |F_1|+|E_C(X,Y)|\ge 5n-15$.

If $c=2$, assume $J_X=\{1,2\}$ and $|X_1|\le |X_2|$. If $|X_1|=1,$ then $|X_2|\ge 4$. Thus, $F_2$ is a 3-extra edge-cut in $G_{n-1}$. Hence, $|F_2|\ge 4(n-1)-12$. Note that $|X_2|$ has at least three external neighbors in $G_n-H_1-H_2$. Therefore, $\left|F\right|\ge |F_1|+|F_2|+|E_C(X,Y)|\ge (n-2)+(4n-16)+3=5n-15$.

If $|X_1|=2$, then $F_1$ is a 1-extra edge-cut in $H_1$; thus, $|F_1|\ge 2(n-1)-4$. Since $\left|X\right|\ge 5$, we have $|X_2|\ge 3$. Then, $F_2$ is a 2-extra edge-cut in $H_2$. By Theorem 2, we have $|F_2|\ge 3(n-1)-7$ or $|F_2|=4(n-1)-12$, $n=5$, and the graph induced by $X_2$ in $H_2$ is a 4-cycle graph. Note that $|X_2|$ has at least $|X_2|-2$ external neighbors in $G_n-H_1-H_2$, that is, $|E_C(X,Y)|\ge |X_2|-2$. Consequently, $|F_2|+|E_C(X,Y)|\ge 3n-9$. Therefore, $\left|F\right|\ge |F_1|+|F_2|+|E_C(X,Y)|\ge (2n-6)+(3n-9)=5n-15$.

If $|X_1|\ge 3$, then $|X_2|\ge |X_1|\ge 3$; thus, $F_1,F_2$ is a 2-extra edge-cut in $H_1,H_2$, respectively. If $n\ge 6$, by Theorem 2, we have $|F_1|\ge 3(n-1)-7$ and $|F_2|\ge 3(n-1)-7$. Therefore, $\left|F\right|\ge |F_1|+|F_2|\ge 2(3n-10)\ge 5n-15$. Now assume $n=5$. If $|F_1|\ne 4(n-1)-12=4$ and $|F_2|\ne 4$, then $|F_1|\ge 3(n-1)-7$ and $|F_2|\ge 3(n-1)-7$. Thus, the conclusion holds. If $|F_1|=4$ or $|F_2|=4$, then $X_1$ induces a 4-cycle $C_4$ in $H_1$ or $X_2$ induces a $C_4$ in $H_2$. We suppose the former one without loss of generality. Note that when $n=5$, $H_1$ is isomorphic to $B_4$. Each $C_4$ in $H_1$ has at most two external neighbors in $H_j$ for $j\ne 1$ (see Figure 3). Thus, $X_1$ has at most two neighbors in $X_2$. Therefore, $|E_C(X_1,Y)|\ge 2$. Then, $\left|F\right|\ge |F_1|+|F_2|+|E_C(X,Y)|\ge 2(4n-12)+2\ge 5n-15$.

If $c=3,4$, assume $J_X=\{1,2,\dots ,c\}$ and $|X_1|\le |X_2|\dots \le |X_c|$. If $|X_2|\ge 2$, for $i\in \{2,\dots ,c\}$, $F_i$ is a 1-extra edge-cut in $H_i$. Note that $|F_1|\ge n-2$. We have $\left|F\right|\ge (n-2)+2(2n-6)\ge 5n-15$. If $|X_2|=1$, then $|X_3|\ge 3$ or $c=4$. In both cases, we have ${\sum }_{i=3}^c\left|F_i\right|\ge 3n-11$. Then, $\left|F\right|\ge {\sum }_{i\in J_0}\left|F_i\right|\ge 2(n-2)+3n-11\ge 5n-15$.

If $c\ge 5$, then $\left|F\right|\ge {\sum }_{i\in J_0}\left|F_i\right|\ge 5(n-2)>5n-15$.  □

4. Conclusions

The extra edge connectivity serves as a crucial reliability measure for interconnection networks. This paper deals with the h-extra edge connectivity of Cayley graphs generated by transposition trees, focusing on the case where h is less than 6. As a key contribution, we establish the h-extra edge connectivity for both the star graph $S_n$ and the bubble-sort graph $B_n$ for all integers $h\le 5$. These results provide fundamental reliability measures that will facilitate future applications of $S_n$ and $B_n$ as topological models for large-scale parallel processing systems. When $h\ge 6$, novel methods will be needed to study these parameters. For a Cayley graph generated by transposition trees, we believe that there exists a close relationship between extra edge connectivity and h-edge connectivity. Observe that ${\lambda }_1\left(S_n\right)={\lambda }^{\left(1\right)}\left(S_n\right)$ and ${\lambda }_5\left(S_n\right)={\lambda }^{\left(2\right)}\left(S_n\right)$, that is, for $h=1,2$, $S_n-F$ has a minimum degree h is equivalent to $S_n-F$ has a minimum connected component which is isomorphic to $S_{h+1}$. Note that $S_{h+1}$ has $(h+1)!$ vertices. We conjecture that ${\lambda }_{(h+1)!-1}\left(S_n\right)={\lambda }^{\left(h\right)}\left(S_n\right)$ for $h\le n-2$.

In addition, there are several other connectivities on graphs, such as component connectivity [26] and algebraic connectivity [37]. To study these parameters for the reliability of Cayley graphs generated by transposition trees is also of interest.