Graph Operations and Neighbor Rupture Degree

Abstract

In a communication network, the vulnerability parameters measure the resistance of the network to disruption of operation after the failure of certain stations or communication links. A vertex subversion strategy of a graph 𝐺, say 𝑆, is a set of vertices in 𝐺 whose closed neighborhood is removed from 𝐺. The survival subgraph is denoted by 𝐺/𝑆. The neighbor rupture degree of 𝐺, Nr(𝐺), is defined to be Nr(𝐺)=max{𝑤(𝐺/𝑆)−|𝑆|−𝑐(𝐺/𝑆)∶𝑆⊂𝑉(𝐺),𝑤(𝐺/𝑆)≥1}, where 𝑆 is any vertex subversion strategy of 𝐺, 𝑤(𝐺/𝑆) is the number of connected components in 𝐺/𝑆 and 𝑐(𝐺/𝑆) is the maximum order of the components of 𝐺/𝑆 (G. Bacak Turan, 2010). In this paper we give some results for the neighbor rupture degree of the graphs obtained by some graph operations.

1. Introduction

A network can be broke down completely or partially with unexpected reasons. If the data is not transmitted to the desired location that means there is a problem on the system. This problem can block a treaty of billions of euros or make a big problem for human’s life. In these days the reliability and the vulnerability of networks are so important. For that reason graphs are taken as a model in the research area of reliability and vulnerability of the networks. Each network center is taken as a vertex and the connections of these vertices are edges of a graph.

A few questions can be asked at this point How can the reliability and the vulnerability of a network be determined? What are the factors of the reliability and the vulnerability? For example, what can be done if there is a problem on the way you are using every day to work? We have two choices; we may give up going to work although we have the risk of dismissal or we can look for another way to work. The question “if there is another way to reach work” may come to our minds. In other words “Has the link connection between home and work completely broken down?”. To answer this question, we must know the dimensions of the problem between home and work. The vulnerability of the graph which represents the way between home and work should be searched. In graph theory some vulnerability parameters are defined to measure the vulnerability value of graphs such as connectivity [1], integrity [2], neighbor integrity [3], rupture degree [4], and neighbor rupture degree [5].

Terminology and notation not defined in this paper can be found in [5]. Let 𝐺 be a simple graph and let 𝑢 be any vertex of 𝐺. The set 𝑁(𝑢)={𝑣∈𝑉(𝐺)∣𝑣≠𝑢;𝑣and𝑢areadjacent} is the open neighborhood of 𝑢, and 𝑁[𝑢]={𝑢}∪𝑁(𝑢) is the closed neighborhood of 𝑢. A vertex 𝑢 in 𝐺 is said to be subverted if the closed neighborhood of 𝑢 is removed from 𝐺. A set of vertices 𝑆={𝑢₁,𝑢₂,…,𝑢_𝑚} is called a vertex subversion strategy of 𝐺 if each of the vertices in 𝑆 has been subverted from 𝐺. If 𝑆 has been subverted from the graph 𝐺, then the remaining graph is called survival graph, denoted by 𝐺/𝑆.

2. Basic Results

In this paper the new vulnerability parameter neighbor rupture degree was studied. The concept of neighbor rupture degree was introduced by Bacak-Turan and Kırlangıc in 2011 [5]. The definition of neighbor rupture degree and some results are given below.

Definition 1 (see [6]). The neighbor rupture degree of a noncomplete connected graph 𝐺 is defined to be (𝐺)𝑤𝐺Nr=max𝑆−|𝑆|𝐺−𝑐𝑆(𝐺)𝐺∶𝑆⊂𝑉,𝑤𝑆≥1, (1) where 𝑆 is any vertex subversion strategy of 𝐺, 𝑤(𝐺/𝑆) is the number of connected components in 𝐺/𝑆, and 𝑐(𝐺/𝑆) is the maximum order of the components of 𝐺/𝑆.

In particular, the neighbor rupture degree of a complete graph 𝐾_𝑛 is defined to be Nr(𝐾_𝑛)=1−𝑛. A set 𝑆⊂𝑉(𝐺) is said to be Nr-𝑠𝑒𝑡 of 𝐺 if (𝐺)𝐺Nr=𝑤𝑆−|𝑆|𝐺−𝑐𝑆. (2) Some known results are listed below.

Theorem 2 (see [6]). (a) Let 𝑃_𝑛 be a path graph with n vertices and 𝑛≥2, Nr𝑃_𝑛=()().0,𝑛≡1mod4−1,𝑛≡0,2,3mod4 (3)
(b) Let 𝐶_𝑛 be a cycle graph with n vertices and 𝑛≥3, Nr𝐶_𝑛=()().−1,𝑛≡0mod4−2,𝑛≡1,2,3mod4 (4)
(c) Let 𝐾_{𝑛1,𝑛2,𝑛3,…,𝑛𝑘} be a 𝑘-partite graph Nr𝐾_{𝑛1,𝑛2,…,𝑛𝑘}=max𝑛₁,𝑛₂,𝑛₃,…,𝑛_𝑘−3. (5)
(d) Let 𝑊_𝑛 be a wheel graph with 𝑛 vertices and 𝑛≥5, Nr𝑊_𝑛=()().−1,𝑛≡1mod4−2,𝑛≡0,2,3mod4 (6)

3. Graph Operations and Neighbor Rupture Degree

In this section some graph operations are operated on graphs and their neighbor rupture degrees are evaluated.

Definition 3 (see [7]). The union graph 𝐺=𝐺₁∪𝐺₂∪⋯∪𝐺_𝑛 has vertex set 𝑉(𝐺)=𝑉(𝐺₁)∪𝑉(𝐺₂)∪⋯∪𝑉(𝐺_𝑛) and edge set 𝐸(𝐺)=𝐸(𝐺₁)∪𝐸(𝐺₂)∪⋯∪𝐸(𝐺_𝑛). If a graph 𝐺 consists of 𝑘(𝑘≥2) disjoint copies of a graph 𝐻, then we write 𝐺=𝑘𝐻.

Theorem 4. Let 𝐺₁,𝐺₂,𝐺₃,…,𝐺_𝑛 be connected graphs. Then Nr𝐺₁∪𝐺₂∪⋯∪𝐺_𝑛≥Nr𝐺₁+Nr𝐺₂+⋯+Nr𝐺_𝑛. (7)

Proof. Let 𝐺=𝐺₁∪𝐺₂∪⋯∪𝐺_𝑛 be the union of 𝐺₁,𝐺₂,…,𝐺_𝑛. Let 𝑆₁,𝑆₂,…,𝑆_𝑛 be Nr-sets of 𝐺₁,𝐺₂,…,𝐺_𝑛 respectively, and let 𝑆=𝑆₁∪𝑆₂∪⋯∪𝑆_𝑛 be a subversion strategy of 𝐺. Then we obtain (𝐺)𝐺Nr≥𝑤𝑆₁∪𝑆₂∪⋯∪𝑆_𝑛−𝑆₁∪𝑆₂∪⋯∪𝑆_𝑛𝐺−𝑐𝑆₁∪𝑆₂∪⋯∪𝑆_𝑛𝐺=𝑤¹𝑆₁𝐺+𝑤²𝑆₂𝐺+⋯+𝑤^𝑛𝑆_𝑛−𝑆₁−𝑆₂−⋯𝑆_𝑛𝑐𝐺−max¹𝑆₁𝐺,𝑐²𝑆₂𝐺,…,𝑐^𝑛𝑆_𝑛𝐺≥𝑤¹𝑆₁𝐺+𝑤²𝑆₂𝐺+⋯+𝑤^𝑛𝑆_𝑛−𝑆₁−𝑆₂−⋯𝑆_𝑛𝐺−𝑐¹𝑆₁𝐺−𝑐²𝑆₂𝐺−⋯−𝑐^𝑛𝑆_𝑛=Nr𝐺₁+Nr𝐺₂+Nr𝐺₃+⋯+Nr𝐺_𝑛. (8) Thus we have Nr(𝐺₁∪𝐺₂∪⋯∪𝐺_𝑛)≥Nr(𝐺₁)+Nr(𝐺₂)+⋯+Nr(𝐺_𝑛).

Theorem 5. Let 𝐾_𝑛1,𝐾_𝑛2,…,𝐾_𝑛𝑚 be complete graphs with 𝑛₁≤𝑛₂≤𝑛_𝑚 where 𝑛_𝑖+1−𝑛_𝑖≥2;∀𝑖∈𝑍⁺. Then Nr𝐾_𝑛1∪𝐾_𝑛2∪⋯∪𝐾_𝑛𝑚=2−𝑚−𝑛₁. (9)

Proof. Let 𝑆 be a subversion strategy of 𝐾_𝑛1∪𝐾_𝑛2∪⋯∪𝐾_𝑛𝑚. Since these are complete graphs, it is obvious that 𝑆 contains at most one vertex from each 𝐾_𝑛̇𝐼.
If |𝑆|=𝑘, then 𝑤((𝐾_𝑛1∪𝐾_𝑛2∪⋯∪𝐾_𝑛𝑚)/𝑆)=𝑚−𝑘 and 𝑐((𝐾_𝑛1∪𝐾_𝑛2∪⋯∪𝐾_𝑛𝑚)/𝑆)≥𝑛_𝑚−𝑘. Thus we have 𝑤((𝐾_𝑛1∪𝐾_𝑛2∪𝐾_𝑛3∪⋯∪𝐾_𝑛𝑚)/𝑆)−|𝑆|−𝑐((𝐾_𝑛1∪𝐾_𝑛2∪𝐾_𝑛3∪⋯∪𝐾_𝑛𝑚)/𝑆)≤𝑚−2𝑘−𝑛_𝑚−𝑘≤𝑚−2𝑘−𝑛₁−2(𝑚−𝑘−1) since 𝑛_𝑖+1−𝑛_𝑖𝑛≥2,_𝑚−𝑘≥𝑛₁+2(𝑚−𝑘−1)=2−𝑚−𝑛₁⟹Nr≤2−𝑚−𝑛₁. (10) There exist 𝑆^∗ such that |𝑆^∗|=𝑚−1, 𝑤((𝐾_𝑛1∪𝐾_𝑛2∪⋯∪𝐾_𝑛𝑚)/𝑆^∗)=1 and 𝑐((𝐾_𝑛1∪𝐾_𝑛2∪⋯∪𝐾_𝑛𝑚)/𝑆^∗)=𝑛₁. Then we have 𝑤𝐾_𝑛1∪𝐾_𝑛2∪⋯∪𝐾_𝑛𝑚𝑆^∗−𝑆^∗𝐾−𝑐^𝑛1∪𝐾_𝑛2∪⋯∪𝐾_𝑛𝑚𝑆^∗=2−𝑚−𝑛₁⟹Nr≥2−𝑚−𝑛₁. (11) From (10) and (11) we obtain Nr(𝐾_𝑛1∪𝐾_𝑛2∪⋯∪𝐾_𝑛𝑚)=2−𝑚−𝑛₁.

The following theorem’s proof is very similar to that of Theorem 5.

Theorem 6. Let 𝐾_𝑛1,𝐾_𝑛2,𝐾_𝑛3,…,𝐾_𝑛𝑚 be complete graphs with 𝑛₁≤𝑛₂≤𝑛₃⋯≤𝑛_𝑚 where 𝑛_𝑖+1−𝑛_𝑖≤2;  for all 𝑖∈𝑍⁺. Then Nr𝐾_𝑛1∪𝐾_𝑛2∪𝐾_𝑛3∪⋯∪𝐾_𝑛𝑚=𝑚−𝑛_𝑚. (12)

Corollary 7. Let Nr𝐾_𝑛∪𝐾_𝑚=2−𝑛,2>𝑛−𝑚,−𝑚,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (13)

Definition 8 (see [7]). The join graph 𝐺=𝐺₁+𝐺₂ has vertex set 𝑉(𝐺)=𝑉(𝐺₁)∪𝑉(𝐺₂) and edge set 𝐸(𝐺)=𝐸(𝐺₁)∪𝐸(𝐺₂)∪{𝑢𝑣∣𝑢∈𝑉(𝐺₁)and𝑣∈𝑉(𝐺₂)}.

In this part, neighbor rupture degree of join of some graphs is given.

Theorem 9. Let 𝐺₁ and 𝐺₂ be two connected graphs. Then Nr𝐺₁+𝐺₂=maxNr𝐺₁,Nr𝐺₂. (14)

Proof. Let 𝑆 be a subversion strategy of 𝐺₁+𝐺₂. There are three cases according to the elements of 𝑆.
Case 1. Let 𝑆=𝑆₁⊂𝑉(𝐺₁) be the Nr-set of 𝐺₁ such that 𝑤(𝐺₁/𝑆₁)−|𝑆₁|−𝑐(𝐺₁/𝑆₁)=Nr(𝐺₁). Since any elements from 𝐺₁ are adjacent to every element of 𝐺₂ in 𝐺₁+𝐺₂, we have 𝑤𝐺₁+𝐺₂𝑆₁−𝑆₁𝐺−𝑐¹+𝐺₂𝑆₁𝐺=𝑤¹𝑆₁−𝑆₁𝐺−𝑐¹𝑆₁=Nr𝐺₁. (15)
Case 2. Let 𝑆=𝑆₂⊂𝑉(𝐺₂) be the Nr-set of 𝐺₂ such that 𝑤(𝐺₂/𝑆₂)−|𝑆₂|−𝑐(𝐺₂/𝑆₂)=Nr(𝐺₂). Since any elements from 𝐺₂ are adjacent to every element of 𝐺₁ in 𝐺₁+𝐺₂, we have 𝑤𝐺₁+𝐺₂𝑆₂−𝑆₂𝐺−𝑐¹+𝐺₂𝑆₂𝐺=𝑤²𝑆₂−𝑆₂𝐺−𝑐²𝑆₂=Nr𝐺₂. (16)
Case 3. Let 𝑆⊂𝑉(𝐺₁)∪𝑉(𝐺₂). Since 𝑆 contains at least one vertex of 𝑉(𝐺₁) which is adjacent to all the vertices of 𝑉(𝐺₂) and 𝑆 contains at least one vertex of 𝑉(𝐺₂) which is adjacent to all the vertices of 𝑉(𝐺₁) in 𝐺₁+𝐺₂, then (𝐺₁+𝐺₂)/𝑆 is empty. It contradicts to the definition of neighbor rupture degree.
By (15) and (16) Nr(𝐺₁+𝐺₂)=max{Nr(𝐺₁),Nr(𝐺₂)}.

For three or more disjoint graphs 𝐺₁,𝐺₂,𝐺₃,…,𝐺_𝑛 sequential join 𝐺₁+𝐺₂+⋯+𝐺_𝑛 is the graph (𝐺₁+𝐺₂)∪(𝐺₂+𝐺₃)∪⋯∪(𝐺_𝑛−1+𝐺_𝑛) [8].

The following theorem’s proof is very similar to that of Theorem 9.

Theorem 10. Let 𝐺₁, 𝐺₂, and 𝐺₃ be connected graphs. Then the neighbor rupture degree of sequential join of  𝐺₁, 𝐺₂, and 𝐺₃ is Nr𝐺₁+𝐺₂+𝐺₃=maxNr𝐺₁∪𝐺₃,Nr𝐺₂. (17)

Corollary 11. If 𝐺₂≅𝐾_𝑛, then Nr𝐺₁+𝐺₂+𝐺₃≥Nr𝐺₁+Nr𝐺₃. (18)

Corollary 12. If 𝑛₁≤𝑛₂≤𝑛₃, then Nr𝐾_𝑛1+𝐾_𝑛2+𝐾_𝑛3=−𝑛₃. (19)

Definition 13 (see [9]). The complement of a simple graph 𝐺 is obtained by taking the vertices of 𝐺 and joining two of them whenever they are not joined in 𝐺 and denoted by 𝐺^𝑐.

Theorem 14. Let 𝑃_𝑛 be a path graph of order 𝑛. Then Nr𝑃^𝑐_𝑛=−1. (20)

Proof. Let 𝑆 be a subversion strategy of 𝑃^𝑐_𝑛 and let 𝑆={𝑢} where 𝑢∈𝑉(𝑃_𝑛).
Case 1.   If deg(𝑢)=1 in 𝑃_𝑛, then 𝑢 is adjacent to all vertices in 𝑃^𝑐_𝑛 except its neighbor in 𝑃_𝑛. It means |𝑁[𝑢]|=𝑛−1 in 𝑃^𝑐_𝑛, then we have 𝑤𝑃^𝑐_𝑛𝑆−|𝑆|𝑃−𝑐^𝑐𝑛𝑆=1−1−1=−1. (21)
Case 2.   If deg(𝑢)=2 in 𝑃_𝑛, then 𝑢 is adjacent to all vertices in 𝑃^𝑐_𝑛 except its neighbors in 𝑃_𝑛. It means |𝑁[𝑢]|=𝑛−2 in 𝑃^𝑐_𝑛 where the remaining two vertices are adjacent. Therefore, 𝑤𝑃^𝑐_𝑛𝑆−|𝑆|𝑃−𝑐^𝑐𝑛𝑆=1−1−2=−2. (22) On the other hand, if we assume 𝑆 is a subversion strategy with |𝑆|≥2, then the remaining graph is empty. Therefore it contradicts to the definition of neighbor rupture degree.
From (21) and (22) we have Nr(𝑃^𝑐_𝑛)=−1.

The following theorem’s proof is very similar to that of Theorem 14.

Theorem 15. Let 𝑊_1,𝑛 be a wheel graph of order 𝑛+1. Then Nr𝑊^𝑐_1,𝑛=−1. (23)

Theorem 16. Let 𝐾_𝑚,𝑛 be a complete bipartite graph. Then Nr𝐾^𝑐_𝑚,𝑛=2−𝑛,2>𝑛−𝑚,−𝑚,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (24)

Proof. It is obvious that 𝐾^𝑐_𝑚,𝑛=𝐾_𝑚∪𝐾_𝑛.
According to Corollary 7 we get the result.

Corollary 17. Let 𝐾_1,𝑛 be a star graph of order 𝑛+1. Then Nr𝐾^𝑐_1,𝑛=−1. (25)

Definition 18 (see [7]). The cartesian product 𝐺=𝐺₁×𝐺₂ has 𝑉(𝐺)=𝑉(𝐺₁)×𝑉(𝐺₂), and two vertices (𝑢₁,𝑢₂) and (𝑣₁,𝑣₂) of 𝐺 are adjacent if and only if either 𝑢₁=𝑣₁,𝑢₂𝑣₂∈𝐸𝐺₂ (26) or 𝑢₂=𝑣₂,𝑢₁𝑣₁∈𝐸𝐺₁. (27)

Theorem 19. Let 𝑃₂×𝑃_3𝑎 be a cartesian product with 𝑎∈𝑍⁺. Then Nr𝑃₂×𝑃_3𝑎=𝑎−1. (28)

Proof. Let 𝑆 be a subversion strategy of 𝑃₂×𝑃_3𝑎 and |𝑆|=𝑟. There are two cases according to the number of elements in 𝑆.
Case 1. Let 1≤𝑟≤𝑎. Then 𝑁[𝑆]≤4𝑟 and 𝑤≤2𝑟.
Since 𝑐((𝑃₂×𝑃_3𝑎)/𝑆)≥(|𝑉(𝑆)|−|𝑁(𝑆)|)/𝑤≥((2.(3𝑎)−4𝑟)/2𝑟)=3𝑎/𝑟−2 and 𝑤𝑃₂×𝑃_3𝑎𝑆−|𝑆|𝑃−𝑐²×𝑃_3𝑎𝑆≤2𝑟−𝑟−3𝑎𝑟−2=𝑟+2−3𝑎𝑟. (29) Let 𝑓(𝑟)=𝑟+2−3𝑎/𝑟. 𝑓 is an increasing function since 𝑓^′(𝑟)=1+3𝑎/𝑟²>0. So it takes its maximum value at 𝑟=𝑎. Then 𝑓(𝑎)=𝑎−3𝑎/𝑎+2=𝑎−1. Hence, Nr𝑃₂×𝑃_3𝑎≤𝑎−1. (30)
Case 2. Let 𝑎≤𝑟≤|𝑉(𝑃₂×𝑃_3𝑎)|. Then 𝑐((𝑃₂×𝑃_3𝑎)/𝑆)≥1 and 𝑤((𝑃₂×𝑃_3𝑎)/𝑆)≤2(3𝑎)−4𝑎−(𝑟−𝑎)=3𝑎−𝑟, thus we obtain 𝑤((𝑃₂×𝑃_3𝑎)/𝑆)−|𝑆|−𝑐((𝑃₂×𝑃_3𝑎)/𝑆)≤3𝑎−𝑟−𝑟−1=3𝑎−2𝑟−1.
Let 𝑓(𝑟)=3𝑎−2𝑟−1. Since 𝑓^′<0, 𝑓 is a decreasing function, so it takes its maximum value at 𝑟=𝑎. Then 𝑓(𝑎)=3𝑎−2𝑎−1=𝑎−1Nr𝑃₂×𝑃_3𝑎≤𝑎−1. (31) From (30) and (31) we have Nr𝑃₂×𝑃_3𝑎≤𝑎−1. (32) It is obvious that there exist 𝑆^∗ such that |𝑆^∗|=𝑎, 𝑤((𝑃₂×𝑃_3𝑎)/𝑆^∗)=2𝑎 and 𝑐((𝑃₂×𝑃_3𝑎)/𝑆^∗)=1 so Nr𝑃₂×𝑃_3𝑎≥𝑎−1. (33) From (32) and (33) we have Nr(𝑃₂×𝑃_3𝑎)=𝑎−1.

The following theorems’ proofs are very similar to that of Theorem 19.

Theorem 20. Let 𝑃₂×𝑃_3𝑎+1 be a cartesian product with 𝑎∈𝑍⁺. Then Nr𝑃₂×𝑃_3𝑎+1=𝑎−2. (34)

Theorem 21. Let 𝑃₂×𝑃_3𝑎+2 be a cartesian product with 𝑎∈𝑍⁺. Then Nr𝑃₂×𝑃_3𝑎+2=𝑎−1. (35)

Theorem 22. Let 𝐾_𝑚 and 𝐾_𝑛 be two complete graphs with (𝑚≤𝑛). Then Nr𝐾_𝑚×𝐾_𝑛=1−𝑛. (36)

Proof. Let 𝑆 be a subversion strategy of 𝐾_𝑚×𝐾_𝑛 and let |𝑆|=𝑟. We have two cases according to the cardinality of 𝑆.
Case 1. If 0≤𝑟<𝑚−1, then 𝑤((𝐾_𝑚×𝐾_𝑛)/𝑆)=1 and 𝑐((𝐾_𝑚×𝐾_𝑛)/𝑆)≥(𝑚−𝑟)(𝑛−𝑟), so we have 𝑤𝐾_𝑚×𝐾_𝑛𝑆−|𝑆|𝐾−𝑐^𝑚×𝐾_𝑛𝑆≤1−𝑟−(𝑚−𝑟)(𝑛−𝑟). (37) Let 𝑓(𝑟)=1−𝑟−𝑚𝑛+(𝑚+𝑛)𝑟−𝑟². Since 𝑓(𝑟) is an increasing function in (0,𝑚−1), it takes its maximum value at 𝑟=𝑚−1 and 𝑓(𝑚−1)=1−𝑛. Thus we get Nr𝐾_𝑚×𝐾_𝑛≤1−𝑛. (38)
Case 2. If 𝑚−1≤𝑟≤(𝑚−1)(𝑛−1), then 𝑤((𝐾_𝑚×𝐾_𝑛)/𝑆)=1 and 𝑐((𝐾_𝑚×𝐾_𝑛)/𝑆)≥𝑛−𝑚+1. So we have 𝑤((𝐾_𝑚×𝐾_𝑛)/𝑆)−|𝑆|−𝑐((𝐾_𝑚×𝐾_𝑛)/𝑆)≤1−𝑟−(𝑛−𝑚+1)=𝑚−𝑛−𝑟.
Let 𝑓(𝑟)=𝑚−𝑛−𝑟. Since 𝑓(𝑟) is a decreasing function, it takes its maximum value at 𝑟=𝑚−1 and 𝑓(𝑚−1)=1−𝑛. Thus we get Nr𝐾_𝑚×𝐾_𝑛≤1−𝑛. (39) From (38) and (39) we have Nr≤1−𝑛. (40) There exist 𝑆^∗ such that 𝑟=𝑚−1, 𝑤((𝐾_𝑚×𝐾_𝑛)/𝑆^∗)=1 and 𝑐((𝐾_𝑚×𝐾_𝑛)/𝑆^∗)=𝑛−𝑚+1, thus we have Nr≥1−𝑛. (41) From (40) and (41) we get Nr=1−𝑛.

Definition 23 (see [9]). The tensor product 𝐺₁⊗𝐺₂ of two simple graphs 𝐺₁ and 𝐺₂ is the graph with 𝑉(𝐺₁⊗𝐺₂)=𝑉₁×𝑉₂ and where in (𝑢₁,𝑢₂) and (𝑣₁,𝑣₂) are adjacent in 𝐺₁⊗𝐺₂ if, and only if, 𝑢₁ is adjacent to 𝑣₁ in 𝐺₁ and 𝑢₂ is adjacent to 𝑣₂ in 𝐺₂.

Theorem 24. Let 𝑃₃⊗𝑃_𝑛 be a tensor product of 𝑃₃ and 𝑃_𝑛  and 𝑛≡0(mod4). Then Nr𝑃₃⊗𝑃_𝑛=𝑛−1. (42)

Proof. Let 𝑆 be a subversion strategy of 𝑃₃⊗𝑃_𝑛 and |𝑆|=𝑟 be the number of removing vertices from 𝑃₃⊗𝑃_𝑛. There are two cases according to the number of elements in 𝑆.
Case 1. If 0≤𝑟≤(𝑛/2), then 𝑤((𝑃₃⊗𝑃_𝑛)/𝑆)≤𝑛+𝑟 and 𝑐((𝑃₃⊗𝑃_𝑛)/𝑆)≥1. Thus we have 𝑤𝑃₃⊗𝑃_𝑛𝑆−|𝑆|𝑃−𝑐³⊗𝑃_𝑛𝑆≤𝑛−1. (43)
Case 2. If (𝑛/2)≤𝑟≤3𝑛, then 𝑤((𝑃₃⊗𝑃_𝑛)/𝑆)≤𝑛+(𝑛/2)−((𝑟−(𝑛/2))=2𝑛−𝑟 and 𝑐((𝑃₃⊗𝑃_𝑛)/𝑆)≥1. Thus we have 𝑤𝑃₃⊗𝑃_𝑛𝑆−|𝑆|𝑃−𝑐³⊗𝑃_𝑛𝑆≤2𝑛−2𝑟−1. (44) Let 𝑓(𝑟)=2𝑛−2𝑟−1 since 𝑓^′(𝑟)<0 the function 𝑓(𝑟) is a decreasing function so it takes its maximum value at 𝑛𝑟=2𝑛,𝑓2=𝑛−1. (45) From (43) and (45) we get Nr𝑃₃⊗𝑃_𝑛≤𝑛−1. (46) There exist 𝑆^∗ such that |𝑆^∗|=(𝑛/2), 𝑤((𝑃₃⊗𝑃_𝑛)/𝑆)=𝑛+(𝑛/2) and 𝑐((𝑃₃⊗𝑃_𝑛)/𝑆)≥1 thus we have Nr𝑃₃⊗𝑃_𝑛≥𝑛−1. (47) From (46) and (47) we get Nr(𝑃₃⊗𝑃_𝑛)=𝑛−1.

The following theorem’s proof is very similar to that of Theorem 24.

Theorem 25. Let 𝑃₃⊗𝑃_𝑛 be a tensor product of 𝑃₃ and 𝑃_𝑛 and 𝑛≠0(mod4). Then Nr𝑃₃⊗𝑃_𝑛()=2𝑛−2𝑛+12−1. (48)

Theorem 26. Let the tensor product of 𝐾_𝑚 and 𝐾_𝑛 is 𝐾_𝑚⊗𝐾_𝑛. Then Nr𝐾_𝑚⊗𝐾_𝑛=Nr𝐾_{𝑚−1,𝑛−1}−1=max{𝑚−5,𝑛−5}. (49)

Proof. Let (𝑎,𝑏) be any vertex of 𝐾_𝑚⊗𝐾_𝑛 such that 𝑎∈𝐾_𝑚 and 𝑏∈𝐾_𝑛. The only vertices that are not adjacent to (𝑎,𝑏) in (𝐾_𝑚⊗𝐾_𝑛) are (𝑎,𝑐_𝑗) with (𝑗=1,2,…,𝑛) and (𝑑_𝑖,𝑏) with (𝑖=1,2,…,𝑚), where 𝑐_𝑗∈𝑉(𝐾_𝑛) and 𝑑_𝑖∈𝑉(𝐾_𝑚).
The vertices (𝑎,𝑐_𝑗) are not adjacent to each other, neither do the vertices (𝑑_𝑖,𝑏). But these are adjacent to each other, so 𝐾_𝑚⊗𝐾_𝑛{𝑎,𝑏}≅𝐾_{𝑚−1,𝑛−1},Nr𝐾_𝑚⊗𝐾_𝑛=Nr𝐾_{𝑚−1,𝑛−1}{}{}.−1=max𝑚−4,𝑛−4−1=max𝑚−5,𝑛−5 (50)

Definition 27 (see [10]). The composition of simple graphs 𝐺 and 𝐻 is the simple graph 𝐺[𝐻] with vertex set 𝑉(𝐺)×𝑉(𝐻), in which (𝑢,𝑣) is adjacent (𝑢^′,𝑣^′) if and only if either 𝑢𝑢^′∈𝐸(𝐺) or 𝑢=𝑢^′ and 𝑣𝑣^′∈𝐸(𝐻).

Theorem 28. Let 𝑃₃[𝑃_𝑛] be the composition of 𝑃₃ and 𝑃_𝑛 with 𝑛≥5. Then neighbor rupture degree of 𝑃₃[𝑃_𝑛] is Nr𝑃₃𝑃_𝑛=(),().1,𝑛≡1mod40,𝑛≡0,2,3mod4 (51)

Proof. Let the vertex set of 𝑃₃[𝑃_𝑛] be labeled as 𝑃^𝑖_𝑛, 𝑃_𝑛^𝑖𝑖, and 𝑃_𝑛^𝑖𝑖𝑖. (52)
It can be easily seen that 𝑃₃[𝑃_𝑛] is the sequential join of three disjoint path graphs, 𝑃₃[𝑃_𝑛]≅𝑃_𝑛+𝑃_𝑛+𝑃_𝑛. Then, according to the Theorem 10 we get Nr𝑃₃𝑃_𝑛=Nr𝑃_𝑛+𝑃_𝑛+𝑃_𝑛=maxNr𝑃_𝑛∪𝑃_𝑛,Nr𝑃_𝑛. (53) By Theorem 2, we have Nr𝑃_𝑛=(),().0,𝑛≡1mod4−1,𝑛≡0,2,3mod4 (54)
To conclude the proof we need to find Nr(𝑃_𝑛∪𝑃_𝑛). Let 𝑆 be a subversion strategy of 𝑃_𝑛∪𝑃_𝑛. Since 𝑃_𝑛∪𝑃_𝑛 has two identical disjoint path graphs, let 𝑥 denote the number of removing vertices of each 𝑃_𝑛 and let |𝑆|=2𝑥.
Case 1. If 𝑥≤⌊(𝑛−1)/4⌋,𝑤((𝑃_𝑛∪𝑃_𝑛)⧵𝑆)≤2𝑥+2 and 𝑐((𝑃_𝑛∪𝑃_𝑛)⧵𝑆)≥⌈(2𝑛−6𝑥)/(2𝑥+2)⌉=⌈(𝑛−3𝑥)/(𝑥+1)⌉, then we get 𝑤𝑃_𝑛∪𝑃_𝑛⧵𝑆−|𝑆|−𝑐𝑃_𝑛∪𝑃_𝑛⧵𝑆≤2−𝑛−3𝑥.𝑥+1 (55) Let 𝑓(𝑥)=2−(𝑛−3𝑥)/(𝑥+1), since 𝑓(𝑥) is an increasing function it takes its maximum value at 𝑥=⌊(𝑛−1)/4⌋. 𝑓(⌊(𝑛−1)/4⌋)=5−(𝑛+3)/(⌊(𝑛−1)/4⌋+1). Since neighbor rupture degree has to be an integer, we have Nr(𝑃_𝑛∪𝑃_𝑛)≤⌈5−(𝑛+3)/(⌊(𝑛−1)/4⌋+1)⌉ and we get Nr𝑃_𝑛∪𝑃_𝑛≤(),().1,𝑛≡1mod40,𝑛≡0,2,3mod4 (56)
Case 2. If 𝑥≥⌊(𝑛−1)/4⌋, 𝑤((𝑃_𝑛∪𝑃_𝑛)⧵𝑆)≤2𝑥, and 𝑐((𝑃_𝑛∪𝑃_𝑛)⧵𝑆)≥1, thus we get Nr𝑃_𝑛∪𝑃_𝑛≤−1. (57) From (56) and (57) we have Nr𝑃_𝑛∪𝑃_𝑛≤(),().1,𝑛≡1mod40,𝑛≡0,2,3mod4 (58) There exist 𝑆^∗ such that |𝑆^∗|=2⌊(𝑛−1)/4⌋, 𝑤((𝑃_𝑛∪𝑃_𝑛)⧵𝑆)=(2⌊(𝑛−1)/4⌋+1) and 𝑐((𝑃_𝑛∪𝑃_𝑛)⧵𝑆)=2 for 𝑛≡0,2,3(mod4), 𝑐((𝑃_𝑛∪𝑃_𝑛)⧵𝑆)=1 for 𝑛≡1(mod4).
Thus we have Nr𝑃_𝑛∪𝑃_𝑛≥(),().1,𝑛≡1mod40,𝑛≡0,2,3mod4 (59) From (58) and (59) we get Nr𝑃_𝑛∪𝑃_𝑛=(),().1,𝑛≡1mod40,𝑛≡0,2,3mod4 (60) By (54) and (60) we conclude the proof.

Theorem 29. Neighbor rupture degree of composition of 𝐾_𝑚 and any graph 𝐺 is Nr𝐾^𝑚[𝐺]=Nr(𝐺). (61)

Proof. Let the vertex set of 𝐾_𝑚[𝐺] be labeled as, 𝐺^𝑖,𝐺^𝑖𝑖,…,𝐺^𝑚, (62) Let 𝑆 be a subversion strategy of 𝐾_𝑚[𝐺]. We have two cases according to the elements of 𝑆.
Case 1. Let we choose one element from any vertex set 𝐺^𝑖(𝑖=1,2,…,𝑚); if 𝑢∈𝑉(𝐺^𝑖) and 𝑆={𝑢}, then it removes all of other vertex sets. So it depends on only 𝐺^𝑖 which we choose one element. Then we have Nr𝐾_𝑚[𝐺]=Nr𝐺^𝑖=Nr(𝐺). (63)
Case 2. Let we choose two elements from any vertex set 𝐺^𝑖(𝑖=1,2,…,𝑚) and 𝐺^𝑗(𝑗=1,2,…,𝑚) with 𝑖≠𝑗. Then (𝐾_𝑚[𝐺])/𝑆 is empty set. It contradicts to the definition of neighbor rupture degree. Thus we obtain Nr𝐾_𝑚[𝐺]=Nr(𝐺). (64)

Definition 30 (see [11]). An 𝑛th power of a graph 𝐺 is formed by adding an edge between all pairs of vertices of 𝐺 with distance at most 𝑛. If 𝑛=2 then it is called a second power of a graph also called a square.

Theorem 31. Neighbor rupture degree of 𝑃²_𝑛(𝑛>6) is Nr𝑃²_𝑛=0,𝑛≡1(mod6),−1,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (65)

Proof. Let 𝑆 be a subversion strategy of 𝑃²_𝑛 and let |𝑆|=𝑟. There are two cases according to the number of elements of 𝑆.
Case 1. If 0≤𝑟≤⌈𝑛/6⌉−1, then 𝑤(𝑃²_𝑛/𝑆)≤𝑟+1, 𝑐(𝑃²_𝑛/𝑆)≥(𝑛−5𝑟)/(𝑟+1); then we get 𝑤𝑃²_𝑛𝑆−|𝑆|−𝑐𝑃²_𝑛⧵𝑆≤𝑟+1−𝑟−𝑛−5𝑟𝑟+1=6−𝑛+5.𝑟+1 (66) Let 𝑓(𝑟)=6−𝑛+5=(),𝑟+10,𝑛≡1mod6<0,otherwise. (67)
Case 2. If =⌈𝑛/6⌉, then 𝑤(𝑃²_𝑛/𝑆)≤𝑟,𝑐(𝑃²_𝑛/𝑆)≥1; then we get 𝑤𝑃²_𝑛𝑆−|𝑆|𝑃−𝑐^2𝑛𝑆≤𝑟−𝑟−1=−1thereforeNr≤−1. (68)
Case 3. If ⌈𝑛/6⌉+1≤𝑟≤𝑛, then 𝑤(𝑃²_𝑛/𝑆)≤𝑟−1, 𝑐(𝑃²_𝑛/𝑆)≥1; then we get 𝑤𝑃²_𝑛𝑆−|𝑆|−𝑐𝑃²_𝑛⧵𝑆≤𝑟−1−𝑟−1=−2thereforeNr≤−2. (69) According to (67), (68), and (69) we have Nr𝑃²_𝑛≤0,𝑛≡1(mod6),−1,otherwise. (70) There exist 𝑆^∗ such that |𝑆^∗|=⌈(𝑛−1)/6⌉, 𝑤(𝑃²_𝑛/𝑆^∗)=⌈𝑛/6⌉, 𝑐(𝑃²_𝑛/𝑆^∗)=1; then 𝑤𝑃²_𝑛𝑆^∗−𝑆^∗𝑃−𝑐^2𝑛𝑆^∗=0,𝑛≡1(mod6),1,otherwise. (71) By (70) and (71) we get the result, 𝑃Nr^2𝑛𝑆𝑤𝑃=max^2𝑛𝑆−|𝑆|𝑃−𝑐^2𝑛𝑆=0,𝑛≡1(mod6),1,otherwise. (72)

4. Conclusion

In this study, we investigate the neighbor rupture degree of graphs obtained by graph operations. The graph operations are used to obtain new graphs. Union, join, complement, composition, power, cartesian product, and tensor product are taken into consideration in this work. These operations are performed to various graphs and their neighbor rupture degrees were determined.