Mostar index of graphs associated to groups

1 Introduction

A connectivity index is a form of molecular attribute whose computation depends upon a chemical graph of a chemical substance in the subject of mathematical chemistry. A large variety of numerical values, also known as topological indices, have been suggested and explored in attempt to distil and collect, or summarie, the content contained in graph connectivity patterns (Todeschini and Conosonni, 2002). Topological indices are numerical quantities that describe the topology of a graph and are generally graph invariant (Qiu and Akl, 1995). For instance, the Wiener index is based on the topological proximity of vertices in a graph, and was defined by Wiener in 1947 to estimate the boiling properties of various alkane isomers. Since then, more than 3,000 topological indices of graphs have been recorded in chemical databases. Another type of topological index aims to measure the nonbalancedness among the bonds of a chemical graph on the base of peripherality of bonds (edges), which is named as the Mostar index (Akhter et al., 2021; Došlić et al., 2018). The Mostar index recently discovered as a bond-additive connectivity index determines the quantity of peripherality of certain edges as well as the graph as a whole (Akhter et al., 2021; Došlić et al., 2018). It is a distinct geometric index which counts the contribution |η _λ − η _μ | of every edge e = λμ in a connected graph, where η _λ is the quantity of vertices that are closer to the vertex λ than the vertex μ, and η _μ is defined in the same way (Ali and Doslic, 2021). Accordingly, this index indicates the degree of specific edges and the degree of peripherality of the graph as a whole. This index attracted many graph theorists to measure the peripherality of various (chemical) graphs. Tepeh addressed the first conjecture about the Mostar index of bicyclic graphs (Tepeh, 2019), which was proposed by Došlić et al. (2018). Further, remarkable work on the Mostar index of carbon nanostructures, trees, and hexagonal chains has been supplied by Arockiaraj et al. (2019), Hayat and Zhou (2019), and Huang et al. (2020).

A classical study of graphs associated with groups attracted many researchers to explore the various theoretical and topological properties of graph. A large number of interesting work have been published by supplying the articles by Abdollahi et al. (2006), Ali et al. (2016), Alolayan et al. (2019), Bhuniya and Bera (2016), Bunday (2006), Cameron and Ghosh (2011), Chakrabarty et al. (2009), and Rahman (2017). Some of the graphs associated to group are defined as follows: Let a group Γ and the center of group Γ be ζ(Γ) = {λ ∈ Γ: λμ = μλ∀μ ∈ Γ}. Γ _G denotes the commuting graph of Γ with the vertex set Γ and two distinct vertices λ and μ ∈ Γ from an edge in Γ _G if and only if λμ = μλ in Γ (Ali et al., 2016; Bunday, 2006). The non-commuting graph of Γ is denoted by G _Γ with the vertex set Γ − ζ(Γ) and two distinct vertices λ and μ ∈ Γ from an edge in G _Γ if and only if λμ ≠ μλ in Γ (Abdollahi et al., 2006; Moghaddamfar et al., 2005; Wei et al., 2020). If λ = gμg ⁻¹ or μ = g ⁻¹ λg for g ∈ Γ, then λ and μ are said to be conjugate of each other. This relation between elements of Γ is an equivalence relation and is called the conjugacy relation. Due to this equivalence relation, Γ is partitioned into disjoint classes each of which is called a conjugacy class. Mathematically, the conjugacy class of λ ∈ Γ is Cl(λ) = {gλg ⁻¹:g ∈ Γ}. G(Γ) denotes a non-conjugate graph with the vertex set Γ and two different vertices λ and μ ∈ Γ from an edge in G(Γ) if and only if λ and μ belong to different conjugacy classes (Alolayan et al., 2019).

Graphs associated with the dihedral group have been considered by Abbas et al. (2021), Salman et al. (2022), and Wei et al. (2020) to study their topological properties such as the Wiener related indices, Harary index, Randić indices, geometric arithmetic indices, atom bond connectivity indices, harmonic index, Hosoya index, and polynomials. This study is aimed to investigate the Mostar index of graphs (commuting, non-commuting, and non-conjugate graphs) associated with the dihedral and semi-dihedral groups.

2 Preliminaries

This section provides partitions of groups under consideration, some basic terminologies of a graph, and the mechanism to compute the Mostar index.

The generating form D _n = 〈a, b|a ⁿ = b ² = e, ab = ba ⁻¹〉 represents the dihedral group of order 2n, which is the collection of symmetries of regular n-polygon. The center of D _n is:

Let us partition D _n as follows:

Ω₁ = {e, a, a ²,…, a ⁿ⁻¹}, Ω₂ = {b, ab, a ² b,…, a ⁿ⁻¹ b}, and Ω₃ = Ω₁− ζ(D _n ), then |Ω₁| = |Ω₂| = n. Further for even n, let Ω 2 = ⋃ i = 0 n 2 − 1 Ω 2 i with Ω 2 i = a i b , a i + n 2 be the subsets according to the commuting elements from Ω₂, and conjugacy classes of D _n are:

The generating form SD_8n = 〈a, b|a ⁴ⁿ = b ² = e, ba = a ²ⁿ⁻¹ b〉 represents the semi-dihedral group of order 8n with:

and

Let us partition SD_8n as follows:

Ω₁ = {e, a, a ²,…, a ⁴ⁿ⁻¹}, Ω₂ = {b, ab, a ² b,…, a ⁴ⁿ⁻¹ b}, and Ω₃ = Ω₁ − ζ(D _n ), then |Ω₁| = |Ω₂| = 4n. For odd n ≥ 3, let Ω 2 = ⋃ i = 0 n − 1 Ω 2 i with Ω 2 i = { a i b , a i + n b , a i + 2 n b , a i + 3 n b } be the subsets according to the commuting elements from Ω₂, and conjugacy classes of SD_8n are:

where ⋃ j = 0 3 Φ 2 j = Ω 2 and ⋃ odd i = 1 n − 1 Φ 3 i ⋃ odd i = 2 n + 1 3 n − 2 Φ 3 i ⋃ even i = 2 2 n − 2 Φ 3 i = Ω 3 .

For even n ≥ 2, let Ω 2 = ⋃ i = 0 2 n − 1 Ω 2 i with Ω 2 i = { a i b , a i + 2 n b } be the subsets according to the commuting elements from Ω₂, and conjugacy classes of SD_8n are:

where ⋃ j = 1 2 Φ 2 j = Ω 2 and ⋃ odd i = 1 n − 1 Φ 3 i ⋃ odd i = 2 n + 1 3 n − 1 Φ 3 i ⋃ even i = 2 2 n − 2 Φ 3 i = Ω 3 .

Let a connected and simple graph G have vertex and edge sets symbolized by V(G) and E(G), respectively. We denote the number of edges(size) of a graph G by S(G). The notation K + H denotes the sum of two graphs K and H with V(K) ∪ V(H) as the vertex set and E(K) ∪ E(H) ∪ {λ ∼ μ: λ ∈ V(K) ∧ μ ∈ V(H)} as the edge set. The number of edges in a shortest path between two distinct vertices λ and μ is defined as the distance across λ and μ, indicated by d(λ, μ). The eccentricity of a vertex μ is the number

For an edge e = λμ, peripheral neighborhoods of e according to its end vertices λ and μ are defined, Arockiaraj et al. (2020), as follows:

Then, η _λ (e|G) = |N _λ (e|Γ _G )| and η _μ (e|G) = |N _μ (e|Γ _G )| are the peripheral degrees of e. Akhter et al. in 2021 and Došlić et al. in 2018 provided the following formula to compute the Mostar index of a graph G

The number of vertices adjacent to a vertex λ in G is called the degree of λ and it is denoted by d(λ). A vertex of degree 1 is known as a leaf in G. Whenever we need to find the size, S(G) of a graph G, we will use the formula ∑ λ ∈ V ( G ) d ( λ ) = 2 S ( G ) provided by the well-known handshake lemma (Rahman, 2017).

Then, N _k′(λ) = N _k (λ) − (∪ _t≤k N _t (μ)). Accordingly, N λ ( e | G ) = ⋃ k = 0 ecc ( λ ) N k ′ ( λ ) .

A neighbor of λ is a vertex adjacent to it in a graph G. The open neighborhood, N(λ), of λ in G is the set of all the neighbors of λ. The closed neighborhood of λ is N[λ] = N(λ) ∪ {λ}. Two vertices λ and μ are false twins in G whenever N(λ) = N(μ), and are true twins whenever N[λ] = N[μ].

3 Commuting graphs

The Mostar index of commuting graphs on the dihedral and semi-dihedral groups is computed in this section.

Case 1 (n is odd):

Let E ( Γ G ) = ⋃ i = 1 3 E i ( Γ G ) with E _i (Γ _G ) = {e ∈ E(Γ _G )|e is of type T _i }.

• Type T ₁: e = λμ ∈ E ₁(Γ _G ) such that λ ∈ Ω₃ and μ ∈ Ω₃.

• Type T ₂: e = λμ ∈ E ₂(Γ _G ) such that λ ∈ Ω₃ and μ ∈ ζ(Γ).

• Type T ₃: e = λμ ∈ E ₃(Γ _G ) such that λ ∈ ζ(Γ) and μ ∈ Ω₂.

Let e is of type T ₁ : Note that N[λ] = ζ(Γ) ∪ Ω₃ = N[μ]. Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₁ be the number of type T ₁ edges. Since Ω₃ induces a complete graph K _n−1, so t 1 = S ( K n − 1 ) = n − 1 2 .

Let e is of type T ₂ : Since ecc(λ) = 2 and ecc(μ) = 1, so N ₀(λ) = {λ}, N ₁(λ) = Ω₁, N ₂(λ) = Ω₂ N ₀(μ) = {μ}, N ₁(μ) = Ω₃ ∪ Ω₂. Accordingly, Remark 1 implies that:

and hence η _λ (e|Γ _G ) = 1 and η _μ (e|Γ _G ) = n + 1. Let t ₂ be the number of type T ₂ edges, then t ₂ = |Ω₃| × |ζ(Γ)| = n − 1.

Let e is of type T ₃ : Note that μ is a leaf in Γ _G , so Proposition 1 yields that η _μ (e|Γ _G ) = 1 and η _λ (e|Γ _G ) = 2n − 1. Let t ₃ be the number of type T ₃ edges, then t ₃ = |ζ(Γ)| × |Ω₂| = n. Now, the Mostar index of Γ _G is:

Case 2 (n is even):

Let E ( Γ G ) = ⋃ i = 1 5 E i ( Γ G ) with E _i (Γ _G ) = {e ∈ E(Γ _G )|e is of type T _i }.

• Type T ₁: e = λμ ∈ E ₁(Γ _G ) such that λ ∈ Ω₃ and μ ∈ Ω₃.

• Type T ₂: e = λμ ∈ E ₂(Γ _G ) such that λ ∈ ζ(Γ) and μ ∈ ζ(Γ).

• Type T ₃: e = λμ ∈ E ₃(Γ _G ) such that λ ∈ Ω 2 i and μ ∈ Ω 2 i .

• Type T ₄: e = λμ ∈ E ₄(Γ _G ) such that λ ∈ Ω₃ and μ ∈ ζ(Γ).

• Type T ₅: e = λμ ∈ E ₅(Γ _G ) such that λ ∈ ζ(Γ) and μ ∈ Ω 2 i .

Let e is of type T ₁ : Note that N[λ] = ζ(Γ) ∪ Ω₃ = N[μ]. Thus λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₁ be the number of type T ₁ edges. Since Ω₃ induces a complete graph K _n−2, so t 1 = S ( K n − 2 ) = n − 2 2 .

Let e is of type T ₂ : Note that N[λ] = ζ(Γ) ∪ Ω₃ ∪ Ω₂ = N[μ]. Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₂ be the number of type T ₂ edges. Since ζ(Γ) induces a complete graph K ₂, so t ₂ = S(K ₂) = 1.

Let e is of type T ₃ : Note that N [ λ ] = ζ ( Γ ) ∪ Ω 2 i = N [ μ ] . Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₃ be the number of type T ₃ edges. Since each Ω 2 i induces the complete graph K ₂ and 0 ≤ i ≤ n 2 − 1 , so t 3 = n 2 × S ( K 2 ) = n 2 .

Let e is of type T ₄ : Since ecc(λ) = 2 and ecc(μ) = 1, so:

Accordingly, Remark 1 implies that:

and hence η _λ (e|Γ _G ) = 1 and η _μ (e|Γ _G ) = n + 1. Let t ₄ be the number of type T ₄ edges, then t ₄ = |Ω₃| × |ζ(Γ)| = 2(n − 2).

Let e is of type T ₅ : As ecc(λ) = 2 and ecc(μ) = 2, so:

Accordingly, Remark 1 implies that:

and hence η _λ (e|Γ _G ) = 2n − 3 and η _μ (e|Γ _G ) = 1. Let t ₅ be the number of type T ₅ edges, then t ₅ = |Ω₂| × |ζ(Γ)| = 2n.

Now, the Mostar index of Γ _G is:

Case 1 (n is odd):

Let E ( Γ G ) = ⋃ i = 1 5 E i ( Γ G ) with E _i (Γ _G ) = {e∈ E(Γ _G )|e is of type T _i }.

• Type T ₁: e = λμ ∈ E ₁(Γ _G ) such that λ ∈ Ω₃ and μ ∈ Ω₃.

• Type T ₂: e = λμ ∈ E ₂(Γ _G ) such that λ ∈ ζ(Γ) and μ ∈ ζ(Γ).

• Type T ₃: e = λμ ∈ E ₃(Γ _G ) such that λ ∈ Ω 2 i and μ ∈ Ω 2 i .

• Type T ₄: e = λμ ∈ E ₄(Γ _G ) such that λ ∈ Ω₃ and μ ∈ ζ(Γ).

• Type T ₅: e = λμ ∈ E ₅(Γ _G ) such that λ ∈ ζ(Γ) and μ ∈ Ω 2 i .

Let e is of type T ₁ : Note that N[λ] = ζ(Γ) ∪ Ω₃ = N[μ]. Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₁ be the number of type T ₁ edges. Since Ω₃ induces a complete graph K _4n−4, so t 1 = S ( K 4 n − 4 ) = 4 n − 4 2 .

Let e is of type T ₂ : Note that N[λ] = ζ(Γ) ∪ Ω₃ ∪ Ω₂ = N[μ]. Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₂ be the number of type T ₂ edges. Since ζ(Γ) induces a complete graph K ₄, so t 2 = S ( K 4 ) = 4 2 .

Let e is of type T ₃ : Note that N [ λ ] = ζ ( Γ ) ∪ Ω 2 i = N [ μ ] . Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₃ be the number of type T ₃ edges. Since each Ω 2 i induces the complete graph K ₄ and 0 ≤ i ≤n − 1, so t ₃ = n × S(K ₄) = 6n.

Let e is of type T ₄ : Since ecc(λ) = 2 and ecc(μ) = 1, so:

Accordingly, Remark 1 implies that:

and hence η _λ (e|Γ _G ) = 1 and η _μ (e|Γ _G ) = 4n + 1. Let t ₄ be the number of type T ₄ edges, then t ₄ = |Ω₃| × |ζ(Γ)| = 4(4n − 4).

Let e is of type T ₅ : Since ecc(λ) = 1 and ecc(μ) = 2, so N ₀(λ) = {λ}, N ₁(λ) = Ω₃ ∪ ζ(Γ) − {λ} ∪ Ω₂, N ₀(μ) = {μ}, N 1 ( μ ) = Ω 2 i − { μ } ∪ ζ ( Γ ) , and N 2 ( μ ) = Ω 3 ∪ Ω 2 − Ω 2 i .

Accordingly, Remark 1 implies that:

and hence η _λ (e|Γ _G ) = 8n − 7 and η _μ (e|Γ _G ) = 1. Let t ₅ be the number of type T ₅ edges, then t ₅ = |Ω₂| × |ζ(Γ)| = 16n.

Now, the Mostar index of Γ _G is:

Case 2 (n is even):

Let E ( Γ G ) = ⋃ i = 1 5 E i ( Γ G ) with E _i (Γ _G ) = {e ∈ E(Γ _G )|e is of type T _i }.

• Type T ₁: e = λμ ∈ E ₁(Γ _G ) such that λ ∈ Ω₃ and μ ∈ Ω₃.

• Type T ₂: e = λμ ∈ E ₂(Γ _G ) such that λ ∈ ζ(Γ) and μ ∈ ζ(Γ).

• Type T ₃: e = λμ ∈ E ₃(Γ _G ) such that λ ∈ Ω 2 i and μ ∈ Ω 2 i .

• Type T ₄: e = λμ ∈ E ₄(Γ _G ) such that λ ∈ Ω₃ and μ ∈ ζ(Γ).

• Type T ₅: e = λμ ∈ E ₅(Γ _G ) such that λ ∈ ζ(Γ) and μ ∈ Ω 2 i .

Let e is of type T ₁ : Note that N[λ] = ζ(Γ) ∪ Ω₃ = N[μ]. Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₁ the number of type T ₁ edges. Since Ω₃ induces a complete graph K _4n−2, so t 1 = S ( K 4 n − 2 ) = 4 n − 2 2 .

Let e is of type T ₂ : Note that N[λ] = V(Γ _G ) = N[μ]. Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₂ be the number of type T ₂ edges. Since ζ(Γ) induces a complete graph K ₂, so t ₂ = S(K ₂) = 1.

Let e is of type T ₃ : Note that N [ λ ] = ζ ( Γ ) ∪ Ω 2 i = N [ μ ] . Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|Γ _G ) = 1 = η _μ (e|Γ _G ). Let t ₃ be the number of type T ₃ edges. Since each Ω 2 i induces the complete graph K _2| and 0 ≤ i ≤ 2n − 1, so t ₃ = 2n × S(K ₂) = 2n.

Let e is of type T ₄ : Since ecc(λ) = 2 and ecc(μ) = 1, so N ₀(λ) = {λ}, N ₁(λ) = (Ω₃ − {λ}) ∪ ζ(Γ), N ₂(λ) = Ω₂, N ₀(μ) = {μ}, and N ₁(μ) = V(Γ _G ) − {μ}. Accordingly, Remark 1 implies that:

and hence n _λ (e|Γ _G ) = 1 and n _μ (e|Γ _G ) = 4n + 1. Let t ₄ be the number of type T ₄ edges, then t ₄ = |Ω₃| × |ζ(Γ)| = 2(4n − 2).

Let e is of type T ₅ : Since ecc(λ) = 1 and ecc(μ) = 2. So N ₀(λ) = {λ}, N ₁(λ) = Ω₃ ∪ (ζ(Γ) − {λ} ∪ Ω₂, N ₀(μ) = {μ}, N 1 ( μ ) = ( Ω 2 i − { μ } ) ∪ ζ ( Γ ) , and N 2 ( μ ) = Ω 3 ∪ ( Ω 2 − Ω 2 i ) .

Accordingly, Remark 1 implies that:

and hence n _λ (e|Γ _G ) = 8n − 3 and n _μ (e|Γ _G ) = 1. Let t ₅ be the number of type T ₅ edges, then t ₅ = |Ω₂| × |ζ(Γ)| = 8n.

Now, the Mostar index of Γ _G is:

4 Non-commuting graphs

In this section, by measuring the amount of peripherality of each edge, we investigate the Mostar index of non-commuting graphs associated with D _n and SD_8n .

Next we discuss the following two cases.

Case 1 (n is odd):

Let E ( Γ G ) = ⋃ i = 1 2 E i ( Γ G ) with E _i (Γ _G ) = {e ∈ E(Γ _G )|e is of type T _i }.

• Type T ₁: e = λμ ∈ E ₁(G _Γ) such that λ ∈ Ω₂ and μ ∈ Ω₂.

• Type T ₂: e = λμ ∈ E ₂(G _Γ) such that λ ∈ Ω₂ and μ ∈ Ω₃.

Let e is of type T ₁ : Note that N[λ] = V(G _Γ) = N[μ]. Thus, λ and μ are true twins, so Proposition 3 yields that η _λ (e|G _Γ) = 1 = η _μ (e|G _Γ). Let t ₁ be the number of type T ₁ edges. Since Ω₂ induces a complete graph K _n , so t 1 = S ( K n ) = n 2 .

Let e is of type T ₂ : Since ecc(λ) = 1 and ecc(μ) = 2, so N ₀(λ) = {λ}, N ₁(λ) = Ω₃ ∪ (Ω₂ − {λ}, N ₀(μ) = {μ}, N ₁(μ) = Ω₂, and N ₂(μ) = Ω₃ − {μ}. Accordingly, Remark 1 implies that:

and hence η _λ (e|G _Γ) = n − 1 and η _μ (e|G _Γ) = 1. Let t ₂ be the number of type T ₂ edges, then t ₂ = |Ω₂| × |Ω₃| = n(n − 1).

Now, the Mostar index of G _Γ is:

Case 2 (n is even):

Let E ( Γ G ) = ⋃ i = 1 2 E i ( Γ G ) with E _i (Γ _G ) = {e ∈ E(Γ _G )|e is of type T _i }.

• Type T ₁: e = λμ ∈ E ₁(G _Γ) such that λ ∈ Ω 2 i and μ ∈ Ω 2 j for i ≠ j.

• Type T ₂: e = λμ ∈ E ₂(G _Γ) such that λ ∈ Ω 2 i and μ ∈ Ω₃.

Let e is of type T ₁ : Since ecc(λ) = 2 and ecc(μ) = 2, so N ₀(λ) = {λ}, N 1 ( λ ) = Ω 3 ∪ ( Ω 2 − Ω 2 i ) , and N 2 ( λ ) = Ω 2 i − { λ } ,

Accordingly, Remark 1 implies that:

and hence η _λ (e|G _Γ) = 2 and η _μ (e|G _Γ) = 2. Let t ₁ be the number of edges of type T ₁. Since the partition of Ω₂ with parts Ω 2 i induces a complete multipartite graph, so for any λ ∈ Ω 2 i , d ( λ ) = 2 n 2 − 1 = n − 2 in the subgraph of G _Γ induced by the set ⋃ i = 0 n 2 − 1 Ω 2 i . Since | Ω 2 i | = 2 and there are n 2 such sets, so by the formula of handshake lemma, t 1 = n ( n − 2 ) 2 .

Let e is of type T ₂ : Since ecc(λ) = 2 and ecc(μ) = 2, so N ₀(λ) = {λ}, N 1 ( λ ) = Ω 3 ∪ ( Ω 2 − Ω 2 i ) , N 2 ( λ ) = Ω 2 i − { λ } , N ₀(μ) = {μ}, N ₁(μ) = Ω₂, and N ₂(μ) = Ω₃ − {μ}.

Accordingly, Remark 1 implies that:

and hence η _λ (e|G _Γ) = n − 2 and η _μ (e|G _Γ) = 2. Let t ₂ be the number of type T ₂ edges, then t ₂ = |Ω₂| × |Ω₃| = n(n− 2).

Now, the Mostar index of G _Γ is:

Case 1 (n is odd):

Let E ( Γ G ) = ⋃ i = 1 2 E i ( Γ G ) with E _i (Γ _G ) = {e ∈ E(Γ _G )|e is of type T _i }.

• Type T ₁: e = λμ ∈ E ₁(G _Γ) such that λ ∈ Ω 2 i and μ ∈ Ω 2 j for i ≠ j.

• Type T ₂: e = λμ ∈ E ₂(G _Γ) such that λ ∈ Ω 2 i and μ ∈ Ω₃.

Let e is of type T ₁ : Since ecc(λ) = 2 and ecc(μ) = 2, so N ₀(λ) = {λ}, N 1 ( λ ) = Ω 3 ∪ ( Ω 2 − Ω 2 i ) , N 2 ( λ ) = Ω 2 i − { λ } , N ₀(μ) = {μ}, N 1 ( μ ) = Ω 3 ∪ ( Ω 2 − Ω 2 j ) , and N 2 ( μ ) = Ω 2 j − { μ } .