Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups

Kemal Toker

doi:10.1515/math-2025-0172

Article Open Access

Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups

Kemal Toker

Published/Copyright: July 10, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

For n ≥ 4 , let OPD n be the orientation-preserving and order-decreasing transformation semigroup on the finite chain X n = { 1 < … < n } . First, we determine the set of two-sided zero-divisors of OPD n , and its cardinality. Then, we let Γ ( OPD n ) be the graph whose vertices are the two-sided zero-divisors of OPD n excluding the zero element θ and distinct two vertices α and β joined by an edge in case α β = θ = β α . In this study, we prove that Γ ( OPD n ) is a connected graph, and we find the diameter, girth, domination number, minimum degree, and maximum degree of Γ ( OPD n ) . Moreover, we give a lower bound for clique number of Γ ( OPD n ) and we prove that Γ ( OPD n ) is an imperfect graph.

Keywords: orientation-preserving and order-decreasing transformations; transformation semigroup; zero-divisor graph; clique number; domination number

MSC 2010: 20M20; 97K30

1 Introduction

In the literature, the zero-divisor graph of a commutative ring was defined by Beck [1]. In Beck’s definition, the zero element is a vertex. Later, Anderson and Livingston redefined the zero-divisor graph without the zero element [2], which is now the standard definition of the zero-divisor graph of a commutative ring. Let R be a commutative ring, 0 be the zero element of R , and Z ( R ) be the set of zero-divisors of R . The zero-divisor graph of R is an undirected graph Γ ( R ) with vertex set Z ( R ) * = Z ( R ) \ { 0 } and distinct two vertices x and y in Z ( R ) * are adjacent vertices in Γ ( R ) if and only if x y = 0 . Similarly, the zero-divisor graph of a commutative semigroup was defined, and some properties of this graph were investigated [3,4]. Since then, zero-divisor graphs of some special commutative semigroups have been investigated (for example [5]). Redmond [6] defined four different zero-divisor graphs on a non-commutative ring. Those graphs can also be considered on a non-commutative semigroup with a zero element. Note that every zero-divisor graph is simple, meaning it has no loops or multiple edges. Let S be a non-commutative semigroup with 0. We assign the following subsets on S:

T ( S ) = { x ∈ S : x y = 0 = z x for some y , z ∈ S \ { 0 } } , Z ( S ) = { x ∈ S : x y = 0 or y x = 0 for some y ∈ S \ { 0 } } , T ( S ) * = T ( S ) \ { 0 } and Z ( S ) * = Z ( S ) \ { 0 } .

We define four different zero-divisor graphs on S as follows:

Γ ( S ) = Γ 1 ( S ) is the undirected graph with vertices T ( S ) * and distinct two vertices x and y are adjacent if and only if x y = 0 = y x ;
Γ 2 ( S ) is the undirected graph with vertices Z ( S ) * and distinct two vertices x and y are adjacent if and only if x y = 0 = y x ;
Γ 3 ( S ) is the undirected graph with vertices Z ( S ) * and distinct two vertices x and y are adjacent if and only if x y = 0 or y x = 0 ; and
Γ 4 ( S ) is the directed graph with vertices Z ( S ) * and for distinct two vertices x and y , x → y is a directed edge if and only if x y = 0 .

In this study, we only consider the zero-divisor graph given in the first definition above.

For n ∈ N , let T n denote the full transformation semigroup on the chain X n = { 1 , … , n } under its natural order. An element α ∈ T n is called order-preserving if x ≤ y implies x α ≤ y α for all x , y ∈ X n , and order-decreasing if x α ≤ x for all x ∈ X n . Then, the subsemigroup consisting of all order-preserving transformations in T n is denoted by O n , and the subsemigroup consisting of all order-decreasing transformations in T n is denoted by D n , and the subsemigroup consisting of all order-preserving and order-decreasing transformations in T n is denoted by C n . Higgins [7] proved that the cardinality of C n is the n th Catalan number, namely, C n = 1 n + 1 2 n n , that is why C n is also known as the n th Catalan monoid. For a sequence ( x 1 , x 2 , … , x r ) on X n , if there exists no more than one subscript i such that x i > x i + 1 , where x r + 1 = x 1 , then ( x 1 , x 2 , … , x r ) is called a cyclic. An element α in T n is called orientation-preserving if ( 1 α , 2 α , … , n α ) is a cyclic. Then, the subsemigroup consisting of all orientation-preserving transformations in T n is denoted by OP n and the subsemigroup consisting of all order-decreasing transformations in OP n is denoted by OPD n . Γ 1 ( C n ) was investigated [8]. Let P n be the partial transformation semigroup on X n , and let SP n = P n \ T n . The undirected graph Γ ( P n ) was studied [9], the undirected graph Γ 3 ( SP n ) and the directed graph Γ 4 ( SP n ) were studied [10]. Recently, Korkmaz defined two undirected graphs on T n and investigated some properties of these two graphs [11]. We refer to [12–14] for other terms in semigroup and graph theories, which are not explained here.

In this study, we investigate some properties of Γ ( OPD n ) . Since OPD n = C n for n = 1 , 2 , we suppose that n ≥ 3 and note that OPD n is a non-commutative semigroup with the zero element θ = 1 2 ⋯ n 1 1 ⋯ 1 . And the identity element of OPD n will be denoted by 1 n . In this study, we prove that Γ ( OPD n ) is a connected graph, and we find the diameter, girth, domination number, minimum degree, and maximum degree of Γ ( OPD n ) . Moreover, we give a lower bound for clique number of Γ ( OPD n ) and prove that Γ ( OPD n ) is an imperfect graph.

2 Zero-divisors of OPD n

For n ≥ 3 , let OPD n * = OPD n \ { θ } , and then we define the following sets:

L = L ( OPD n ) = { α ∈ OPD n : α β = θ for some β ∈ OPD n * } , R = R ( OPD n ) = { α ∈ OPD n : γ α = θ for some γ ∈ OPD n * } , and T = T ( OPD n ) = { α ∈ OPD n : α β = θ = γ α for some β , γ ∈ OPD n * } = L ∩ R ,

which are called the set of left, right, and two-sided zero-divisors of OPD n , respectively. In this section, we determine the left, right, and two-sided zero-divisors of OPD n , and then, find their cardinalities. Let us remember known result from Korkmaz [11].

Proposition 2.1

[11, Proposition 1] For any α , β ∈ T n , α β = θ if and only if im ( α ) ⊆ 1 β − 1 . In particular, α 2 = θ if and only if im ( α ) ⊆ 1 α − 1 .

Let us now define some specific mappings that will be useful throughout this study. For each 2 ≤ k ≤ n , let

(1) β k = 1 ⋯ n − 1 n 1 ⋯ 1 k and

(2) γ k = 1 ⋯ k − 1 k k + 1 ⋯ n 1 ⋯ 1 2 1 ⋯ 1 .

For each 1 ≤ r ≤ n − 1 , let OPD ( n , r ) = { α ∈ OPD n : ∣ im ( α ) ∣ ≤ r } , and recall that Narayana number N ( n , r ) is defined by N ( n , r ) = 1 n n r n r − 1 . It is known that ∑ r = 1 n N ( n , r ) = C n (for example [15]). It is shown [16, Lemma 1 ( i )] that ∣ OPD ( n , r ) ∣ = − n + 1 + ∑ m = 1 r C m + ∑ m = r + 1 n ∑ k = 1 r N ( m , k ) . Moreover, since OPD ( n , n − 1 ) = OPD n \ { 1 n } , we conclude that ∣ OPD n ∣ = 1 − n + ∑ m = 1 n C m for n ≥ 3 . Thus, we have the following result.

Lemma 2.2

For n ≥ 3 , L = OPD n \ { 1 n } , and so ∣ L ∣ = − n + ∑ m = 1 n C m .

Proof

For any α ∈ OPD n \ { 1 n } , let A = X n \ im ( α ) . Since A ≠ ∅ and 1 ∈ im ( α ) , we have max ( A ) = k for some 2 ≤ k ≤ n . If we consider γ k as defined in (2), then it is clear that γ k ∈ OPD n * and α γ k = θ , and so α is a left-zero divisor element of OPD n . Since 1 n is the identity, we have L = OPD n \ { 1 n } , and so ∣ L ∣ = − n + ∑ m = 1 n C m .□

For any α ∈ T n and ∅ ≠ Y ⊆ X n , we denote the restriction map of α to Y by α ∣ Y . For any α ∈ OP n , the order-preserving degree of α is defined by

opd ( α ) = max { m : α ∣ Y ∈ O m } .

Thus, opd ( α ) = n for all α ∈ C n and 2 ≤ opd ( α ) ≤ n − 1 for all α ∈ OPD n \ C n since 1 α = 1 for all α ∈ D n . For any α ∈ OPD n \ C n , if opd ( α ) = m , then it is clear that α has the following tabular form

α = 1 2 ⋯ m m + 1 … n 1 2 α ⋯ m α 1 … 1

with the property that 1 ≤ 2 α ≤ … ≤ m α ≤ m .

Lemma 2.3

For n ≥ 3 , R = { α ∈ OPD n : ∣ 1 α − 1 ∣ ≥ 2 } , or equivalently, R = ( OPD n \ C n ) ∪ { α ∈ C n : 2 α = 1 } , and so,

∣ R ∣ = 1 − n + C n + ∑ m = 1 n − 2 C m .

Proof

Let A = { α ∈ OPD n : ∣ 1 α − 1 ∣ ≥ 2 } , and for any α ∈ A , let opd ( α ) = m and suppose that m ≠ n . If we consider β n as defined in (1), then since n α = 1 , we have β n α = θ , and so α is a right-zero divisor element of OPD n . Suppose that m = n . Since α ∈ C n and ∣ 1 α − 1 ∣ ≥ 2 , we must have 1 α = 2 α = 1 . If we consider β 2 as defined in (1), then we have β 2 α = θ , and so α is a right-zero divisor element of OPD n .

For any α ∈ OPD n , let ∣ 1 α − 1 ∣ = 1 , that is 1 α − 1 = { 1 } . If β α = θ for some β ∈ OPD n , then it follows from Proposition 2.1 that im ( β ) ⊆ 1 α − 1 = { 1 } , and so β = θ . Thus, α cannot be a right-zero divisor. Therefore, we have

R = { α ∈ OPD n : ∣ 1 α − 1 ∣ ≥ 2 } = ( OPD n \ C n ) ∪ { α ∈ C n : 2 α = 1 } .

Moreover, if we define the mapping ψ : { α ∈ C n : 2 α = 2 } → C n − 1 by

α ψ = 1 2 3 ⋯ n − 1 1 3 α − 1 4 α − 1 ⋯ n α − 1

for all α ∈ { α ∈ C n : 2 α = 2 } , then it is clear that ψ is a well-defined mapping. For any α 1 , α 2 ∈ { α ∈ C n : 2 α = 2 } , suppose that α 1 ψ = α 2 ψ . Then, since 1 α 1 = 1 = 1 α 2 , 2 α 1 = 2 = 2 α 2 and k α 1 = k α 2 for 3 ≤ k ≤ n , it follows that α 1 = α 2 , and so ψ is one to one. For any β ∈ C n − 1 , define

β ˆ = 1 2 3 ⋯ n 1 2 2 β + 1 ⋯ ( n − 1 ) β + 1 .

Then, it is clear that β ˆ ∈ { α ∈ C n : 2 α = 2 } , and that β ˆ ψ = β . Thus, ψ is onto, and so ψ is a bijection. Moreover, since { α ∈ C n : 2 α = 1 } = C n \ { α ∈ C n : 2 α = 2 } , it follows that

∣ { α ∈ C n : 2 α = 1 } ∣ = ∣ C n ∣ − ∣ { α ∈ C n : 2 α = 2 } ∣ = C n − C n − 1 .

Therefore, from [16, Lemma 1 ( i )], we have

∣ R ∣ = ∣ OPD n \ C n ∣ + ∣ { α ∈ C n : 2 α = 1 } ∣ = 1 − n + ∑ m = 1 n − 1 C m + ( C n − C n − 1 ) = 1 − n + C n + ∑ m = 1 n − 2 C m ,

as required.□

Since R ⊆ L , we have the following immediate corollary.

Corollary 2.4

For n ≥ 3 , T = R .

3 Zero-divisor graph of OPD n

We denote the vertex set and the edge set of a simple graph G by V ( G ) and E ( G ) , respectively. For any n + 1 different vertices u = v 0 , v 1 , … , v n = v in V ( G ) , if there exists an edge v i − v i + 1 in E ( G ) for each 0 ≤ i ≤ n − 1 , then u = v 0 − v 1 − … − v n − 1 − v n = v is called a path between u and v , and n is called the length of the path. The length of a shortest path between u and v in G is denoted by d G ( u , v ) . If there exist a path for all two distinct vertices in G , then G is called a connected graph. The eccentricity of a vertex v in a connected simple graph G , denoted by ecc ( v ) , is defined by

ecc ( v ) = max { d G ( u , v ) : u ∈ V ( G ) } .

The diameter of a connected simple graph G , denoted by diam ( G ) , is defined by

diam ( G ) = max { ecc ( v ) : v ∈ V ( G ) } .

Observe that V ( Γ ( OPD 3 ) ) = 1 2 3 1 1 2 , 1 2 3 1 1 3 , 1 2 3 1 2 1 and E ( Γ ( OPD 3 ) ) contains only one edge, namely, 1 2 3 1 1 3 − 1 2 3 1 2 1 . Next we let

(3) δ k = 1 ⋯ k − 1 k k + 1 ⋯ n 1 ⋯ 1 k 1 ⋯ 1 for 2 ≤ k ≤ n ,

(4) λ k = 1 ⋯ k k + 1 ⋯ n 1 ⋯ 1 k + 1 ⋯ k + 1 for 2 ≤ k ≤ n − 1 ,

(5) ρ = 1 2 3 … n 1 1 3 … n , and

(6) τ = 1 2 3 … n 1 1 2 … n − 1 .

For convenience, we use the notation Γ instead of Γ ( OPD n ) . For any v ∈ V ( G ) , the neighborhood of v is denoted by N ( v ) and defined by

N ( v ) = { u ∈ V ( G ) : u − v ∈ E ( G ) } .

Moreover, for α ∈ V ( Γ ) , we let

N 1 ( α ) = N ( α ) ∩ C n and N 2 ( α ) = N ( α ) ∩ ( OPD n \ C n ) .

Then, we state and prove one of the main results in this study.

Theorem 3.1

For n ≥ 4 , Γ is connected and diam ( Γ ) = 4 .

Proof

For any distinct α , β ∈ V ( Γ ) , we consider three cases; both of them in C n , just one of them in C n and none of them in C n .

Case 1: Suppose that α , β ∈ C n . First note that 2 α = 1 = 2 β since α , β ∈ V ( Γ ) = ( OPD n \ C n ) ∪ { α ∈ C n : 2 α = 1 } . Then, we consider three subcases; none of the image sets contains n , just one of the image sets contains n , and both the image sets contain n .

Subcase ( i ). Suppose that n α , n β < n . If one of them is β 2 as defined in (1), then we have the path α − β ; and if none of them is β 2 , then we have the path α − β 2 − β .

Subcase ( i i ). Without loss of generality, suppose that n α = n and n β < n . If k = min ( X n \ im ( α ) ) , then since 2 ≤ k ≤ n − 1 , we consider γ k , β 2 , and β 3 as defined in (1), and (2). Suppose that β ∉ { β 2 , β 3 } . If k ≠ 2 , then α − γ k − β 2 − β is a path in Γ , and if k = 2 , then α − γ 2 − β 3 − β 2 − β is a path in Γ , and so, d Γ ( α , β ) ≤ 4 . If β ∈ { β 2 , β 3 } , then it is similarly shown that α and β are connected, and d Γ ( α , β ) ≤ 3 .

Subcase ( i i i ). Suppose that n α = n and n β = n . If k = min ( X n \ im ( α ) ) and l = min ( X n \ im ( β ) ) , then since 2 ≤ k , l ≤ n − 1 , we consider δ 2 , γ k , γ l , and β n as defined in (1), (2), and (3). If k = l = 2 , then α − δ 2 − β is a path in Γ since 1 α = 2 α = 1 = 1 β = 2 β . Without loss of generality, suppose that k ≠ 2 and l = 2 . Similarly, if α , β ∉ { β n } , then α − γ k − β n − δ 2 − β is a path in Γ and if α = β n or β = β n , then it is clear that d Γ ( α , β ) ≤ 2 . Now, suppose k , l ≥ 3 . If k = l , then α − γ k − β is a path in Γ , and if k ≠ l , then α − γ k − γ l − β is a path in Γ .

Case 2: Suppose that α ∈ C n and β ∈ OPD n \ C n . Let k = min ( X n \ im ( α ) ) . If n α = n , α ≠ β n and β ≠ γ k , then α − γ k − β n − β is a path in Γ . If n α = n and α = β n or n α = n and β = γ k , then it is clear that α and β are adjacent vertices in Γ . If n α < n , α ≠ β 2 and β ≠ δ 3 , then α − β 2 − δ 3 − β n − β is a path in Γ . If n α < n and α = β 2 or n α < n and β = δ 3 , then it is also clear that d Γ ( α , β ) ≤ 3 .

Case 3: Suppose that both α and β in OPD n \ C n . Since n α = 1 = n β , it follows that α − β n − β is a path in Γ .

Therefore, Γ is a connected graph and d Γ ( α , β ) ≤ 4 for all α , β ∈ V ( Γ ) . If we consider ρ and τ as defined in (5) and (6), then we note that ρ and τ are non-adjacent vertices in Γ . Moreover, if ρ η = η ρ = θ and τ μ = μ τ = θ , then we have η = δ 2 and μ = β 2 . Finally, since β 2 δ 2 ≠ θ , it follows that d Γ ( ρ , τ ) ≥ 4 , and so diam ( Γ ) = 4 .□

For any simple graph G and v ∈ V ( G ) , the degree of v , denoted by deg G ( v ) , is defined as the number of adjacent vertices to v in G . Moreover, the minimum degree of G , denoted by δ ( G ) , is defined by

δ ( G ) = min { deg G ( v ) : v ∈ V ( G ) } ,

and the maximum degree of G , denoted by Δ ( G ) , is defined by

Δ ( G ) = max { deg G ( v ) : v ∈ V ( G ) } .

Thus, if G is connected, then δ ( G ) ≥ 1 . Moreover, since deg Γ ( ρ ) = deg Γ ( τ ) = 1 , we have the following immediate corollary.

Corollary 3.2

For n ≥ 4 , δ ( Γ ) = 1 .

To find Δ ( Γ ) , we need some preliminary results. Let U 1 = { α ∈ C n : 2 α = 2 } and U 2 = { α ∈ C n : n α = n } . As shown in the proof of Lemma 2.3 that ∣ U 1 ∣ = C n − 1 , one can easily show that ∣ U 2 ∣ = C n − 1 and ∣ U 1 ∩ U 2 ∣ = ∣ { α ∈ C n : 2 α = 2 and n α = n } ∣ = C n − 2 . Since { α ∈ C n : 2 α = 1 and n α < n } = C n \ ( U 1 ∪ U 2 ) , we conclude that the cardinality of the set { α ∈ C n : 2 α = 1 and n α < n } is C n − 2 C n − 1 + C n − 2 (see, also [8, proof of Lemma 3.3]).

For 2 ≤ m ≤ n , let Q m = { α ∈ OPD n : opd ( α ) = m } . Then, it is shown [16, Lemma 1] that

( i ) OPD n = ⋃ m = 2 n Q m , ( i i ) Q m ∩ Q m ′ = ∅ for all 2 ≤ m ≠ m ′ ≤ n , and ( i i i ) Q m ∪ { θ } and C m are isomorphic for each 2 ≤ m ≤ n .

By using these results, we prove the following lemma.

Lemma 3.3

For n ≥ 4 , deg Γ ( β 2 ) = C n − C n − 1 + C n − 2 − n − 1 .

Proof

For n ≥ 4 , let

A = { α ∈ V ( Γ ) ∩ C n : n α < n } , ℬ = { α ∈ OPD n \ C n : 2 α = 1 } , and C = { α ∈ OPD n \ C n : 2 α = 2 } .

Then, it is clear that β 2 ∈ A , ℬ ∪ C = OPD n \ C n , and ℬ ∩ C = ∅ . Since A ∪ { θ } = { α ∈ C n : 2 α = 1 and n α < n } , it follows that ∣ A ∣ = C n − 2 C n − 1 + C n − 2 − 1 . For 2 ≤ m ≤ n − 1 , if Q m ′ = { α ∈ OPD n : opd ( α ) = m and 2 α = 2 } , then C is a disjoint union of Q 2 ′ , … , Q n − 1 ′ , and that for each 2 ≤ m ≤ n − 1 , Q m ′ and { α ∈ C m : 2 α = 2 } are isomorphic semigroups, and ∣ { α ∈ C m : 2 α = 2 } ∣ = C m − 1 , it follows that ∣ C ∣ = ∑ m = 2 n − 1 C m − 1 = ∑ m = 1 n − 2 C m . Moreover, from the proof of Lemma 2.3, we have ∣ ℬ ∣ = ∣ OPD n \ C n ∣ − ∣ C ∣ = C n − 1 − n + 1 . Therefore, since any α in V ( Γ ) is an adjacent vertex to β 2 if and only if α ∈ ( A ∪ ℬ ) \ { β 2 } , it follows that deg Γ ( β 2 ) = ∣ A ∣ − 1 + ∣ ℬ ∣ = C n − C n − 1 + C n − 2 − n − 1 .□

It is known that the Catalan numbers satisfy the recurrence relation C n = ∑ m = 0 n − 1 C m C n − 1 − m , where C 0 = 1 [17]. Therefore, we have the following proposition.

Proposition 3.4

For n ≥ 4 , deg Γ ( β 2 ) > ∑ i = 2 n − 1 ( C i − 1 ) .

Proof

For n ≥ 4 , since C n − 2 ≥ 2 , it follows from Lemma 3.3 that

deg Γ ( β 2 ) = ∑ m = 0 n − 1 C m C n − 1 − m − C n − 1 + C n − 2 − n − 1 = ∑ m = 1 n − 1 C m C n − 1 − m + C n − 2 − n − 1 ≥ ∑ m = 1 n − 1 C m C n − 1 − m − n + 1 > ∑ m = 1 n − 1 C m − n + 1 = ∑ m = 1 n − 1 ( C m − 1 ) = ∑ m = 2 n − 1 ( C m − 1 ) ,

as required.□

Proposition 3.5

For n ≥ 4 and 3 ≤ k ≤ n , deg Γ ( β k ) < deg Γ ( β 2 ) .

Proof

First, we show that deg Γ ( β n ) < deg Γ ( β 2 ) . It is clear that N 1 ( β n ) = ∅ and N 2 ( β n ) = OPD n \ C n , and so from Proposition 3.4, deg Γ ( β n ) = ∑ i = 2 n − 1 ( C i − 1 ) < deg Γ ( β 2 ) .

For all α ∈ V ( Γ ) ∩ C n , we recall 2 α = 1 , and we let 3 ≤ k ≤ n − 1 . Since λ k − 1 ∈ N 1 ( β 2 ) \ N 1 ( β k ) , it follows that ∣ N 1 ( β k ) ∣ < ∣ N 1 ( β 2 ) ∣ . Moreover, if α ∈ N 2 ( β k ) , then α has the following tabular form

α = 1 2 ⋯ k − 1 k k + 1 ⋯ n − 1 n 1 2 α ⋯ ( k − 1 ) α 1 ( k + 1 ) α ⋯ ( n − 1 ) α 1 .

If we let

(7) α ˜ = 1 2 3 ⋯ k k + 1 ⋯ n − 1 n 1 1 2 α ⋯ ( k − 1 ) α ( k + 1 ) α ⋯ ( n − 1 ) α 1 ,

then we have α ˜ ∈ N 2 ( β 2 ) . If we define the mapping φ k : N 2 ( β k ) → N 2 ( β 2 ) by α φ k = α ˜ for all α ∈ N 2 ( β k ) , then it is clear that φ k is a well-defined one to one mapping, and so ∣ N 2 ( β k ) ∣ ≤ ∣ N 2 ( β 2 ) ∣ . Therefore, we conclude that deg Γ ( β k ) < deg Γ ( β 2 ) for each 3 ≤ k ≤ n .□

Proposition 3.6

For n ≥ 4 and 2 ≤ m ≤ n − 1 , we have deg Γ ( γ m ) < deg Γ ( β 2 ) .

Proof

Let 2 ≤ m ≤ n − 1 . It is clear that N 2 ( γ m ) ⊆ N 2 ( β 2 ) for all 2 ≤ m ≤ n − 1 . Moreover, since γ m ∈ N 2 ( β 2 ) if m > 2 , and since 1 2 3 ⋯ n − 1 n 1 1 2 ⋯ 2 1 ∈ N 2 ( β 2 ) \ N 2 ( γ 2 ) , it follows that ∣ N 2 ( γ m ) ∣ + 1 ≤ ∣ N 2 ( β 2 ) ∣ for all 2 ≤ m ≤ n − 1 .

For 2 ≤ m ≤ n , let J ( m ) = { α ∈ C n : 2 α = 1 and m ∉ im ( α ) } . Then, we show that ∣ J ( m ) ∣ < ∣ J ( n ) ∣ for all 2 ≤ m ≤ n − 1 . For 3 ≤ m ≤ n and α ∈ J ( m − 1 ) , let α ˆ : X n → X n be defined by

x α ˆ = x α x α ≠ m m − 1 x α = m

for all x ∈ X n . Similarly, the mapping ψ m : J ( m − 1 ) → J ( m ) defined by α ψ m = α ˆ for all α ∈ J ( m − 1 ) is one to one, and so ∣ J ( m − 1 ) ∣ ≤ ∣ J ( m ) ∣ for all 3 ≤ m ≤ n . Moreover, if we consider the transformation

β = 1 2 3 ⋯ n − 1 n 1 1 3 ⋯ n − 1 n − 1 ,

then it is clear that β ∈ J ( n ) \ J ( n − 1 ) and β ≠ α ˆ for all α ∈ J ( n − 1 ) , and so ψ n is not onto, i.e., ∣ J ( n − 1 ) ∣ < ∣ J ( n ) ∣ . Therefore, we have ∣ J ( m ) ∣ < ∣ J ( n ) ∣ for all 2 ≤ m ≤ n − 1 . Since

N 1 ( γ m ) = J ( m ) \ { θ } and N 1 ( β 2 ) ∪ { β 2 } = J ( n ) \ { θ } ,

it follows that ∣ N 1 ( γ m ) ∣ − 1 < ∣ N 1 ( β 2 ) ∣ for all 2 ≤ m ≤ n − 1 . Therefore, deg Γ ( γ m ) < deg Γ ( β 2 ) for all 2 ≤ m ≤ n − 1 .□

Theorem 3.7

Δ ( Γ ) = C n − C n − 1 + C n − 2 − n − 1 for n ≥ 4 .

Proof

Let α ∈ V ( Γ ) . For α ∈ V ( Γ ) ∩ C n , let k = min ( im ( α ) \ { 1 } ) , consider β k and suppose α ≠ β k . For any ξ ∈ N ( α ) , it follows from Proposition 2.1 that im ( ξ ) ⊆ 1 α − 1 ⊆ X n − 1 = 1 β k − 1 and im ( β k ) ⊆ im ( α ) ⊆ 1 ξ − 1 , and so ξ β k = θ = β k ξ . Thus, if ξ ≠ β k , then ξ ∈ N ( β k ) , and if ξ = β k , then α ∈ N ( β k ) . Therefore, deg Γ ( α ) ≤ deg Γ ( β k ) , and so from Proposition 3.5, deg Γ ( α ) ≤ deg Γ ( β 2 ) .

If α ∈ V ( Γ ) ∩ ( OPD n \ C n ) and opd ( α ) = m , then α has the following tabular form

α = 1 2 ⋯ m − 1 m m + 1 ⋯ n 1 2 α ⋯ ( m − 1 ) α k 1 ⋯ 1

with the property 1 ≤ 2 α ≤ … ≤ ( m − 1 ) α ≤ k ≤ m ≤ n − 1 and k ≥ 2 . If we consider α ¯ = 1 ⋯ m − 1 m m + 1 ⋯ n 1 ⋯ 1 k 1 ⋯ 1 , then we similarly have deg Γ ( α ) ≤ deg Γ ( α ¯ ) . Now, consider γ m . If k = 2 , then α ¯ = γ m , and so suppose k ≥ 3 . If ζ ∈ N 1 ( α ¯ ) , then it is clear that ζ ∈ N 1 ( γ m ) , and so, N 1 ( α ¯ ) ⊆ N 1 ( γ m ) . Moreover, we have 1 2 3 ⋯ n 1 1 2 ⋯ 2 ∈ N 1 ( γ m ) \ N 1 ( α ¯ ) since m ≥ k ≥ 3 . Thus, ∣ N 1 ( α ¯ ) ∣ ≤ ∣ N 1 ( γ m ) ∣ − 1 .

If ζ ∈ N 2 ( α ¯ ) , then k ζ = 1 , and so, ζ has the following tabular form:

ζ = 1 2 ⋯ k − 1 k k + 1 ⋯ n − 1 n 1 2 ζ ⋯ ( k − 1 ) ζ 1 ( k + 1 ) ζ ⋯ ( n − 1 ) ζ 1 .

Thus, we have i ζ ≠ m for i ∈ { 2 , 3 , … , k − 1 , k + 1 , … , n − 1 } since ζ and α ¯ are adjacent vertices in Γ . If we define ζ ˜ = 1 2 3 ⋯ k k + 1 ⋯ n − 1 n 1 1 2 ζ ⋯ ( k − 1 ) ζ ( k + 1 ) ζ ⋯ ( n − 1 ) ζ 1 as in (7), then we have im ( ζ ˜ ) ⊆ 1 γ m − 1 and im ( γ m ) ⊆ 1 ζ ˜ − 1 , and so, ζ ˜ ∈ N 2 ( γ m ) if ζ ˜ ≠ γ m . If we define the mapping f : N 2 ( α ¯ ) → N 2 ( γ m ) ∪ { γ m } by ζ f = ζ ˜ for all ζ ∈ N 2 ( α ¯ ) , then it is clear that f is a well-defined one to one mapping, and so, ∣ N 2 ( α ¯ ) ∣ ≤ ∣ N 2 ( γ m ) ∣ + 1 . Thus, from Proposition 3.6, deg Γ ( α ) ≤ deg Γ ( α ¯ ) ≤ deg Γ ( γ m ) < deg Γ ( β 2 ) . Therefore, from Lemma 3.3, we have Δ ( Γ ) = deg Γ ( β 2 ) = C n − C n − 1 + C n − 2 − n − 1 .□

The length of a shortest cycle contained in a graph G is called the girth of G and it is denoted by gr ( G ) . Moreover, if G does not contain any cycles, then its girth is defined as infinity. Thus, if G is a simple connected graph, then the length of a shortest cycle must be at least 3, and so gr ( G ) ≥ 3 . Then, we have the following corollary.

Corollary 3.8

For n ≥ 4 , gr ( Γ ) = 3 .

Proof

If we consider α = 1 2 … n − 2 n − 1 n 1 1 … 1 2 2 , β 2 and γ 3 as defined in (1) and (2), then α − β 2 − γ 3 − α is a cycle in Γ , and so gr ( Γ ) = 3 .□

Let D be a non-empty subset of the vertex set V ( G ) of a graph G . For each v ∈ V ( G ) , if v ∈ D , or if there exists u ∈ D such that u − v is an edge in E ( G ) , then D is called a dominating set for G . The domination number of G , denoted by Y ( G ) , is defined by

Y ( G ) = min { ∣ D ∣ : D is a dominating set for G } .

To find Y ( Γ ) , we also consider the following vertices in V ( Γ ) :

μ 1 = 1 2 3 … n 1 1 3 … n , μ k = 1 2 3 … k + 1 k + 2 … n 1 1 2 … k k + 2 … n ( 2 ≤ k ≤ n − 2 ) , μ n − 1 = 1 2 3 … n 1 1 2 … n − 1 , and μ n = 1 2 … n − 1 n 1 2 … n − 1 1 .

With these notations, we have the following theorem.

Theorem 3.9

For n ≥ 4 , Y ( Γ ) = n .

Proof

For each 1 ≤ k ≤ n − 1 , it is easy to check that μ k α = α μ k = θ if and only if α = γ k + 1 , and that μ n α = α μ n = θ if and only if α = β n . Thus, deg Γ ( μ k ) = 1 for each 1 ≤ k ≤ n . Now, let U = { γ 2 , γ 3 , … , γ n , β n } . If D is a dominating set for Γ , then either μ k or its unique adjacent vertex in D , and so Y ( Γ ) ≥ n .

Next we show that U is a dominating set for Γ . For any α ∈ V ( Γ ) \ U , we consider two cases: either α ∈ OPD n \ C n or α ∈ C n . In the first case, since n α = 1 , it is clear that α and β n are adjacent vertices in Γ . In the second case, we have two subcases, either n α < n or n α = n . If α ∈ C n and n α < n , then α and γ n are adjacent vertices in Γ . If α ∈ C n and n α = n , then there exists at least one k ∈ X n \ im ( α ) such that 2 ≤ k ≤ n − 1 . Then, it is easy to see that α and γ k are adjacent vertices in Γ . Therefore, U is a dominating set of Γ , and so Y ( Γ ) = n .□

Let G be a simple graph and C be a non-empty subset of V ( G ) . If every two distinct vertices in C are adjacent, then C is called a clique in G , and moreover, the subgraph whose vertex set is C and edge set contains all edges of G , which have endpoints in C is called the subgraph of G induced by C . Thus, C is a clique if and only if the subgraph of G induced by C is a complete graph. Moreover, the number of vertices of any maximal clique in G is called the clique number of G , and it is denoted by ω ( G ) .

For any real number x , we denote the smallest integer greater than or equal to x by ⌈ x ⌉ . For any p , q , r ∈ N with p < q , let

Y q , p = { p + 1 , p + 2 , … , q } and X r = { 1 , 2 , … , r }

under their natural order. Moreover, let O * denote the set of all order-preserving mappings from Y q , p to X r . For any α ∈ O * , it is clear that ( ∣ 1 α − 1 ∣ , ∣ 2 α − 1 ∣ , … , ∣ r α − 1 ∣ ) is a solution of the equation

x 1 + x 2 + … + x r = q − p , where all x i ∈ N ∪ { 0 } .

Conversely, for each solution of this equation ( c 1 , c 2 , … , c r ) , there exists unique α ∈ O * such that ∣ i α − 1 ∣ = c i for each 1 ≤ i ≤ r . Therefore, since the number of solutions of the above equation is the same with the cardinality of O * , we have

(8) ∣ O * ∣ = q − p + r − 1 r − 1 ,

[17]. Now, we give a lower bound for clique number of Γ in the following theorem.

Theorem 3.10

For n ≥ 4 , if ⌈ n 2 ⌉ = s , then ω ( Γ ) ≥ n s − ( n − s + 1 ) .

Proof

For 2 ≤ r ≤ n − 2 , if we let

V r = { α ∈ OPD n : { 1 } ≠ im ( α ) ⊆ X r ⊆ 1 α − 1 } ,

then it is clear that ∅ ≠ V r ⊆ V ( Γ ) . Moreover, let Λ be the subgraph of Γ induced by V r . Then, for any distinct two vertices α and β in V r , since im ( α ) ⊆ 1 β − 1 and im ( β ) ⊆ 1 α − 1 , it follows from Proposition 2.1 that α β = θ = β α , and so Λ is a complete graph.

Moreover, for any α ∈ V r , if opd ( α ) = m , then we observe that r + 1 ≤ m ≤ n , x α = 1 for all x ∈ X r ∪ Y n , m , and that the restriction α ∣ Y m , r is an order-preserving transformation from Y m , r to X r . From (8), there exist m − 1 r − 1 many order-preserving mappings from Y m , r to X r , and one of them is θ ∣ Y m , r , and so

∣ V r ∣ = ∑ m = r + 1 n m − 1 r − 1 − 1 = ∑ k = 0 n − r r − 1 + k r − 1 − ( n − r + 1 ) = n r − ( n − r + 1 ) .

If ⌈ n 2 ⌉ = s , then it is clear that n r − ( n − r + 1 ) ≤ n s − ( n − s + 1 ) for all 2 ≤ r ≤ n − 2 , and so ω ( Γ ) ≥ n s − ( n − s + 1 ) .□

For n = 4 , it is easy to see that ω ( Γ ) = 4 2 − ( 4 − 2 + 1 ) = 3 . In particular, the subgraph of Γ induced by 1 2 3 4 1 1 1 2 , 1 2 3 4 1 1 2 1 , 1 2 3 4 1 1 2 2 is a maximal complete subgraph of Γ . However, we have not proved ω ( Γ ) = n s − ( n − s + 1 ) in general.

If we color all the vertices in G with the rule of no two adjacent vertices have the same color, then the minimum number of colors needed to color of G is called the chromatic number of G , and it is denoted by χ ( G ) . If ω ( H ) = χ ( H ) for every induced subgraph H of G , then G is called a perfect graph, otherwise it is called imperfect graph.

Theorem 3.11

Γ is an imperfect graph for n ≥ 4 .

Proof

For n ≥ 4 , let ξ = 1 2 … n − 2 n − 1 n 1 1 … 1 n − 2 1 ∈ V ( Γ ) and H = { ξ , γ n − 2 , λ n − 2 , β n − 2 , β n } . If Π is the subgraph of Γ induced by H , then it is a routine matter to check that Π is a cycle graph with the cycle

λ n − 2 − γ n − 2 − β n − ξ − β n − 2 − λ n − 2 .

Therefore, since ω ( Π ) = 2 and χ ( Π ) = 3 , Γ is an imperfect graph.□

Acknowledgments

We would like to thank the referees for their valuable comments which helped to improve the manuscript. My sincere thanks are due to Prof. Dr. Hayrullah Ayık for his helpful suggestions and encouragement.

Funding information: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and manuscript preparation.
Conflict of interest: The author states no conflicts of interest.

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Received: 2024-11-01

Revised: 2025-04-21

Accepted: 2025-06-06

Published Online: 2025-07-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2025-0172

Keywords for this article

orientation-preserving and order-decreasing transformations; transformation semigroup; zero-divisor graph; clique number; domination number

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