Startseite Mathematik Note on structural properties of graphs
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Note on structural properties of graphs

  • Luis D. Arreola-Bautista , Gerardo Reyna , Jesús Romero-Valencia und José M. Sigarreta EMAIL logo
Veröffentlicht/Copyright: 22. Februar 2022
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 20 Heft 1

Abstract

In this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order of a finite group G , by means of its commuting graph. The order of these subgroups attains the bound c = ∣ [ x 1 ] ∣ + ⋯ + ∣ [ x m ] ∣ , where [ x i ] denotes the conjugacy class of x i in G and m is the smallest integer j such that ∣ [ x 1 ] ∣ + ⋯ + ∣ [ x j ] ∣ ≥ ∣ C G ( x j ) ∣ , where C G ( x j ) is the centralizer of x j in G .

Keywords: Abelian subgroup; commuting graph; clique; chromatic number; weakly perfect graph
MSC 2010: 05C75

1 Introduction

Over the years, graphs generated from a group or a semigroup have been extensively studied. For example, in 1964, Bosak [1] studied graphs induced by semigroups; in [2] the intersection graphs of non-trivial subgroups of abelian groups with finite order were studied by Zelinka; in [3,4,5] the directed graph defined over the elements of a group, the well-known Cayley digraph, was studied; in the books in [6,7,8], numerous valuable applications of this kind of graph are presented, hence the importance of researching them. Another best-known graph is the so-called directed power graph, whose definition was given by Kelarev and Quinn [9], which is defined in such a way that it is possible to apply it also to semigroups, and thus, it was in [10,11,12] that power graphs of semigroups were considered for the first time. These papers use only the short-term “power graph.” However, it makes sense for both directed and undirected power graphs. In addition, Kelarev and Quinn [11] defined an interesting class of directed graphs over semigroups, the semigroup divisibility graphs. There is also one more example, the well-known hyperbolic graphs, mentioned in the works [13,14,15], whose main objects of study were initially Cayley graphs associated with finite groups, nowadays these graphs have several applications in Physics and Geometry. For more information on structural properties associated with a graph, see [16,17,18].

It is said that group theory started with Galois (1811–1832), who showed that the best way to understand polynomials is by relating them to certain groups of permutations of their roots. From then onward, group theory has become a useful tool for different fields of mathematics, such as combinatorics, geometry, logic, number theory, and topology.

At the end of the nineteenth century, there were two main streams of group theory: on the one hand topological groups (especially Lie groups); on the other hand, finite groups. The latter since its beginnings in the nineteenth century has grown to become an extensive and diverse part of algebra, in particular, the theory of locally finite groups and the theory of nilpotent, solvable groups [19,20]. In the early 1980s, this development culminated in the classification of finite simple groups, impressively and convincingly demonstrating the strength of its methods and results. Thanks to those works, all finite groups that are constructible from these simple groups are now known.

Graph theory is an increasing mathematical discipline containing deep and strong results of high applicability. Its rapid development in the last few decades is not only due to its status as the main structure on which currently applied mathematics (computer science, combinatorial optimization, and operations research) is supported, but also due to its increasing connections in the applied sciences.

In this paper, we are concerned with the commuting graph G G of a group G . It is defined as the graph with G as the set of vertices and where two of them form an edge if and only if they commute. This graph has been studied from several perspectives, for example, in [21] the chromatic and clique numbers are obtained for the commuting graphs of the dihedral-type groups; in [22] it is proved that the commuting graph of a finite minimal non-solvable group has diameter ≥ 3 ; in [23] the authors obtain the number of spanning trees of the commuting graph for some specific groups, as well as the classification of finite groups for which the power graph and the commuting graph coincide; and in [24] properties of this graph, as coloring and independence number, are used to prove results about finite groups.

The existence of abelian subgroups of maximal order of a given group G is a topic widely studied in different papers. For example, in [25] it was shown that if m is the maximal order of an abelian subgroup of a finite group G , then ∣ G ∣ divides m ! and in [26] some results are presented on m . Moreover, in [27] a classification of the abelian subgroups of maximal order of finite irreducible Coxeter groups is obtained, the geometry of these subgroups is studied and some applications of such classification to statistical physics are given.

In [24] Bertram, by using the commuting graph G G , gave an upper bound for the order of an abelian subgroup of a finite group G . In addition, he proposed to find necessary and sufficient conditions for the existence of an abelian subgroup whose order attains the bound ∣ [ x 1 ] ∣ + ∣ [ x 2 ] ∣ + ⋯ + ∣ [ x m ] ∣ . In this paper, we give a solution to this problem and present some related results.

In [28] the following result appears.

Lemma 1.1

If, for some q the number of vertices of degree ≥ q is ≤ q , then the graph G can be q -colored.

And it is used in [24] to prove the following theorem.

Theorem 1.2

Let G be a finite group with conjugacy classes indexed by cardinality:

1 = ∣ [ x 1 ] ∣ ≤ ∣ [ x 2 ] ∣ ≤ ⋯ ,

and let C G ( x ) denote the centralizer of x . If m is the smallest number i satisfying

∣ [ x 1 ] ∣ + ∣ [ x 2 ] ∣ + ⋯ + ∣ [ x i ] ∣ ≥ ∣ C G ( x i ) ∣ ,

then each abelian subgroup A of G satisfies

(1) ∣ [ x 1 ] ∣ + ∣ [ x 2 ] ∣ + ⋯ + ∣ [ x m ] ∣ ≥ ∣ A ∣ .

The problem proposed in [24] is the following:

Find necessary and sufficient conditions on G in order that inequality (1) in Theorem 1.2 becomes an equality for some abelian subgroup A < G .

In this paper, we use the following notation and known facts. Let G be an undirected simple graph, whose vertex set is V = { x 1 , x 2 , … , x n } . The degree of a vertex x is the number of edges incident with x and it is denoted by d ( x ) . We say that the vertices of G can be k -colored when there exists a partition of V into k subsets, with no two vertices in the same subset connected by an edge of G . The minimum number k for which G can be k -colored is called the chromatic number of G and it is denoted by χ ( G ) . A complete subgraph of G is a subgraph when every pair of vertices is connected by an edge. A maximal complete subgraph of G is called a clique; we also use the term k -clique for a clique consisting of k vertices. The clique number ω ( G ) is the maximal size of a clique contained in G . Note that if G can be k -colored, then the number of vertices in each clique (as well as in each complete subgraph) has cardinality ≤ k ; hence, we always have ω ( G ) ≤ χ ( G ) . A graph G is called weakly perfect if ω ( G ) = χ ( G ) (see [29]).

A subset X of a finite group G is called a commuting set if x y = y x for any x , y ∈ X . The commuting number κ ( G ) of a finite group G is the maximum cardinality of a commuting set. A subset Y ⊆ G is called an anti-commuting set if x y = y x implies x = y for any x , y ∈ Y . The Λ -number of G is the minimal number k such that G can be partitioned into k anti-commuting subsets; it is denoted by Λ ( G ) .

From the proof of Theorem 1.2, we get the following remark.

Remark 1.3

The graph G G is ∣ [ x 1 ] ∣ + ∣ [ x 2 ] ∣ + ⋯ + ∣ [ x m ] ∣ -colorable, thus,

χ ( G G ) ≤ ∣ [ x 1 ] ∣ + ∣ [ x 2 ] ∣ + ⋯ + ∣ [ x m ] ∣ .

2 Results

Throughout this paper we denote by c the sum ∣ [ x 1 ] ∣ + ∣ [ x 2 ] ∣ + ⋯ + ∣ [ x m ] ∣ , the left hand side of inequality (1). Note that

ω ( G G ) ≤ χ ( G G ) ≤ c .

2.1 G G parameters and the bound c

Theorem 2.1

A finite group G contains an abelian subgroup of order c if and only if G G contains a c -clique.

Corollary 2.2

A finite group G contains an abelian subgroup A of order c if and only if G G is a weakly perfect graph with χ ( G G ) = c .

Proof

If G contains an abelian subgroup of order c , then c ≤ ω ( G G ) ≤ χ ( G G ) ≤ c by Theorem 2.1, and hence χ ( G G ) = c .

For the converse, since ω ( G G ) = χ ( G G ) = c , the graph G G contains a c -clique, which implies that G contains an abelian subgroup A of order c .□

Corollary 2.3

A finite group G contains an abelian subgroup A of order c if and only if κ ( G ) = Λ ( G ) = c .

Proof

If G G is the commuting graph of G , then κ ( G ) = ω ( G G ) , Λ ( G ) = χ ( G G ) and Theorem 2.2 applies.□

Proposition 2.4

Let G be a finite group. Then the following statements are equivalent.

  1. G contains a commuting set with c elements.

  2. G contains an abelian subgroup of order c .

  3. G G contains a c -clique.

  4. G G is a weakly perfect graph and χ ( G G ) = c .

  5. G G contains a complete subgraph on c vertices.

  6. χ ( G G ) = κ ( G ) = c .

  7. ω ( G G ) = Λ ( G ) = c .

Proof

  1. Note that if X is a commuting set of G , then ⟨ X ⟩ is a complete subgraph of G G on c vertices. ⟨ X ⟩ is in fact a c -clique and we have already seen that this implies that X is an abelian subgroup.

  2. If G contains a commuting set with c elements, G G contains a c -clique. By Theorem 2.1, G contains an abelian subgroup of order c .

  3. If G G contains a c -clique, G contains an abelian subgroup of order c , by Corollary 2.2 G G is weakly perfect and χ ( G G ) = c .

  4. G G weakly perfect implies ω ( G G ) = χ ( G G ) , thus there is a c -clique.

  5. G G can be c -colored and does not contain a complete subgraph in more than c vertices and therefore G does not contain a commutative set with more than c vertices. From (5), G G cannot be k -colored for k < c and G contains a commutative set with c elements. It follows at once that χ ( G G ) = κ ( G ) = c .

  6. Suppose χ ( G G ) = κ ( G ) = c , since κ ( G ) = ω ( G G ) and Λ ( G ) = χ ( G G ) we get the result.

  7. ω ( G G ) = c implies the existence of a c -clique, thus, there is commuting set of cardinality c .□

The following lemma given in [28] together with the fact that the degree of a vertex x ∈ G G equals the order of its centralizer in G minus 1 lead to Proposition 2.6.

Lemma 2.5

Let G be a graph with chromatic number χ ( G ) = q + 1 and without any ( q + 1 ) -cliques. Let T = { x ∈ G : d ( x ) > q } , then

(2) ∑ x ∈ T ( d ( x ) − q ) ≥ q − 2 .

Proposition 2.6

Let G be a finite group. If χ ( G G ) = c and the set S = { x ∈ G : ∣ C G ( x ) ∣ > c } satisfies

(3) ∑ x ∈ S ( ∣ C G ( x ) ∣ − c ) < c − 3 ,

then G contains an abelian subgroup of order c .

Proof

If G does not contain such an abelian subgroup, Theorem 2.1 says that G G does not contain any c -clique, hence we may apply Lemma 2.5 with q = c − 1 , if T = { x ∈ V ( G G ) : d ( x ) > c − 1 } , then

(4) ∑ x ∈ T ( d ( x ) − c + 1 ) ≥ c − 3 .

But d ( x ) = ∣ C G ( x ) ∣ − 1 for each x . Hence, S = T and inequality (4) becomes

∑ x ∈ S ( ∣ C G ( x ) ∣ − c ) ≥ c − 3 . □

Proposition 2.7

Let G be a finite group. If G satisfies the hypothesis of Proposition 2.6, then G is abelian of order > 3 .

Proof

If S = ∅ , then left hand side of inequality (3) is 0 and therefore we must have c > 3 . Moreover, note that

∣ G ∣ = ∣ C G ( 1 ) ∣ ≤ c = χ ( G ) ≤ ∣ V ( G ) ∣

and G is abelian (otherwise some pair of vertices of G G could be colored with the same color).

If S is non-empty, then 1 ∈ G must be in S and hence ∣ C G ( 1 ) ∣ − c = ∣ G ∣ − c < c − 3 . This implies that ∣ G ∣ < 2 c − 3 . On the other hand, since G contains an abelian subgroup of order c , we have ∣ G ∣ = c k for some integer k and c ( k − 2 ) < − 3 . But this inequality holds just for k = 1 and c > 3 , in particular we must have k = 1 and hence G is abelian of order c > 3 .□

From Propositions 2.6 and 2.7, we obtain the following result.

Theorem 2.8

Let G be a finite group of order > 3 with Λ ( G ) = c and consider S = { x ∈ G : ∣ C G ( x ) ∣ > c } . Then G is abelian if and only if

(5) ∑ x ∈ S ( ∣ C G ( x ) ∣ − c ) < c − 3 .

Proof

If G is abelian, then ∣ C G ( x ) ∣ = ∣ G ∣ = Λ ( G ) = c for all x ∈ G , hence the left hand side in inequality (3) is 0, while the right hand side is ∣ G ∣ − 3 , a number greater than 0.

For the converse, suppose that G is a non-abelian group of order > 3 . According to Proposition 2.7, the hypothesis of Proposition 2.6 is not satisfied by G , so we must have χ ( G G ) ≠ c or

∑ x ∈ S ( ∣ C G ( x ) ∣ − c ) ≥ c − 3 .

Since χ ( G G ) = Λ ( G ) = c , the result follows.□

2.2 Groups for which the bound is attained

The following results show a relation between the expression of the order of finite groups and the existence of abelian subgroups of order c . In order to prove Theorems 2.9, 2.10 and 2.13 we use some known results from group theory.

Theorem 2.9

Let p be a prime and G a group of order p 3 , then G has an abelian subgroup of order c .

Proof

If G is abelian, we are done. If not, it is known that ∣ Z ( G ) ∣ = p and the number N of conjugacy classes is

N = p + 1 p 3 ∑ ∣ C G ( x ) ∣ ,

where the sum ∑ ∣ C G ( x ) ∣ is taken over the non-central elements.

Note that ∣ C G ( x ) ∣ = p 2 for each non-central element, otherwise there exists some non-central elements such that ∣ C G ( x ) ∣ = p . But Z ( G ) ≤ C G ( x ) also has order p ; hence, Z ( G ) = C G ( x ) . Since g ∈ C G ( x ) , x ∈ Z ( G ) and this is a contradiction. Therefore, the number of conjugacy classes is N = p + ( p 2 − 1 ) ; p with just one element and ( p 2 − 1 ) with p elements.

Finally, if m = 2 p − 1 we have c = 1 + 1 + ⋯ + 1 + p + ⋯ + p = p 2 . Note that G contains a subgroup of order p 2 , which is necessarily an abelian subgroup.□

Theorem 2.10

If p is a prime number and P is a non-abelian group of order p 3 then, P × Z p contains an abelian subgroup of order c .

Proof

First note that Z ( P × Z p ) = Z ( P ) × Z ( Z p ) = Z ( P ) × Z p . Now, since P is non-abelian, ∣ Z ( P ) ∣ = p and therefore ∣ Z ( P × Z p ) ∣ = p 2 . Also, we have C P × Z p ( x , y ) = C P ( x ) × C Z p ( y ) = C P ( x ) × Z p for each ( x , y ) ∈ P × Z p . If x ∈ P is a non-central element, we have already seen in the proof of Theorem 2.9 that ∣ C P ( x ) ∣ = p 2 , in particular, C P ( x ) is an abelian subgroup of P . Hence, if ( x , y ) ∈ P × Z p is a non-central element, then C P × Z p ( x , y ) is an abelian subgroup of order p 3 .

According to Burnside’s lemma, there are p 2 conjugacy classes of order 1 and p 3 − p conjugacy classes of order p . Hence, m = 2 p 2 − p in Theorem 1.2 and consequently c = p 2 + ( p 2 − p ) p = p 3 . If x ∈ P − Z ( P ) and y ∈ Z p is any element, then ( x , y ) is a non-central element of P × Z p and C P × Z p ( x , y ) is an abelian subgroup of order c .□

Next we analyze special cases when the order is p n q . The following lemmas (see [30] and [31]) will be useful.

Lemma 2.11

If G is a group of order p n q , where p > q are primes, then G contains a unique Sylow p -subgroup P .

Lemma 2.12

If G is a group which is not abelian, then G / Z ( G ) is not cyclic. In particular, [ G : Z ( G ) ] can never be a prime number.

Theorem 2.13

Let G be a non-abelian group of order p n q , where p and q are prime numbers:

  1. If p > q and the unique Sylow p -subgroup of G is abelian, then c = p n and G contains an abelian subgroup of order c . In particular, for n = 1 , 2 the group G contains an abelian subgroup of order c .

  2. If p < q and n = 2 , then either c = 4 , c = p q or c = q . In all these cases G contains an abelian subgroup of order c .

Moreover, if p 2 does not divide q − 1 , the case c = q is not possible.

Proof

(1) Let P be the unique Sylow p -subgroup of G , note that all elements of this Sylow p -subgroup have the smallest conjugacy classes with one or q elements and note also that actually there exist elements x ∈ P with ∣ [ x ] ∣ = q (otherwise Z ( G ) = G or ∣ Z ( G ) ∣ = p n contradicting hypothesis or Lemma 2.12 in any case).

Finally, let [ x 1 ] , [ x 2 ] , … , [ x k ] be the conjugacy classes of the elements in P ordered increasingly by size. Since P is the union of these conjugacy classes we have

p n = ∣ P ∣ = ∑ i = 1 k ∣ [ x i ] ∣ = ∣ C G ( x k ) ∣ = c .

If n = 1 , 2 the unique Sylow p -subgroup of G is abelian.

(2) First, we examine the possible numbers of Sylow q -subgroups and Sylow p -subgroups. Let n q and n p be the numbers of Sylow q -subgroups and Sylow p -subgroups, respectively. Since q > p , q ∤ ( p − 1 ) , it follows that n q ≠ p . Now, if n q = p 2 , then necessarily q ∣ ( p + 1 ) , and hence q = p + 1 . But the only prime numbers satisfying this condition are p = 2 and q = 3 . Consequently, ∣ G ∣ = 12 and then G ≅ A 4 , D 12 , Z 3 ⋊ Z 4 . In the first case c = 4 and in the other cases c = 6 . A glance to the subgroups of G shows the existence of an abelian subgroup of order c .

The other possibility is n q = 1 , let Q be such a Sylow q -subgroup. Let us now consider the possible values for n p . Note that such a number cannot be equal to 1, otherwise G is the direct product of these Sylow subgroups, and being each abelian, G must be abelian. This yields to n p = q .

For analyzing the order of Z ( G ) , Lemma 2.12 ensures that ∣ Z ( G ) ∣ ≠ p q , p 2 . On the other hand, if Z ( G ) has order q , then this is the unique Sylow q -subgroup, so all the elements of the different Sylow p -subgroups, except for the identity, are non-central elements. Hence, their centralizers have order p 2 . But ∣ Z ( G ) ∣ divides the order of such centralizers, which is impossible. The only remaining possible values for ∣ Z ( G ) ∣ are p or 1. Suppose that ∣ Z ( G ) ∣ = p . Since each Sylow p -subgroup is a centralizer, Z ( G ) is contained in each Sylow p -subgroup. Thus, the sum of ∣ Z ( G ) ∣ and the number of non-central elements in each Sylow ( p or q )-subgroup is p + q ( p 2 − p ) + ( q − 1 ) = p 2 q − p q + p + q − 1 . Subtracting this sum from p 2 q we get p q − p − q + 1 , which corresponds to the number of non-central elements of G that do not belong to any Sylow ( p or q )-subgroup.

Now, the non-central elements of Sylow p -subgroups and Sylow q -subgroups have centralizers of order p 2 and p q , respectively. The remaining p q − p − q + 1 elements have centralizers of order p q . Applying Burnside’s lemma we get:

N = p + 1 p 2 q ( ( q − 1 ) p q + ( p q − p − q + 1 ) p q + q ( p 2 − p ) p 2 ) = p + ( q − 1 ) + p ( p − 1 ) ,

where p , q − 1 and p ( p − 1 ) are the number of conjugacy classes of cardinally 1, p and q , respectively. Thus, the sum of the cardinals of the first p conjugacy classes and the following q − 1 conjugacy classes whose cardinals are p is c = p q . If x is an element not belonging to any Sylow ( p or q )-subgroup, then ∣ ⟨ x ⟩ ∣ = p q .

Suppose ∣ Z ( G ) ∣ = 1 , for x ∈ Q , x ≠ 1 , we may observe that Q ≤ C G ( x ) , thus, ∣ C G ( x ) ∣ = q or p q . If ∣ C G ( x ) ∣ = p q there exists y ∈ C G ( x ) of order p and, therefore, it must belong to a Sylow p -subgroup P , which implies that ∣ P ∩ Q ∣ ≥ 2 , a contradiction. Hence, ∣ C G ( x ) ∣ = q and Q = C G ( x ) . Moreover, the intersection of all Sylow p -subgroups is trivial, otherwise we may obtain a subgroup of order p contained in all of them and the centralizer of a non-trivial element of this subgroup would have order > p 2 (it commutes with every element of these Sylow p -subgroups). Now, if we take the sum over the number of elements in all Sylow subgroups we obtain q ( p 2 − 1 ) + ( q − 1 ) + 1 = p 2 q , which means that G is the union of its Sylow subgroups. Finally, calculating N we obtain

N = 1 + 1 p 2 q ( ( q − 1 ) q + q ( p 2 − 1 ) p 2 ) = 1 + q − 1 p 2 + ( p 2 − 1 ) .

So G has one conjugacy class with a unique element, q − 1 p 2 conjugacy classes each with p 2 elements and p 2 − 1 conjugacy classes each with q elements. This yields c = q and the order of the unique Sylow q -subgroup attains the bound. Notice that this case is not possible if p 2 does not divide q − 1 .□

3 Comparative analysis and conclusion

The main goal of our research was to determine under what conditions a finite group contains an abelian subgroup of maximal order. This interest is motivated by [24], where Bertram, relying on the commuting graph of a finite group, establishes that the order of every abelian subgroup of a finite group has an upper bound c . A question that immediately arises is “under what conditions equality is attained?.” Precisely, Bertram proposed it in his work without finding the solution, he only mentioned as an example the solvable groups with a number of conjugation classes less than or equal to 7, except for G = Sym ( 4 ) (the symmetric group on 4 letters), where the bound c coincides with the order of a centralizer, in fact, the largest centralizer other than G . In this paper, we give a solution to this problem, that is, we find necessary and sufficient conditions on a finite group G to contain an abelian subgroup with order the bound c . In addition, we find some results that relate structural properties of the commuting graph G G to those of the underlying group G , as well as some families of non-abelian groups attaining the upper bound c .

Over the years, other works have studied, in a certain way, abelian subgroups of maximal order of a finite group. For example, in [25] and [26] some arithmetic properties on the maximum order of an abelian subgroup are established, as well as their relation to the order of the group. In [25], it is shown that for any upper bound k for the order of the abelian subgroups of a finite group G occurs that ∣ G ∣ divides to k ! . If no number less than c satisfies this property we conclude that G contains an abelian subgroup of order c . In [27], abelian subgroups of maximal order of finite irreducible Coxeter groups are classified. Based on the development of our research, from the commuting graph of such Coxeter groups, it could be known which of these maximal order subgroups attains the bound c .

  1. Funding information: This article was supported by the Perfil Deseable del Programa para el Desarrollo Profesional Docente (PRODEP), a federal institution of the México’s government, the Universidad Autónoma de Guerrero (UAGro). J.M.S. was supported by a grant from Agencia Estatal de Investigación, Spain (PID2019-106433GB-I00/AEI/10.13039/501100011033).

  2. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-06-09
Revised: 2021-09-09
Accepted: 2021-12-05
Published Online: 2022-02-22

© 2022 Luis D. Arreola-Bautista et al., published by De Gruyter

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

Artikel in diesem Heft

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2021-0137
Schlagwörter für diesen Artikel
Abelian subgroup; commuting graph; clique; chromatic number; weakly perfect graph
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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