scholarly journals Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs

10.37236/8071 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Matteo Cavaleri ◽  
Alfredo Donno ◽  
Andrea Scozzari
Keyword(s):  
Wreath Product ◽  
Wiener Index ◽  
Wreath Products ◽  
Centrality Measures ◽  
Star Graph ◽  
Graph Products ◽  
Lexicographic Product ◽  
Star Graphs ◽  
Total Distance ◽  
Conditional Expectations

In the last decades much attention has turned towards centrality measures on graphs. The Wiener index and the total distance are key tools to investigate the median vertices, the distance-balanced property and the opportunity index of a graph. This interest has recently been addressed to graphs obtained via classical graph products like the Cartesian, the direct, the strong and the lexicographic product. We extend this study to a relatively new graph product, that is, the wreath product. In this paper, we compute the total distance for the vertices of an arbitrary wreath product graph $G\wr H$ in terms of the total distances in $H$ and of some distance-based indices of $G$. We explicitly compute these indices for the star graph $S_n$, providing a closed formula for the total distances in $S_n\wr H$ when $H$ is complete or a star. As a consequence, we obtain the Wiener index of these graphs, we characterize the median and the central vertices, and finally we give an upper and a lower bound for the opportunity index of $S_n\wr S_m$ in terms of tail conditional expectations of an associated binomial distribution.

scholarly journals EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

2012 ◽  
Vol 92 (1) ◽  
pp. 127-136 ◽  
Author(s):  
CHERYL E. PRAEGER ◽  
CSABA SCHNEIDER
Keyword(s):  
Permutation Group ◽  
Wreath Product ◽  
Automorphism Groups ◽  
Wreath Products ◽  
Error Correcting Codes ◽  
Permutation Groups ◽  
Graph Products ◽  
Product Action ◽  
Hamming Graphs

AbstractWe consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals WREATH PRODUCTS, NILPOTENT ORBITS AND SYMPLECTIC DEFORMATIONS

2007 ◽  
Vol 18 (05) ◽  
pp. 473-481
Author(s):  
BAOHUA FU
Keyword(s):  
Wreath Product ◽  
Wreath Products ◽  
Nilpotent Orbit ◽  
Nilpotent Orbits ◽  
Symplectic Resolutions

We recover the wreath product X ≔ Sym 2(ℂ2/± 1) as a transversal slice to a nilpotent orbit in 𝔰𝔭6. By using deformations of Springer resolutions, we construct a symplectic deformation of symplectic resolutions of X.

Automorphisms of Iterated Wreath Product p-Groups

2012 ◽  
Vol 55 (2) ◽  
pp. 390-399 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Wreath Products ◽  
Cyclic Groups ◽  
Important Family

AbstractWe determine the order of the automorphism group Aut(W) for each member W of an important family of finite p-groups that may be constructed as iterated regular wreath products of cyclic groups. We use a method based on representation theory.

Diagnosability of Bubble-Sort Star Graphs with Missing Edges

2019 ◽  
Vol 19 (02) ◽  
pp. 1950002 ◽  
Author(s):  
SHIYING WANG ◽  
YINGYING WANG
Keyword(s):  
Multiprocessor System ◽  
Star Graph ◽  
Star Graphs ◽  
Comparison Model ◽  
Model Following ◽  
Strong Property

The diagnosability of a multiprocessor system plays an important role. The bubble-sort star graph BSn has many good properties. In this paper, we study the diagnosis on BSn under the comparison model. Following the concept of the local diagnosability, the strong local diagnosability property is discussed. This property describes the equivalence of the local diagnosability of a node and its degree. We prove that BSn (n ≥ 5) has this property, and it keeps this strong property even if there exist (2n − 5) missing edges in it, and the result is optimal with respect to the number of missing edges.

Simple groups and irreducible lattices in wreath products

2020 ◽  
pp. 1-12 ◽  
Author(s):  
ADRIEN LE BOUDEC
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Free Group ◽  
Wreath Products ◽  
Simple Groups ◽  
Finitely Generated ◽  
Locally Compact ◽  
Regular Tree ◽  
Finitely Generated Groups ◽  
A Finite Group

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group.

W*-superrigidity for wreath products with groups having positive first ℓ2-Betti number

2015 ◽  
Vol 26 (01) ◽  
pp. 1550003 ◽  
Author(s):  
Mihaita Berbec
Keyword(s):  
Wreath Product ◽  
Betti Number ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Wreath Products ◽  
Hyperbolic Groups ◽  
Free Products ◽  
Von Neumann ◽  
Product Group ◽  
Group Von Neumann Algebra

In [M. Berbec and S. Vaes, W*-superrigidity for group von Neumann algebras of left–right wreath products, Proc. London Math. Soc.108 (2014) 1116–1152] we have proven that, for all hyperbolic groups and for all nontrivial free products Γ, the left–right wreath product group 𝒢 ≔ (ℤ/2ℤ)(Γ) ⋊ (Γ × Γ) is W*-superrigid, in the sense that its group von Neumann algebra L𝒢 completely remembers the group 𝒢. In this paper, we extend this result to other classes of countable groups. More precisely, we prove that for weakly amenable groups Γ having positive first ℓ2-Betti number, the same wreath product group 𝒢 is W*-superrigid.

scholarly journals On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 185
Author(s):  
Ram Band ◽  
Sven Gnutzmann ◽  
August Krueger
Keyword(s):  
Existence Of Solutions ◽  
Stationary Waves ◽  
Star Graph ◽  
Oscillation Theorem ◽  
Nodal Domains ◽  
Nodal Structure ◽  
Star Graphs ◽  
Spectral Curves ◽  
Nonlinear Quantum ◽  
Cubic Nonlinear Schrödinger Equation

We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e., the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting, such as the classic Sturm oscillation theorem.

2-Distance chromatic number of some graph products

2020 ◽  
Vol 12 (02) ◽  
pp. 2050021
Author(s):  
Ghazale Ghazi ◽  
Freydoon Rahbarnia ◽  
Mostafa Tavakoli
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Graph Products ◽  
Lexicographic Product ◽  
Graph Product ◽  
Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

scholarly journals Wreath decompositions of finite permutation groups

1989 ◽  
Vol 40 (2) ◽  
pp. 255-279 ◽  
Author(s):  
L. G. Kovács
Keyword(s):  
Permutation Group ◽  
Wreath Product ◽  
Transitive Group ◽  
Wreath Products ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Formal Definition ◽  
The Third ◽  
Product Action ◽  
Factor Decomposition

There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition’ concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition.If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition’ concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle.Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G.

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