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Junyang Zhang

Publications and source records attributed to Junyang Zhang.

29 records · Page 2Linked to original sources
arXiv2023

Compound WKI-SP hierarchy and multiloop soliton solutions

The generalized hierarchies of compound WKI-SP (Wadati-Konno-Ichikawa and short pulse) equations are presented. The proposed integrable nonlinear equations include the WKI-type equations, the SP-type equations and the compound generalized WKI-SP equations. A chain of hodograph transformations are established to relate the compound WKI-SP equations with the MKdV-SG (modified Korteweg-de Vries and sine-Gordon) equations. As applications, the multiloop soliton solutions of one compound WKI-SP equation are obtained. We emphasize on showing abundant solitonic behaviors of two loop solitons. The role of each parameter plays in the movement of two-loop solion are shown detailedly in a table.

nlin.SI↗
arXiv2022

A note on regular sets in Cayley graphs

A subset $R$ of the vertex set of a graph $Γ$ is said to be $(κ,τ)$-regular if $R$ induces a $κ$-regular subgraph and every vertex outside $R$ is adjacent to exactly $τ$ vertices in $R$. In particular, if $R$ is a $(κ,τ)$-regular set of some Cayley graph on a finite group $G$, then $R$ is called a $(κ,τ)$-regular set of $G$. Let $H$ be a non-trivial normal subgroup of $G$, and $κ$ and $τ$ a pair of integers satisfying $0\leqκ\leq|H|-1$, $1\leqτ\leq|H|$ and $\gcd(2,|H|-1)\midκ$. It is proved that (i) if $τ$ is even, then $H$ is a $(κ,τ)$-regular set of $G$; (ii) if $τ$ is odd, then $H$ is a $(κ,τ)$-regular set of $G$ if and only if it is a $(0,1)$-regular set of $G$.

math.CO↗
arXiv2022

Characterizing subgroup perfect codes by 2-subgroups

A perfect code in a graph $Γ$ is a subset $C$ of $V(Γ)$ such that no two vertices in $C$ are adjacent and every vertex in $V(Γ)\setminus C$ is adjacent to exactly one vertex in $C$. Let $G$ be a finite group and $C$ a subset of $G$. Then $C$ is said to be a perfect code of $G$ if there exists a Cayley graph of $G$ admiting $C$ as a perfect code. It is proved that a subgroup $H$ of $G$ is a perfect code of $G$ if and only if a Sylow $2$-subgroup of $H$ is a perfect code of $G$. This result provides a way to simplify the study of subgroup perfect codes of general groups to the study of subgroup perfect codes of $2$-groups. As an application, a criterion for determining subgroup perfect codes of projective special linear groups $\mathrm{PSL}(2,q)$ is given.

math.CO↗
arXiv2022

Skew-Morphisms of Elementary Abelian p-Groups

A skew-morphism of a finite group $G$ is a permutation $σ$ on $G$ fixing the identity element, and for which there exists an integer function $π$ on $G$ such that $σ(xy)=σ(x)σ^{π(x)}(y)$ for all $x,y\in G$. It has been known that given a skew-morphism $σ$ of $G$, the product of $\langle σ\rangle$ with the left regular representation of $G$ forms a permutation group on $G$, called the skew-product group of $σ$. In this paper, the skew-product groups of skew-morphisms of finite elementary abelian $p$-groups are investigated. Some properties, characterizations and constructions about that are obtained.

math.CO↗
arXiv2022

Nowhere-zero 3-flows in Cayley graphs on supersolvable groups

Tutte's 3-flow conjecture asserts that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow $2$-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.

math.CO↗
arXiv2021

Perfect codes in vertex-transitive graphs

Given a graph $Γ$, a perfect code in $Γ$ is an independent set $C$ of vertices of $Γ$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code in $Γ$ is a set $C$ of vertices of $Γ$ such that every vertex of $Γ$ is adjacent to a unique vertex in $C$. To study (total) perfect codes in vertex-transitive graphs, we generalize the concept of subgroup (total) perfect code of a finite group introduced in \cite{HXZ18} as follows: Given a finite group $G$ and a subgroup $H$ of $G$, a subgroup $A$ of $G$ containing $H$ is called a subgroup (total) perfect code of the pair $(G,H)$ if there exists a coset graph $Cos(G,H,U)$ such that the set consisting of left cosets of $H$ in $A$ is a (total) perfect code in $Cos(G,H,U)$. We give a necessary and sufficient condition for a subgroup $A$ of $G$ containing $H$ to be a (total) perfect code of the pair $(G,H)$ and generalize a few known results of subgroup (total) perfect codes of groups. We also construct some examples of subgroup perfect codes of the pair $(G,H)$ and propose a few problems for further research.

math.CO↗
arXiv2021

On subgroup perfect codes in Cayley graphs

A perfect code in a graph $Γ= (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if there exists a Cayley graph of $G$ which admits $H$ as a perfect code. Equivalently, $H$ is a subgroup perfect code of $G$ if there exists an inverse-closed subset $A$ of $G$ containing the identity element such that $(A, H)$ is a tiling of $G$ in the sense that every element of $G$ can be uniquely expressed as the product of an element of $A$ and an element of $H$. In this paper we obtain multiple results on subgroup perfect codes of finite groups, including a few necessary and sufficient conditions for a subgroup of a finite group to be a subgroup perfect code, a few results involving $2$-subgroups in the study of subgroup perfect codes, and several results on subgroup perfect codes of metabelian groups, generalized dihedral groups, nilpotent groups and $2$-groups.

math.CO↗
arXiv2018

Smooth skew-morphisms of the dihedral groups

A skew-morphism $φ$ of a finite group $A$ is a permutation on $A$ such that $φ(1)=1$ and $φ(xy)=φ(x)φ^{π(x)}(y)$ for all $x,y\in A$ where $π:A\to\mathbb{Z}_{|φ|}$ is an integer function. A skew-morphism is smooth if $π(φ(x))=π(x)$ for all $x\in A$. The concept of smooth skew-morphisms is a generalization of that of $t$-balanced skew-morphisms. The aim of the paper is to develop a general theory of smooth skew-morphisms. As an application we classify smooth skew-morphisms of the dihedral groups.

math.GR↗
arXiv2012

A Classification of Orientable Regular Embeddings of Complete Multipartite Graphs

Let $K_{m[n]}$ be the complete multipartite graph with $m$ parts, while each part contains $n$ vertices. The orientably-regular embeddings of complete graphs $K_{m[1]}$ have been determined by Biggs (1971) \cite{Big1}, James and Jones (1985) \cite{JJ}. During the past twenty years, several papers such as Du et al.(2007, 2010) \cite{DJKNS1,DJKNS2}, Jones et al. (2007, 2008) \cite{JNS1,JNS2}, Kwak and Kwon (2005, 2008) \cite{KK1,KK2} and Nedela et al. (1997, 2002)\cite{NS,NSZ} contributed to the orientably-regular embeddings of complete bipartite graphs $K_{2[n]}$ and the final classification was given by Jones \cite{Jon1} in 2010. Based on our former paper \cite{ZD}, this paper gives a complete classification of orientably-regular embeddings of graphs $K_{m[n]}$ for the general cases $m\ge 3$ and $n\ge 2$.

math.CO↗