arXiv subjects

Liyun Wu

Publications and source records attributed to Liyun Wu.

4 recordsLinked to original sources

arXiv2023

A new class of generalized inverses in semigroups and rings with involution

Let $S$ be a $*$-semigroup and let $a,w,v\in S$. The initial goal of this work is to introduce two new classes of generalized inverses, called the $w$-core inverse and the dual $v$-core inverse in $S$. An element $a\in S$ is $w$-core invertible if there exists some $x\in S$ such that $awx^2=x$, $xawa=a$ and $(awx)^*=awx$. Such an $x$ is called a $w$-core inverse of $a$. It is shown that the core inverse and the pseudo core inverse can be characterized in terms of the $w$-core inverse. Several characterizations of the $w$-core inverse of $a$ are derived, and the expression is given by the inverse of $w$ along $a$ and $\{1,3\}$-inverses of $a$ in $S$. Also, the connections between the $w$-core inverse and other generalized inverses are given. In particular, when $S$ is a $*$-ring, the existence criterion for the $w$-core inverse is given by units. The dual $v$-core inverse of $a$ is defined by the existence of $y\in S$ satisfying $y^2va=y$, $avay=a$ and $(yva)^*=yva$. Dual results for the dual $v$-core inverse also hold.

math.RA↗

arXiv2023

A new class of partial orders

Let $R$ be a unital $*$-ring. For any $a,w,b\in R$, we apply the defined $w$-core inverse to define a new class of partial orders in $R$, called the $w$-core partial order. Suppose $a,b\in R$ are $w$-core invertible. We say that $a$ is below $b$ under the $w$-core partial order, denoted by $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, if $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$ and $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$, where $a_w^{\tiny{\textcircled{\#}}}$ denotes the $w$-core inverse of $a$. Characterizations of the $w$-core partial order are given. Also, the relationships with several types of partial orders are considered. In particular, we show that the core partial order coincides with the $a$-core partial order, and the star partial order coincides with the $a^*$-core partial order.

math.RA↗

arXiv2021

Connectivity keeping paths in $k$-connected bipartite graphs

In 2010, Mader [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.] proved that every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor+m-1$ contains a path $P$ of order $m$ such that $G-V(P)$ is still $k$-connected. In this paper, we consider similar problem for bipartite graphs, and prove that every $k$-connected bipartite graph $G$ with minimum degree at least $k+m$ contains a path $P$ of order $m$ such that $G-V(P)$ is still $k$-connected.

math.CO↗

arXiv2019

The connectivity of a bipartite graph and its bipartite complementary graph

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph $G$ and the same invariant in the complement $G^c$ of $G$ is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph $G=G[X,Y]$ with bipartition ($X,Y$), its bipartite complementary graph $G^{bc}$ is a bipartite graph with $V(G^{bc})=V(G)$ and $E(G^{bc})=\{xy:\ x\in X,\ y\in Y$ and $xy \notin E(G)\}$. In this paper, we obtain the Nordhaus-Gaddum type inequalities for connectivity of bipartite graphs and its bipartite complementary graphs. Furthermore, we prove that these inequalities are best possible.

math.CO↗