arXiv subjects

A-Ming Liu

Publications and source records attributed to A-Ming Liu.

14 recordsLinked to original sources

arXiv2024

The structure of a finite group and the maximum $π$-index of its elements

Given a set of primes $π$, the $π$-index of an element $x$ of a finite group $G$ is the $π$-part of the index of the centralizer of $x$ in $G$. If $π=\{p\}$ is a singleton, we just say the $p$-index. If the $π$-index of $x$ is equal to $p_1^{k_1}\ldots p^{k_s}$, where $p_1,\ldots,p_s$ are distinct primes, then we set $\exp_π(x)=k_1+\ldots+k_s$. In this short note, we study how the number $ε_π(G)=\max\{ε_π(x):x\in G\}$ restricts the structure of the factor group $G/Z(G)$ of $G$ by its center. First, for a finite group $G$, we prove that $ϕ_p(G/Z(G))\leqε_p(G)$, where $ϕ_p(G/Z(G))$ is the Frattini length of a Sylow $p$-subgroup of $G/Z(G)$. Second, for a $π$-separable finite group $G$, we prove that $l_π(G/Z(G))\leqε_π(G)$, where $l_π(G/Z(G))$ is the $π$-length of $G/Z(G)$.

math.GR↗

arXiv2024

A contribution to the theory of $σ$-properties of a finite group

We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every maximal subgroup $V$ of the generalized Fitting subgroup $F^{*}(G)$ of $G$; a finite soluble group $G$ is a $PST$-group (that is, Sylow permutability is a transitive relation on $G$) if and only if $G$ has a normal subgroup $D$ such that $G/D$ is nilpotent and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every subnormal subgroup $A$ of $G$.

math.GR↗

arXiv2023

The spectral radius of minor free graphs

In this paper, we present a sharp upper bound for the spectral radius of an $n$-vertex graph without $F$-minor for sufficient large $n$, where $F$ is obtained from the complete graph $K_r$ by deleting disjointed paths. Furthermore, the graphs which achieved the sharp bound are characterized. This result may be regarded to be an extended revision of the number of edges in an $n$-vertex graph without $F$-minor.

math.CO↗

arXiv2023

On the $A_α$-spectral radius of graphs without linear forests

Let $A(G)$ and $D(G)$ be the adjacency and degree matrices of a simple graph $G$ on $n$ vertices, respectively. The \emph{$A_α$-spectral radius} of $G$ is the largest eigenvalue of $A_α(G)=αD(G)+(1-α)A(G)$ for a real number $α\in[0,1]$. In this paper, for $α\in (0,1)$, we obtain a sharp upper bound for the $A_α$-spectral radius of graphs on $n$ vertices without a subgraph isomorphic to a liner forest for $n$ large enough and characterize all graphs which attain the upper bound. As a result, we completely obtain the maximum signless Laplacian spectral radius of graphs on $n$ vertices without a subgraph isomorphic to a liner forest for $n$ large enough.

math.CO↗

arXiv2022

Spectral extremal results on the $α$-index of graphs without minors and star forests

Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_α(G)$ of $A (G)$ and $D (G) $ by $$A_α(G)=αD(G)+(1-α)A(G)$$ for any real number $0\leqα\leq1$. The \emph{$α$-index} of $G$ is the largest eigenvalue of $A_α(G)$. In this paper, we determine the maximum $α$-index and characterize all extremal graphs for $K_r$ minor-free graphs, $K_{s,t}$ minor-free graphs, and star-forest-free graphs for any $0<α<1$ by unified eigenvector approach, respectively.

math.CO↗

arXiv2021

The signless Laplacian spectral radius of graphs without intersecting odd cycles

Let $F_{a_1,\dots,a_k}$ be a graph consisting of $k$ cycles of odd length $2a_1+1,\dots, 2a_k+1$, respectively which intersect in exactly a common vertex, where $k\geq1$ and $a_1\ge a_2\ge \cdots\ge a_k\ge 1$. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all $F_{a_1,\dots,a_k}$-free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.

math.CO↗

arXiv2021

On the spectral radius of graphs without a star forest

In this paper, we present two sharp upper bounds for the spectral radius of (bipartite) graphs with forbidden a star forest and characterize all extremal graphs. Moreover, the minimum least eigenvalue of the adjacency matrix of graph with forbidden a star forest and all extremal graphs for graphs are obtained.

math.CO↗

arXiv2021

The Cohen-Macaulay Property of $f$-ideals

For positive integers $d<n$, let $[n]_d=\{A\in 2^{[n]}\mid |A|=d\}$ where $[n]=:\{1,2,\ldots, n\}$. For a pure $f$-simplicial complex $Δ$ such that ${\rm dim}(Δ)={\rm dim}(Δ^c)$ and $\mathcal{F}(Δ)\cap \mathcal{F}(Δ^c)=\emptyset$, we prove that the facet ideal $I(Δ)$ is Cohen-Macaulay if and only if it has linear resolution. For a $d$-dimensional pure $f$-simplicial complex $Δ$ such that $Δ'=:\langle F\mid F\in [n]_d\smallsetminus \mathcal F(Δ)\rangle$ is an $f$-simplicial complex, we prove that $I(Δ^c)$ is Cohen-Macaulay if and only if $I(Δ')$ has linear resolution.

math.AC↗

arXiv2021

A $G$-covering subgroup system of a finite group for some classes of $σ$-soluble groups

Let ${\frak F}$ be a class of group and $G$ a finite group. Then a set $Σ$ of subgroups of $G$ is called a \emph{$G$-covering subgroup system} for the class ${\frak F}$ if $G\in {\frak F}$ whenever $Σ\subseteq {\frak F}$. We prove that: {\sl If a set of subgroups $Σ$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $Σ$ is a $G$-covering subgroup system for the classes of all $σ$-soluble and all $σ$-nilpotent groups, and for the class of all $σ$-soluble $PσT$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook.

math.GR↗

arXiv2020

The signless Laplacian spectral radius of graphs with forbidding linear forests

Turán type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Turán type extremal problem is how to maximize (signless Laplacian) spectral radius over all graphs which do not contain fixed subgraphs. In this paper, we first present a stability result for $k\cdot P_3$ in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or $k\cdot P_3$ as a subgraph, respectively.

math.CO↗

arXiv2018

A construction of sequentially Cohen-Macaulay graphs

For every simple graph $G$, a class of multiple clique cluster-whiskered graphs $G^{md}$ is introduced, and it is shown that all graphs $G^{md}$ are vertex decomposable, thus the independence simplicial complex ${\rm Ind}\,G^{md}$ is sequentially Cohen-Macaulay; the properties of the graphs $G^{md}$ and the clique-whiskered graph $G^π$ are studied, including the enumeration of facets of the complex ${\rm Ind}\, G^π$ and, the calculation of Betti numbers of the cover ideal $I_c(G^{md})$.

math.AC↗

arXiv2018

Spectral extremal results with forbidding linear forests

The Turán type extremal problem asks to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, the spectral Turán type extremal problem asks to maximize spectral radius of all graphs which do not contain fixed subgraphs. In this paper, we determine the maximum spectral radius of all graphs without containing a linear forest as a subgraph and characterize all corresponding extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without containing $k\cdot P_3$ as a subgraph are obtained and all extremal graphs are also characterized. Moreover, some relations between Tuán type extremal problems and spectral Turán type extremal problems are discussed.

math.CO↗

arXiv2016

Boolean graphs are unmixed and vertex decomposable

For each Boolean graph $B_n$, it is proved that both $B_n$ and its complement graph $\overline{B_n}$ are vertex decomposable. It is also proved that $B_n$ is an unmixed graph, thus it is also Cohen-Macaulay.

math.AC↗

arXiv2016

Minimal free resolution of a graded ideal with linear quotients

Let $I$ be a graded ideal of $K[x_1,\ldots,x_n]$ generated by homogeneous polynomials of a same degree $d$, and assume that $I$ has linear quotients. In this note, we use Horseshoe Lemma to give a relatively direct inductive construction of a minimal free resolution of $I$, which is called a $d$-linear resolution.

math.AC↗