Suppose that $𝒢$ is a finite, connected graph and $X$ is a lazy random walk on $𝒢$. The lamplighter chain $X^{\diamond }$ associated with $X$ is the random walk on the wreath product ${𝒢}^{\diamond }={𝐙}_2\wr 𝒢$, the graph whose vertices consist of pairs $(\underline{f},x)$ where $f$ is a labeling of the vertices of $𝒢$ by elements of ${𝐙}_2=\{0,1\}$ and $x$ is a vertex in $𝒢$. There is an edge between $(\underline{f},x)$ and $(\underline{g},y)$ in ${𝒢}^{\diamond }$ if and only if $x$ is adjacent to $y$ in $𝒢$ and $f_z=g_z$ for all $z\ne x,y$. In each step, $X^{\diamond }$ moves from a configuration $(\underline{f},x)$ by updating $x$ to $y$ using the transition rule of $X$ and then...

Suppose that $A$ is a real symmetric matrix of order $n$. Denote by $m_A\left(0\right)$ the nullity of $A$. For a nonempty subset $\alpha$ of $\{1,2,...,n\}$, let $A\left(\alpha \right)$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $\alpha$. When $m_{A\left(\alpha \right)}\left(0\right)=m_A\left(0\right)+\left|\alpha \right|$, we call $\alpha$ a P-set of $A$. It is known that every P-set of $A$ contains at most $\lfloor n/2\rfloor$ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As...

Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and $m_k\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{m_k\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left(S_r\right)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

Let $k$ be a field of characteristic $p\>0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that...

The generalized $k$-connectivity ${\kappa }_k\left(G\right)$ of a graph $G$ was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized $k$-edge-connectivity which is defined as ${\lambda }_k\left(G\right)=min\{\lambda \left(S\right):S\subseteq V\left(G\right)$ and $\left|S\right|=k\}$, where $\lambda \left(S\right)$ denotes the maximum number $\ell$ of pairwise edge-disjoint trees $T_1,T_2,...,T_{\ell }$ in $G$ such that $S\subseteq V\left(T_i\right)$ for $1\le i\le \ell$. In this paper we prove that for any two connected graphs $G$ and $H$ we have ${\lambda }_3(G\square H)\ge {\lambda }_3\left(G\right)+{\lambda }_3\left(H\right)$, where $G\square H$ is the Cartesian product of $G$ and $H$. Moreover, the bound is sharp. We also...

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\le n$ and $m\ge 2n$, and for the case that $F_m$ has no odd component. Moreover, we...

Given a graph $G$, let $f\left(G\right)$ denote the maximum number of edges in a bipartite subgraph of $G$. Given a fixed graph $H$ and a positive integer $m$, let $f(m,H)$ denote the minimum possible cardinality of $f\left(G\right)$, as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. In this paper we prove that $f(m,{\theta }_{k,s})\ge \frac{1}{2}m+\Omega \left(m^{(2k+1)/(2k+2)}\right)$, which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write $K_k^{\text{'}}$ and $K_{t,s}^{\text{'}}$ for the subdivisions of $K_k$ and $K_{t,s}$. We show that $f(m,K_k^{\text{'}})\ge \frac{1}{2}m+\Omega \left(m^{(5k-8)/(6k-10)}\right)$ and $f(m,K_{t,s}^{\text{'}})\ge \frac{1}{2}m+\Omega \left(m^{(5t-1)/(6t-2)}\right)$, improving a result of Q. Zeng, J. Hou. We also give lower bounds on...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

The Ramsey number $R(G,H)$ for a pair of graphs $G$ and $H$ is defined as the smallest integer $n$ such that, for any graph $F$ on $n$ vertices, either $F$ contains $G$ or $\overline{F}$ contains $H$ as a subgraph, where $\overline{F}$ denotes the complement of $F$. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers $R(K_1+L_n,P_m)$ and $R(K_1+L_n,C_m)$ for some integers $m$, $n$, where $L_n$ is...

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...