Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G\cong Sz\left(q\right)$ if and only if $𝒮\left(G\right)\cong 𝒮\left(Sz\right(q\left)\right)$, where $q=2^{2m+1}\ge 8$.

Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G{\cong }^2{G}_2\left(3^{2n+1}\right)$ if and only if $𝒮\left(G\right)\cong 𝒮{(}^2{G}_2\left(3^{2n+1}\right))$. As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group ${}^2{G}_2\left(3^{2n+1}\right)$.

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...

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...

For a fixed positive integer $k$ and $G=(V,E)$ a connected graph of order $n$, whose minimum vertex degree is at least $k$, a set $S\subseteq V$ is a total $k$-dominating set, also known as a $k$-tuple total dominating set, if every vertex $v\in V$ has at least $k$ neighbors in $S$. The minimum size of a total $k$-dominating set for $G$ is called the total $k$-domination number of $G$, denoted by ${\gamma }_{kt}\left(G\right)$. The total $k$-domination problem is to determine a minimum total $k$-dominating set of $G$. Since the exact problem is in general quite difficult...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

A nonincreasing sequence $\pi =(d_1,...,d_n)$ of nonnegative integers is a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. In this case, $G$ is referred to as a realization of $\pi$. Given two graphs $G_1$ and $G_2$, A. Busch et al. (2014) introduced the potential-Ramsey number of $G_1$ and $G_2$, denoted by $r_{\mathrm{pot}}(G_1,G_2)$, as the smallest nonnegative integer $m$ such that for every $m$-term graphic sequence $\pi$, there is a realization $G$ of $\pi$ with $G_1\subseteq G$ or with $G_2\subseteq \overline{G}$, where $\overline{G}$ is the complement of $G$. For $t\ge 2$ and $0\le k\le \lfloor \frac{t}{2}\rfloor$, let $K_t^{-k}$ be the graph...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

Let $T$ be a tree with $n$ vertices. To each edge of $T$ we assign a weight which is a positive definite matrix of some fixed order, say, $s$. Let $D_{ij}$ denote the sum of all the weights lying in the path connecting the vertices $i$ and $j$ of $T$. We now say that $D_{ij}$ is the distance between $i$ and $j$. Define $D:=\left[D_{ij}\right]$, where $D_{ii}$ is the $s\times s$ null matrix and for $i\ne j$, $D_{ij}$ is the distance between $i$ and $j$. Let $G$ be an arbitrary connected weighted graph with $n$ vertices, where each weight is a positive definite matrix of order...

Let $ex(n,G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_i$ denote a path consisting of $i$ vertices and let $m{P}_i$ denote $m$ disjoint copies of $P_i$. In this paper we count $ex(n,3P_4)$.