Degree Sequence of Tight Distance Graphs

A graph G on n vertices is a tight distance graph if there exists a set $D\subseteq \{1,2,\dots ,n-1\}$ such that $V(G)=\{0,1,\dots ,n-1\}$ and $ij\in E(G)$ if and only if $|i-j|\in D$. A characterization of the degree sequences of tight distance graphs is given. This characterization yields a fast method for recognizing and realizing degree sequences of tight distance graphs.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

Solvable PST-groups,strong Sylow bases and mutually permutable products

A subgroup H of a group G is said to permute with the subgroup K of G if $HK=KH$. Subgroups H and K are mutually permutable (totally permutable) in G if every subgroup of H permutes with K and every subgroup of K permutes with H (if every subgroup of H permutes with every subgroup of K). If H and K are mutually permutable and $H\cap K=1$, then H and K are totally permutable. A subgroup H of G is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a PST-group if S-permutability is a transitive relation in G. Let $\{p_1,\dots ,p_n,p_{n+1},\dots ,p_k\}$ be the set of prime divisors of the order of a finite group G with $\{p_1,\dots ,p_n\}$ the set of prime divisors of the order of the normal subgroup N of G. A set of Sylow subgroups $\{P_1,\dots ,P_n,P_{n+1},\dots ,P_k\}$, $P_i\in {\mathrm{Syl}}_{p_i}(G)$, form a strong Sylow system with respect to N if $P_i{P}_j$ is a mutually permutable product for all $i\in \{1,2,\dots ,n\}$ and $j\in \{1,2,\dots ,k\}$. We show that a finite group G is a solvable PST-group if and only if it has a normal subgroup N such that $G/N$ is nilpotent and G has a strong Sylow system with respect to N. It is also shown that G is a solvable PST-group if and only if G has a normal solvable PST-subgroup N and $G/N^{″}$ is a solvable PST-group.     

Enumeration of simple complete topological graphs

A simple topological graph $T=(V(T),E(T))$ is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is $2^{\mathrm{\Theta }(n^4)}$. We also show that the number weak isomorphism classes of simple complete topological graphs with n vertices and $\left(\begin{array}{c}\hfill n\hfill \\ \hfill 4\hfill \end{array}\right)$ crossings is at least $2^{n(\log n-O(1))}$, which improves the estimate of Harborth an Mengersen.     

An exact upper bound for sums of element orders in non-cyclic finite groups

Denote the sum of element orders in a finite group G by $\psi (G)$ and let $C_n$ denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that $\psi (G)\le \frac{7}{11}\psi (C_n)$ and $\psi (G)<\frac{1}{q?1}\psi (C_n)$. The first result is best possible, since for each $n=4k$, k odd, there exists a group G of order n satisfying $\psi (G)=\frac{7}{11}\psi (C_n)$ and the second result implies that if G is of odd order, then $\psi (G)<\frac{1}{2}\psi (C_n)$. Our results improve the inequality $\psi (G)<\psi (C_n)$ obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some $\psi (G)$-based sufficient conditions for the solvability of G.