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.     

Classification of some 3-subgroups of the finite groups of Lie type E6

We consider the finite exceptional group of Lie type $G=E_6^{\epsilon }(q)$ (universal version) with $3|q?\epsilon$, where $E_6^{+1}(q)=E_6(q)$ and $E_6^{?1}(q){=}^2{E}_6(q)$. We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local $M<G$ which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing $Z(G)$, all extraspecial subgroups containing $Z(G)$, and all cyclic groups of order 9 containing $Z(G)$. These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.     

Upper bounds for the sum of Laplacian eigenvalues of graphs

Let G be a graph with n vertices and $e(G)$ edges, and let ${\mu }_1(G)?{\mu }_2(G)???{\mu }_n(G)=0$ be the Laplacian eigenvalues of G. Let $S_k(G)={\sum }_{i=1}^k{\mu }_i(G)$, where $1?k?n$. Brouwer conjectured that $S_k(G)?e(G)+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for $1?k?n$. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for $S_k(G)$, and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.     

A lower bound for on-line ranking number of a path

A k-ranking of a graph G is a mapping $\varphi :V(G)\to \{1,\dots ,k\}$ such that any path with endvertices x and y satisfying $x\ne y$ and $\varphi (x)=\varphi (y)$ contains a vertex z with $\varphi (z)>\varphi (x)$. The ranking number ${\chi }_{\mathrm{r}}(G)$ of G is the minimum k admitting a k-ranking of G. The on-line ranking number ${\chi }_{\mathrm{r}}^{*}(G)$ of G is the corresponding on-line invariant; in that case vertices of G are coming one by one so that a partial ranking has to be chosen by considering only the structure of the subgraph of G induced by the present vertices. It is known that $\lfloor {\log }_{2}n\rfloor +1={\chi }_{\mathrm{r}}(P_n)⩽{\chi }_{\mathrm{r}}^{*}(P_n)⩽2\lfloor {\log }_{2}n\rfloor +1$. In this paper it is proved that ${\chi }_{\mathrm{r}}^{*}(P_n)>1.619{\log }_{2}n-1$.