Degree Conditions and Degree Bounded Trees

In this paper, we give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and $A\subseteq V(G)$. We denote by ${\sigma }_k(A)$ the minimum value of the degree sum in G of any k pairwise nonadjacent vertices of A, and by $w(G-A)$ the number of components of the subgraph $G-A$ of G induced by $V(G)-A$. Our main results are the following: (i) If ${\sigma }_k(A)⩾|G|-1$, then G contains a tree T with maximum degree ⩽k and $A\subseteq V(T)$. (ii) If ${\sigma }_{k-w(G-A)}(A)⩾|A|-1$, then G contains a spanning tree T with $d_T(x)⩽k$ for any $x\in A$. These are generalizations of the result by S. Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Seminar Univ. Humburg 43 (1975) 263–267] and degree conditions are sharp.     

A sharp lower bound on Steiner Wiener index for trees with given diameter

Let $G$ be a connected graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. For a subset $S$ of $V\left(G\right)$, the Steiner distance$d\left(S\right)$ of $S$ is the minimum size of a connected subgraph whose vertex set contains $S$. For an integer $k$ with $2\le k\le n?1$, the Steiner$k$-Wiener index${\text{SW}}_k\left(G\right)$ is ${\sum }_{S?V\left(G\right),\left|S\right|=k}d\left(S\right)$. In this paper, we introduce some transformations for trees that do not increase their Steiner $k$-Wiener index for $2\le k\le n?1$. Using these transformations, we get a sharp lower bound on Steiner $k$-Wiener index for trees with given diameter, and obtain the corresponding extremal graph as well.     

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.