A dominating set $D\subseteq V\left(G\right)$ is a weakly connected dominating set in $G$ if the subgraph $G{\left[D\right]}_w=(N_G\left[D\right],E_w)$ weakly induced by $D$ is connected, where $E_w$ is the set of all edges having at least one vertex in $D$. Weakly connected domination number${\gamma }_w\left(G\right)$ of a graph $G$ is the minimum cardinality among all weakly connected dominating sets in $G$. A graph $G$ is said to be weakly connected domination stable or just ${\gamma }_w$-stable if ${\gamma }_w\left(G\right)={\gamma }_w(G+e)$ for every edge $e$ belonging to the complement $\overline{G}$ of $G.$ We provide a constructive characterization of weakly connected domination...

The Wiener index, W, is the sum of distances between all pairs of vertices in a graph G. The quadratic line graph is defined as L(L(G)), where L(G) is the line graph of G. A generalized star S is a tree consisting of Δ ≥ 3 paths with the unique common endvertex. A relation between the Wiener index of S and of its quadratic graph is presented. It is shown that generalized stars having the property W(S) = W(L(L(S)) exist only for 4 ≤ Δ ≤ 6. Infinite families of generalized stars with this property...

The Wiener index of a connected graph is defined as the sum of the distances between all unordered pairs of its vertices. We characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut-vertices.