For a graph $G$, by $\chi_2(G)$ we denote the minimum integer $k$, such that there is $k$-coloring of vertices $G$ in which at distance most 2 receive distinct colors. Equivalently, chromatic number square $G$. In 1977 Wegner conjectured if planar and has maximum degree $\Delta$, then $\chi_2(G) \leq 7$ $\Delta 3$, \Delta+5$ $4 \Delta 7$, $\lfloor 3\Delta/2 \rfloor +1$ \geq 8$. Despite extensiv...

Generalized Tur\'an problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing number cliques fixed order graph with vertices and bounded maximum degree, was recently completely resolved by Chase. Kirsch Radcliffe raised natural variant this problem where edges is instead vertices. In paper, we determine resolving conjectu...

An L(2, 1)-labeling of a graph G is an assignment of a nonnegative integer to each vertex of G such that adjacent vertices receive integers that differ by at least two and vertices at distance two receive distinct integers. The span of such a labeling is the difference between the largest and smallest integers used. The λ-number of G, denoted by λ(G), is the minimum span over all L(2, 1)-labeli...

The Wiener index of a graph is the sum of all pairwise distances of vertices of the graph. In this paper we characterize the trees which minimize the Wiener index among all trees of given order and maximum degree and the trees which maximize the Wiener index among all trees of given order that have only vertices of two di erent degrees.