An independent transversal of a graph G $G$ with vertex partition P ${\mathscr{P}}$ is an set intersecting each block in single vertex. Wanless and Wood proved that if has size at least t $t$ the average degree vertices most ∕ 4 $t\unicode{x02215}4$ , then exists. We present construction showing this result optimal: for any ε > 0 $\varepsilon \gt 0$ sufficiently large there forest into parts su...

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance $G$ keeping largest nonzero entries in each row and column leaving zeros remaining ones. $\mathcal{E}$-eigenvalues are eigenvalues $\mathcal{E}(G)$. modulus an eigenvalue $\mathcal{E}$-spectral radius $G$. $\mathcal{E}$-energy sum absolute values all In this article, we study some extrem...

Expanding parameters of graphs (magnification constant, edge and vertex cutset expansion) are related by very simple inequalities to forwarding parameters (edge and vertex forwarding indices). This shows that certain graphs have eccentricity close to the diameter. Connections between the forwarding indices and algebraic parameters like the smallest eigenvalue of the Laplacian or the genus of th...