Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce novel notion graphs. Using this, solve problem determining \emph{maximal safety distance} two players can keep at all times while traversing a graph. Moreover, their moves must be made with respect to certain move rules. For this purpose, different variants given connected A...

In recent years, the eigenvalues of distance matrix a graph have attracted lot attention mathematicians, since there is close connection between its spectrum and structural properties graph. Thus, quite recently an interesting result was obtained, relating Hamiltonicity to spectral radius graph, on basis which more general conjecture about formulated. We confirm this put forward for k -connecte...

As highly discriminant distance-based topological indices, the Balaban index and the sum-Balaban index of a graph $G$ are defined as $J(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)D_{G}(v)}}$ and $SJ(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)+D_{G}(v)}}$, respectively, where $D_{G}(u)=sumlimits_{vin V}d(u,v)$ is the distance sum of vertex $u$ in $G$, $m$ is the n...