SKREKOVSKI , Riste

{Recently Hansen and Vukicevic proved that the inequality $M_1/n \leq M_2/m$, where $M_1$ and $M_2$ are the first and second Zagreb indices, holds for chemical graphs, and Vukicevic and Graovac proved that this also holds for trees. In both works is given a distinct counterexample for which this inequality is false in general. Here, we present...

A plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially 11-colourable. As a consequence, we derive that every 2-connected plane graph with maximum face-size at most 7...

A vertex coloring of a plane graph is ℓ-facial if every two distinct vertices joined by a facial walk of length at most ℓ receive distinct colors. It has been conjectured that every plane graph has an ℓ-facial coloring with at most 3ℓ+1 colors. We improve the currently best known bound and show that every plane graph has an ℓ-facial coloring...

The Wiener index of a graph $G$, denoted by $W(G)$, is the sum of distances between all pairs of vertices in $G$. In this paper, we consider the relation between the Wiener index of a graph, $G$, and its line graph, $L(G)$. We show that if $G$ is of minimum degree at least two, then $W(G) ≤ W(L(G))$. We prove that for every non-negative integer...