A vertex z in a connected graph G resolves two vertices u and v in G if d G (u, z) ̸ = d G (v, z). A set of vertices R G {u, v} is a set of all resolving vertices of u and v in G. For every two distinct vertices u and v in G, a resolving function f of G is a real function f : V (G) → [0, 1] such that f (R G {u, v}) ≥ 1. The minimum value of f (V (G)) from all resolving functions f of G is called the fractional metric dimension of G. In this paper, we consider a graph which is obtained by the comb product between two connected graphs G and H, denoted by G £o H. For any connected graphs G, we determine the fractional metric dimension of G £o H where H is a connected graph having a stem or a major vertex.

The paper deals with the problem of labeling the vertices and edges of a plane graph in such a way that the labels of the vertices and edges surrounding that face add up to a weight of that face. A labeling of a plane graph is called d-antimagic if for every positive integer s, the s-sided face weights form an arithmetic progression with a difference d. Such a labeling is called super if the smallest possible labels appear on the vertices. In the paper we examine the existence of such labelings for several families of plane graphs.

In the paper the numerical model based on the exponential approximation of commodity stock exchanges was derived. The price prognoses of aluminium on the London Metal Exchange were determined as numerical solution of the Cauchy initial problem for the 1 st order ordinary differential equation. To make the numerical model more accurate the idea of the modification of the initial condition value by the stock exchange was realized. By having analyzed the forecasting success of the chosen initial condition drift types, the initial condition drift providing the most accurate prognoses for the commodity price movements was determined. The suggested modification of the original model made the commodity price prognoses more accurate.

In mathematical models, for forecasting prices on commodity exchanges different mathematical methods are used. In the paper the numerical model based on the exponential approximation of commodity stock exchanges was derived. The price prognoses of aluminium on the London Metal Exchange were determined as numerical solution of the Cauchy initial problem for the 1st order ordinary differential equation. To make the numerical model more accurate the idea of the modification of the initial condition value by the stock exchange was realized. The derived numerical model was observed to determine the influence of the decreased size of the limiting value error causing the modification of the initial condition value by the chosen stock exchange on the accuracy of the obtained prognoses. The advantage of the chosen sizes of the limiting value error 7 % and 8 % within different movements of aluminium prices was studied.