In this paper, we show the values of crossing numbers for join products of graph G on five vertices with the discrete graph D n and the path P n on n vertices. The proof is done with the help of software. The software generates all cyclic permutations for a given number n. For cyclic permutations, P 1 -P m will create a graph in which to calculate the distances between all vertices of the graph. These distances are used in proof of crossing numbers of presented graphs.

We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length are rank-one, as it was shown in [6] (Shue, Anderson, Dey 1998). We establish a bound on the transient after which any product of matrices whose length exceeds that bound becomes rankone.

C++ Implementation of the Cyclic-Order Graph Algorithm is presented together with the source code. This program helps in determination of crossing numbers for join products of graphs with paths of given numbers of vertices necessary for the proofs in the Graph Theory. Comparison with MATLAB (GNU Octave) is provided together with outlooks for the future development.

Airports are one of the segments within air transport and their serviceability ensures air transport efficiency. However, airports are among the largest contributors to the negative environmental impact of this kind of transport. Individual activities at airports leave a negative environmental footprint. By optimizing processes, it is possible to reduce the negative environmental impacts of these subjects. Airport winter maintenance is one of the important activities accompanying airports in geographical areas where winter season is expected due to the climate zone and weather conditions. Effective winter maintenance ensures optimal airport operation and has a significant impact on safety. However, the winter maintenance of the airport causes an excessive burden on the environment around the airports, in particular by applying de-icing agents to aircraft and movement areas, or by emissions and noise caused by equipment for snow and ice removal. The aim of the present article is to optimize the winter maintenance of the airport, namely the deployment of winter maintenance equipment with respect to meteorological conditions while maintaining the priorities for winter maintenance between individual airport areas. The aim of optimization is to achieve the saving of maintenance time and reduce the negative environmental impact of winter maintenance by effectively deploying equipment and devices.

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn, where the graph G consists of one 5-cycle and of one isolated vertex, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn and with the path Pn on n vertices for other two graphs.