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Introduction/purpose: Chemical structures are conveniently represented by graphs where atoms are nodes (vertices) and chemical bonds are branches (lines) in the graph. A graphical representation of a molecule provides a lot of useful information about the chemical properties of the molecule. It is known that numerous physical and chemical properties of molecules are highly correlated with theoretical invariants of graphs, which we call topological indices. One such theoretical invariant is the dominance number. The aim of this research is to determine the k-dominance number for cyclooctane chains 𝐶𝑂𝐶𝑛 1 , 𝐶𝑂𝐶𝑛 2 , 𝐶𝑂𝐶𝑛 3 and 𝐶𝑂𝐶𝑛 4 , for k ∈ {1,2,3}, n ∈ 𝑁. Methods: The cyclooctane chain is a chain of octagons connected by a single line. The vertices of the octagon are treated as nodes of the graph, and the sides and the line connecting them, as branches in the graph. Using mathematical methods, k-dominance was determined on one octagon, k∈{1,2,3}. Then, by representing the cyclooctane chains 𝐶𝑂𝐶𝑛 1 , 𝐶𝑂𝐶𝑛 2 , 𝐶𝑂𝐶𝑛 3 and 𝐶𝑂𝐶𝑛 4 , in a convenient, isomorphic way, we determined their k-dominance number, k∈{ 1,2,3}. Results: Determining k-dominance, k∈{1,2,3}, for 4 cyclooctane chains 𝐶𝑂𝐶𝑛 1 , 𝐶𝑂𝐶𝑛 2 , 𝐶𝑂𝐶𝑛 3 and 𝐶𝑂𝐶𝑛 4 , we obtained 12 different formulas to calculate their k-dominance number. All formulas are composed of several alternative algebraic expressions, the selection of which is conditioned by the divisibility of the number n by the number 2, 3 or 4, depending on the type of cyclooctane chain and k-dominance to be determined. The results of the research are fully presented in the paper through mathematically proven theorems and graphical representations. Conclusion: The results show that the k-dominance numbers, k∈{1,2,3}, on cyclooctane chains 𝐶𝑂𝐶𝑛 1 , 𝐶𝑂𝐶𝑛 2 , 𝐶𝑂𝐶𝑛 3 and 𝐶𝑂𝐶𝑛 4 , are determined and explicitly expressed by mathematical expressions. They also indicate the possibility of their application in molecular graphs of cyclooctane rings, in computational chemistry, chemical and biological industry.

Introduction/purpose: A graph as a mathematical object occupies a special place in science. Graph theory is increasingly used in many spheres of business and scientific fields. This paper analyzes pentagonal cactus chains, a special type of graphs composed of pentagonal cycles in which two adjacent cycles have only one node in common. The aim of the research is to determine the dominant set and the dominance number on ortho and meta pentagonal cactus chains. Methods: When the corresponding destinations are treated as graph nodes and the connections between them as branches in the graph, the complete structure of the graph is obtained, to which the laws of graph theory are applied. The vertices of the pentagon are treated as nodes of the graph and the sides as branches in the graph. By applying mathematical methods, the dominance was determined on one pentagon, then on two pentagons with a common node, and then on ortho and meta pentagonal cactus chains. Results: The research has shown that the dominance number on the ortho chain 𝑂ℎ of the length h ≥ 2 is equal to the value of the expression ⌈ 3ℎ 2 ⌉ while on the meta chain 𝑀ℎ it is equal to the value of the expression h+1, which was proven in this paper. Conclusion: The results show that the dominant sets and the dominance numbers on ortho and meta pentagonal cactus chains are determined and explicitly expressed by mathematical expressions. They also point to the possibility of their application in the fields of science as well as in the spheres of business in which these structures appear.

The twenty-first century has brought the growing use of information and communication technologies in many spheres of human life. Also in education. In the learning process, students are increasingly focused on the use of modern technologies and software packages. Exploring the quality of maths teaching in primary school, we found students' knowledge of using modern technologies but also the lack of visual-logical approach in solving mathematical problems. In this paper, we show how the establishment of figurative numbers and selected examples that demonstrate the observation of legality, can direct students to a visual-logical approach to solving tasks with sets of numbers. We organized a three-hour exercise in experimental and control group, during which the experimental group was working with figurative numbers and the control group with selected examples. On the pre-test both groups showed a low degree of ability to observe legality among the numbers. On the post-test, both groups achieved progress compared to the pre-test, although the experimental group achieved greater progress. The research has shown that dealing with figurative numbers and selected examples that demonstrate the observation of legality, contributes to the development of pupils' abilities to perceive the laws of numbers and to apply the laws observed to solving problems with sets and arrays of numbers.