We introduce two new graph characteristics, the edge [Formula: see text]-irregularity strength and the vertex [Formula: see text]-irregularity strength of a graph. We estimate the bounds of these parameters and determine their exact values for several families of graphs namely, paths, ladders and fans.

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

A simple graphGadmits anH-covering if every edge inE(G)belongs to a subgraph ofGisomorphic toH. The graphGis said to be (a,d)-H-antimagic if there exists a bijection from the vertex setV(G)and the edge setE(G)onto the set of integers1, 2, …,VG+E(G)such that, for all subgraphsH′ofGisomorphic toH, the sum of labels of all vertices and edges belonging toH′constitute the arithmetic progression with the initial termaand the common differenced.Gis said to be a super (a,d)-H-antimagic if the smallest possible labels appear on the vertices. In this paper, we study super tree-antimagic total labelings of disjoint union of graphs.

The development of the transport segment is currently an essential process which affects several other industries. The transport infrastructure and the services provided in this sector influence economic growth, the efforts aimed at increasing competitiveness, as well as prosperity of the society. One of the key problems Slovakia is facing is the long-term growth of differences between individual regions. The present article deals with the evaluation and comparison of selected transport infrastructure indicators in eight regions of Slovakia. The evaluation was carried out by applying basic statistical methods and multiple-criteria statistical methods. Every region was characterised by 20 selected variables describing its uniqueness (e.g. population, area, GDP per capita, road infrastructure etc.). The evaluation of similarities between individual regions in terms of selected variables was carried out by applying the Principal Component Analysis (PCA) and Hierarchical Cluster Analysis. Within the PCA, the original input variables were replaced with three principal components describing as much as 86.68% of the cumulative variance. The average linkage method, as one of the hierarchical methods, was applied to create a dendrogram representing the similarities between the regions of Slovakia. The cophenetic correlation coefficient value of CC=0.936 confirmed the proper selection of the average linkage method. The output of the cluster analysis was that 8 regions of Slovakia were divided into five similar homogenous clusters based on the examined variables. The final analysis indicated that the transport infrastructure and the development thereof significantly affect the differences between individual regions of Slovakia and, as a matter of fact, they belong to the factors creating such differences.

J. Šafarik in Košice. He received the PhD.-degree at the Mathematic-Physics Faculty, University of J.A. Comenius in Bratislava, and he was habilitated at the Faculty of Mechanical Engineering, University of Žilina, in the scientific branch of applied mathematics. His scientific focus is oriented to the modelling of processes in the biomedical engineering, to the research of materials, as well as to the measurement and diagnostic equipment, prognostics and evaluation of medical proceed. He is a co-author of one monograph, one academic textbook and author of several textbooks. He has published more than 60 papers in the scientific journals and in the proceedings of the conferences and he has more than 80 quotations concerning his professional works. He was also incorporated in various grant research projects and industrial projects.