Ján Plesník

Comenius University in Bratislava, Presburg, Bratislavský, Slovakia

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Publications (9)1.62 Total impact

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    ABSTRACT: It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist(see [20] or [5]). Furthermore, for degree 2, it is shown that for k 3 there are nodigraphs of order `close" to, i.e., one less than, Moore bound [18]. In this paper, weshall consider digraphs of diameter k, degree 3 and number of vertices one less thanMoore bound. We give a necessary condition for the existence of such digraphs and,using this condition, we deduce that such digraphs do not exist for infinitely ...
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    Ján Plesník
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    ABSTRACT: A simple method is proposed to find the orthogonal projection of a given point to the solution set of a system of linear equations. This is also a direct method for solving systems of linear equations. The output of the method is either the projection or inconsistency of the system. Moreover, in the process also linearly dependent equations are recognized. This paper is constrained for giving theoretical foundations, computational complexity and some numerical experiments with dense matrices although the method allows to employ sparsity. The raw method could not compete with best software packages in solving linear equations for general matrices, but it was competitive in finding projections for matrices with small number of rows relative to the number of columns.
    Linear Algebra and Its Applications - LINEAR ALGEBRA APPL. 01/2007; 422(2):455-470.
  • Ján Plesník
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    ABSTRACT: A strong digraph is k-geodetically connected (k-GC) if the removal of at least k vertices is required to increase the distance between at least one pair of vertices or reduce it to a single vertex. Such digraphs can serve as models of distance invulnerable networks (immune to k – 1 or fewer vertex failures) for a system with one-way communications. For every integer n, we determine the minimum size 2-GC digraphs of order n. Further, we find the minimum size of a k-GC digraph of order n if n ≡ 0 (mod k) and give good bounds for all n. Also, several operations on and constructions of k-GC digraphs are presented. We show that the problem of finding a minimum size k-GC spanning subdigraph of a given digraph is NP-hard. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(4), 243–253 2004
    Networks 09/2004; 44(4):243 - 253. · 0.65 Impact Factor
  • Ján Plesník
    Networks. 01/2003; 41:73-82.
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    Edy Tri Baskoro, Mirka Miller, Ján Plesník
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    ABSTRACT: The nonexistence of digraphs with order equal to the Moore bound Mddk = 1 + d+: : : +d k for dd k > 1 has lead to the study of the problem of the existence oalmost' Moore digraphs, namely digraphs with order close to the Moore bound. In 1], it was shown that almost Moore digraphs of order Mddk ; 1, degree d, diameter k (dd k 3) contain either no cycle of length k or exactly one such cycle. In this paper we shall derive some further necessary conditions for the existence of almost Moore digraphs for degree and diameter greater than 1. As a consequence, for diameter 2 and degree d, 2 d 12, we show that there are no almost Moore digraphs of order Mdd2 ; 1 with one vertex in a C2 except the digraphs with every vertex in C2 .
    Ars Comb. 01/2000; 56.
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    ABSTRACT: Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem,whichis to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree # 15 and diameter # 10 have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. # This research started when J. Plesnk and J. Siran were visiting the Department of Computer Science and Software Engineering of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. 1 the electronic journal of combinatorics 5 (1998), #R9 2 This opens up a new possible direction in the search for large vertex-transitive graphs of given degree and di...
    Electr. J. Comb. 01/1998; 5.
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    Edy Tri Baskoro, Mirka Miller, Ján Plesník
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    ABSTRACT: The Moore bound for a diregular digraph of degree d and diameter k is M d;k =1 + d + : : : + dk. It is known that digraphs of order M d;k do not exist for d ? 1and k ? 1 ([24] or [6]). In this paper we study digraphs of degree d, diameter kand order M d;k \Gamma 1, denoted by (d; k)-digraphs. Miller and Fris showed that (2; k)-digraphs do not exist for k 3 [22]. Subsequently, we gave a necessary condition ofthe existence of (3; k)-digraphs, namely, (3; k)-digraphs do not exist if k...
    Graphs and Combinatorics 01/1998; 14:109-119. · 0.35 Impact Factor
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    ABSTRACT: It is known tht Moore digraphs of degree d > 1 and diameter k > 1 do not exist (see [20] or [5]). Furthermore, for degree 2, it is shown tht for l ≥ 3 there are no digraphs of order “close” to, i.e., one less than Moore bound [18]. In this paper, we shall consider digraphs of diameter k, degree 3 and number of vertices one less than Moore bound. We give a necessary condition for the existence of such digraphs and, using this condition, we deduce that such digraphs do not exist for infinitely many values of the diameter. © 1995 John Wiley & Sons, Inc.
    Journal of Graph Theory 10/1995; 20(3):339 - 349. · 0.63 Impact Factor

Publication Stats

44 Citations
1.62 Total Impact Points

Institutions

  • 1995–2007
    • Comenius University in Bratislava
      • • Department of Mathematical Analysis and Numerical Mathematics
      • • Faculty of Mathematics, Physics and Informatics
      • • Department of Computer Science
      Presburg, Bratislavský, Slovakia