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Featured researches published by Andrzej Włoch.


Discrete Mathematics | 2008

On the existence and on the number of (k,l)-kernels in the lexicographic product of graphs

Waldemar Szumny; Iwona Włoch; Andrzej Włoch

In [G. Hopkins, W. Staton, Some identities arising from the Fibonacci numbers of certain graphs, Fibonacci Quart. 22 (1984) 225-228.] and [I. Wloch, Generalized Fibonacci polynomial of graphs, Ars Combinatoria 68 (2003) 49-55] the total number of k-independent sets in the generalized lexicographic product of graphs was given. In this paper we study (k,l)-kernels (i.e. k-independent sets being l-dominating, simultaneously) in this product and we generalize some results from [A. Wloch, I. Wloch, The total number of maximal k-independent sets in the generalized lexicographic product of graphs, Ars Combinatoria 75 (2005) 163-170]. We give the necessary and sufficient conditions for the existence of (k,l)-kernels in it. Moreover, we construct formulas which calculate the number of all (k,l)-kernels, k-independent sets and l-dominating sets in the lexicographic product of graphs for all parameters k,l. The result concerning the total number of independent sets generalizes the Fibonacci polynomial of graphs. Also for special graphs we give some recurrence formulas.


Discrete Mathematics | 1997

On ( k,l )-kernels in generalized products

Andrzej Włoch; Iwona Włoch

Abstract We prove some theorems, which describe the parameters k , l of ( k , l )-kernels in the generalized Cartesian product with respect to digraphs and in the generalized lexicographical product with respect to digraphs and graphs.


Applied Mathematics and Computation | 2013

Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers

Andrzej Włoch

In this paper we study some properties of the generalized Fibonacci numbers and the generalized Lucas numbers. These numbers are equal to the total numbers of k-independent sets in special graphs. We give some identities for the generalized Fibonacci numbers and the generalized Lucas numbers, which can be useful also in problems of counting of k-independent sets in graphs.


Discrete Applied Mathematics | 2010

Generalized sequences and k-independent sets in graphs

Iwona Włoch; Andrzej Włoch

In this paper we give a generalization of known sequences and then we give their graph representations. We generalize Fibonacci numbers, Lucas numbers, Pell numbers, Tribonacci numbers and we prove that they are equal to the total number of k-independent sets in special graphs.


Discrete Applied Mathematics | 2009

Fibonacci numbers and Lucas numbers in graphs

Mariusz Startek; Andrzej Włoch; Iwona Włoch

A subset S@?V(G) is independent if no two vertices of S are adjacent in G. In this paper we study the number of independent sets in graphs with two elementary cycles. In particular we determine the smallest number and the largest number of these sets among n-vertex graphs with two elementary cycles. The extremal values of the number of independent sets are described using Fibonacci numbers and Lucas numbers.


Discussiones Mathematicae Graph Theory | 2007

On (k,l)-kernels in D-join of digraphs

Waldemar Szumny; Andrzej Włoch; Iwona Włoch

In [5] the necessary and sufficient conditions for the existence of (k, l)-kernels in a D-join of digraphs were given if the digraph D is without circuits of length less than k. In this paper we generalize these results for an arbitrary digraph D. Moreover, we give the total number of (k, l)-kernels, k-independent sets and l-dominating sets in a D-join of digraphs.


Discrete Applied Mathematics | 2013

On a new type of distance Fibonacci numbers

Iwona Włoch; Urszula Bednarz; Dorota Bród; Andrzej Włoch; Małgorzata Wołowiec-Musiał

In this paper, we define a new kind of Fibonacci numbers generalized in the distance sense. This generalization is related to distance Fibonacci numbers and distance Lucas numbers, introduced quite recently. We also study distinct properties of these numbers for negative integers. Their representations and interpretations in graphs are also studied.


Journal of Applied Mathematics | 2014

On Types of Distance Fibonacci Numbers Generated by Number Decompositions

Anetta Szynal-Liana; Andrzej Włoch; Iwona Włoch

We introduce new types of distance Fibonacci numbers which are closely related with number decompositions. Using special decompositions of the number we give a sequence of identities for them. Moreover, we give matrix generators for distance Fibonacci numbers and their direct formulas.


Discrete Applied Mathematics | 2012

On generalized Fibonacci numbers and k-distance Kp-matchings in graphs

Andrzej Włoch


Ars Combinatoria | 2005

The total number of maximal k-independent sets in the generalized lexicographical product of graphs.

Iwona Włoch; Andrzej Włoch

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Iwona Włoch

Rzeszów University of Technology

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Anetta Szynal-Liana

Rzeszów University of Technology

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Dorota Bród

Rzeszów University of Technology

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Urszula Bednarz

Rzeszów University of Technology

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