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1

On Fibonacci numbers in edge coloured trees

Bednarz U., Bród D., Szynal-Liana A., Włoch I., Wołowiec-Musiał M.

Opuscula Mathematica

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2017

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Vol. 37, no. 4

479--490

EN

In this paper we show the applications of the Fibonacci numbers in edge coloured trees. We determine the second smallest number of all (A, 2B)-edge colourings in trees. We characterize the minimum tree achieving this second smallest value.

2

On two-parameters generalization of Fibonacci Numbers

Bród D.

Mathematica Applicanda

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2017

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Vol. 45, No. 1

81--92

EN

In this paper we introduce a new two-parameters generalization of Fibonacci numbers – distance s-Fibonacci numbers Fs (k, n). We generalize the known distance Fibonacci numbers by adding an additional integer parameter s. We give combinatorial and graph interpretations of these numbers. Moreover, we present some properties of distance s-Fibonacci numbers, which generalize known properties of classical Fibonacci and Padovan numbers.

PL

W artykule wprowadzono dwuparametrowe uogólnienie klasycznych liczb Fibonacciego - odległościowe s-liczby Fibonacciego. Przedstawiono kombinatoryczne i grafowe interpretacje tych liczb. Pokazane zostały także pewne ich własności, które uogólniają znane własności liczb Fibonacciego i liczb Padovana. Wyznaczona została także funkcja tworząca rozważanego ciągu.

3

On the existence of (k,k-1) - kernels in directed graphs

Bród D., Włoch A., Włoch I.

Journal of Mathematics and Applications

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2006

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Vol. 28

7-12

EN

We calI a subset J of vertices of a digraph D as a (k, k-1) - kerneI of D, for a fixed k ≥ 2, if all distanees between vertices from J are at Ieast k and the distance from each vertex not belonging to J to the set J is at most k - 1. Some theorems concerning the existence of (k, k - 1) - kernels are proved. The resuIts generalize the well - known Riehardson theorem [9], which says: A digraph without odd circuits has a kernel.

4

The number of all stable sets in r-ary tree

Bród D., Włoch A.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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2000

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z. 24 [181]

21-26

EN

The total number of all stable sets of graph Pn is represented by Fibonacci numbers Fn, see [2]. In this paper we calculate the number of all stable sets in special kinds of trees. This number is given by recurence relations and presented results generalized theorems from[2] and [5].