On a two-parameter generalization of Jacobsthal numbers and its graph interpretation

In this paper we introduce a two-parameter generalization of the classical Jacobsthal numbers ((s,p)-Jacobsthal numbers). We present some properties of the presented sequence, among others Binet’s formula, Cassini’s identity, the generating function. Moreover, we give a graph interpretation of (s,p)-Jacobsthal numbers, related to independence in graphs

On split r-Jacobsthal quaternions

In this paper we introduce a one-parameter generalization of the split Jacobsthal quaternions, namely the split r-Jacobsthal quaternions. We give a generating function, Binet formula for these numbers. Moreover, we obtain some identities, among others Catalan, Cassini identities and convolution identity for the split r-Jacobsthal quaternions

The number of maximum matchings in a tree

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) $m(T)$ of a tree $T$ of given order. While the trees that attain the lower bound are easily characterised, the trees with largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of maximum matchings in a tree of order $n$ is at most $O(1.391664^n)$ (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by G\'orska and Skupie\'n on the number of maximal matchings (maximal with respect to set inclusion).Comment: 38 page

A new hybrid generalization of Fibonacci and Fibonacci-Narayana polynomials

The hybrid numbers are generalization of complex, hyperbolic and dual numbers. The hybrinomials are polynomials which generalize hybrid numbers. In this paper, we introduce and study the distance Fibonacci hybrinomials, i.e. hybrinomials with coefficients being distance Fibonacci polynomials

On a new two-parameter generalization of dual-hyperbolic Jacobsthal numbers

In this paper we introduce two-parameter generalization of dualhyperbolic Jacobsthal numbers: dual-hyperbolic (s,p)-Jacobsthal numbers. We present some properties of them, among others the Binet formula, Catalan, Cassini, d’Ocagne identities. Moreover, we give the generating function, matrix generator and summation formula for these numbers

On two-parameters generalization of Fibonacci numbers

In this paper we introduce  a new two-parameters generalization ofFibonacci numbers - distance s-Fibonacci numbers  F_s(k,n). We generalize 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

On some properties of split Horadam quaternions

In this paper we introduce and study the split Horadam quaternions. We give some identities, among others Binet’s formula, Catalan’s, Cassini’s and d’Ocagne’s identities for these numbers

(2, k)-Distance Fibonacci Polynomials

In this paper we introduce and study (2,k)-distance Fibonacci polynomials which are natural extensions of (2,k)-Fibonacci numbers. We give some properties of these polynomials—among others, a graph interpretation and matrix generators. Moreover, we present some connections of (2,k)-distance Fibonacci polynomials with Pascal’s triangle

O nowym uogólnieniu kokwaternionów Pella

In this paper we introduce and study a generalization of the split Pell quaternions - split r-Pell quaternions. We give some identities, among others Binet's formula, Catalan's, Cassini's and d'Ocagne's identity for these numbers.W artykule wprowadzono pewne jednoparametrowe uogólnienie kokwaternionów Pella - kokwaterniony r-Pella. Przedstawione zostały różne tożsamości dla tych liczb m.in. wzór Bineta, Catalana i Cassiniego

Charakteryzacje drzew ekstremalnych ze względu na dominowanie : praca doktorska /

Praca doktorska. Akademia Górniczo-Hutnicza im. Stanisława Staszica (Kraków), 2006.Bibliogr. k. 66-69.Zbiory dominujące w literaturze, przykłady zastosowań, drzewa ekstremalne ze względu na pewne własności, na liczbę maksymalnych, największych zbiorów niezależnych, liczba zbiorów dominujących grafów, wzory ogólne na wyznaczanie liczby zbiorów dominujących dowolnego grafu, zależności rekurencyjne określające liczbę zbiorów dominujących pewnych podklas drzew, charakteryzacja drzew z najmniejszą, największą liczbą zbiorów dominujących, lasy, drzewa z największą liczbą zbiorów dominujących, o istnieniu efektywnych zbiorów dominujących, liczba efektywnych zbiorów dominujących pewnych podklas drze