$(k,{\alpha }_{n-1})$-Fibonacci Numbers and $P_k$-Matchings in Multigraphs

Abstract

In this paper we generalize the Fibonacci numbers and the Lucas numbers with respect to $n$, respectively $n+1$ parameters. Using these definitions we count special subfamilies of the set of $n$ integers. Next we give the graph interpretations of these numbers with respect to the number of $P_k$,-matchings in special graphs and we apply it for proving some identity and also for counting other subfamilies of the set of n integers.