\((k, \alpha_{n-1})\)-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.