\(k\)-Lucas numbers and associated bipartite graphs
For a positive integer \(k\geq 2\), the \(k\)-Fibonacci sequence \(\{g_n^{(k)}\}\) is defined as: \(g_1^{(k)}=\dots=g_{k-2}^{(k)}=0\), \(g_{k-1}^{(k)}=g_k^{(k)}=1\) and for \(n>k\), \(g_n^{(k)}=g_{n-1}^{(k)}+g_{n-2}^{(k)}+\dots+g_{n-k}^{(k)}\). Moreover, the \(k\)th Lucas number is defined as \(l_n^{(k)}=g_{n-1}^{(k)}+g_{n+k-1}^{(k)}\) for \(n\geq 1\). The author constructs classes of bipartite graphs, whose numbers of \(1\)-factors are equal to \(k\)th Lucas number.
scientific article; zbMATH DE number 1556772