The Kirchhoff index of folded hypercubes and some variant networks (Q1718236)

Summary: The \(n\)-dimensional folded hypercube \(F Q_n\) is an important and attractive variant of the \(n\)-dimensional hypercube \(Q_n\), which is obtained from \(Q_n\) by adding an edge between any pair of vertices complementary edges. \(F Q_n\) is superior to \(Q_n\) in many measurements, such as the diameter of \(F Q_n\)which is \(\lceil n / 2 \rceil\), about a half of the diameter in terms of \(Q_n\). The Kirchhoff index \(\text{K} \text{f}(G)\) is the sum of resistance distances between all pairs of vertices in \(G\). In this paper, we established the relationships between the folded hypercubes networks \(F Q_n\) and its three variant networks \(l(F Q_n)\), \(s(F Q_n)\), and \(t(F Q_n)\) on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes of \(F Q_n\), \(l(F Q_n)\), \(s(F Q_n)\), and \(t(F Q_n)\) were proposed, respectively.