The Kirchhoff index of some combinatorial networks (Q1723295)

Summary: The Kirchhoff index Kf(\(G\)) is the sum of the effective resistance distances between all pairs of vertices in \(G\). The hypercube \(Q_n\) and the folded hypercube \(F Q_n\) are well known networks due to their perfect properties. The graph \(G^{\ast}\), constructed from \(G\), is the line graph of the subdivision graph \(S(G)\). In this paper, explicit formulae expressing the Kirchhoff index of \((Q_n)^{\ast}\) and \((F Q_n)^{\ast}\) are found by deducing the characteristic polynomial of the Laplacian matrix of \(G^{\ast}\) in terms of that of \(G\).