Resistance distance and Kirchhoff index in dihedral Cayley graphs

作者：Huang, Jing^*

来源：Discrete Applied Mathematics, 2022, 307: 125-134.

DOI：10.1016/j.dam.2021.10.009

• Dihedral Cayley graph
• Eigenvalue
• Eigenvector
• Resistance distance
• Kirchhoff index

Let D-2n = < a, b vertical bar a(n) = b(2) = 1, bab = a(-1)>. = {1, a,..., a(n-1), b, ba,..., ba(n-1)} be the dihedral group of order 2n and let S be a symmetric subset of D-2n. The dihedral Cayley graph Cay(D-2n, S) is the graph with vertex set V-Cay( D2n,V-S) = D-2n and edge set E-Cay(D2n,E-S) = {{g, h} vertical bar gh(-1) is an element of S, g, h is an element of D-2n}. In this paper, the complete information for the eigenvalues and their corresponding orthonormal eigenvectors of Cay(D-2n, S) is provided. The closed-form formulae of the Kirchhoff index and the resistance distances between any two vertices of Cay(D-2n, S) are then derived.