Kemeny's constant of a cilinder octagonal chain

Abstract

If A(G) is the adjacency matrix of a graph G with n vertices and D−1/2(G) is the diagonal matrix of reciprocals of square roots of vertex degrees, then the Kemeny’s constant of G is K(G) = n i=2 1 1−λi , where λ2,λ3,...,λn are all but the largest eigenvalue of D−1/2(G)A(G)D−1/2(G). We use an approach based on determinants of particular tridiagonal matrices admitting certain periodicity to provide a closed formula for the Kemeny’s constant of a cylinder octagonal chain graph, where a graph in question is obtained from a linear octagonal chain graph by identifying the lateral edges. In this way we present the correct result of [S. Zaman, A. Ullah, Kemeny’s constant and global mean first passage time of random walks on octagonal cell network, Math. Meth. Appl. Sci., 46 (2023), 9177–9186] that for the graphs in question calculated the multiple of Kirchhoff index instead.