Kemeny's Constant of a Cylinder Octagonal Chain
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Authors: B. ALSHAMARY, M. ANđELIć AND Z. STANIć
DOI: 10.46793/KgJMat2609.1467A
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) = ∑ i=2n
, 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.
Keywords:
Kemeny’s constant, octagonal chain, tridiagonal matrix.
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