Extremal polygonal chains with respect to the Kemeny’s constant

Let G be a connected graph. The Kemeny’s constant of G , denoted by κ ( G ) , is the expected time to travel from a fixed starting vertex to a random destination vertex chosen according to the stationary distribution. This constant is an important graph invariant arising from Markov chain theory that has attracted considerable interest in graph theory. In this paper, by combining electrical network approaches with comparison results on Kemeny’s constant of S , T -isomers, we characterize the maximum and minimum polygonal chains with respect to the Kemeny’s constant. This extends the results of Yang and Li (2026) on hexagonal chains and Liao et al. (2026) on pentagonal chains to general polygonal chains with an arbitrary prescribed ordered face-length sequence.