Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index

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In this paper, the extremal problems on the phenylene chains with respect to some graph invariants are studied. All the graphs minimizing (resp. maximizing) the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index among all the phenylene chains each of which contains n four-membered rings are identified.