The q-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width L y and arbitrary length L x has the form Z (G, q, vLx , where v is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet (v = −1) is the chromatic polynomial P (G, q). Using coloring and transfer matrix methods, we give general formulas for C X,G = N X,G,λ j=1 c X,G,j for X = Z, P on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) ), where U n (x) is the Chebyshev polynomial of the second kind, we determine the number of λ Z,G,j 's with coefficient c (d) in Z(G, q, v) for these cyclic strips of width L y to befor 0 ≤ d ≤ L y and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues λ Z,G,j is calculated to be N Z,Ly,λ = 2Ly Ly . Results are also presented for the analogous numbers n P (L y , d) and N P,Ly,λ for P (G, q). We find that n P (L y , 0) = n P (L y − 1, 1) = M Ly−1 (Motzkin number), n Z (L y , 0) = C Ly (the Catalan number), and give an exact expression for N P,Ly,λ . Our results for N Z,Ly,λ and N P,Ly,λ apply for both the cyclic and Möbius strips of both the square and triangular lattices; we also point out the interesting relations N Z,Ly,λ = 2N DA,tri,Ly and N P,Ly,λ = 2N DA,sq,Ly , where N DA,Λ,n denotes the number of directed lattice animals on the lattice Λ. We find the asymptotic growths N Z,Ly,λ ∼ L
