Abstract: The spectra of matching polynomials which are useful in the computations of resonance energy and grand canonical partition functions of molecular’s. It also present other properties for certain classes of graphs and lattices. In [1] Balasubramanian calculates several matching polynomials and matching roots of several molecular graphs. He found that the matching polynomial of C₆, C₁₀, C₁₄, C₁₈ and C₂₂ are divided by x²-2. In this note,we prove that x²-2 divides MC_4k+2（x）, k = 1, 2,..., n and obtain some other properties of matching polynomials of paths and cycles. 

Key words: Graph algorithms, Matching polynomial, Matching roots, Combinatorial problems