A cell-centered finite volume scheme for the 2D evolutionary diffusion equation on arbitrary polygonal meshes is constructed. We apply the backward Euler scheme to discrete the time derivative term, and employ the vertex unknowns as auxiliary ones to discrete the diffusion operator, by solving an underdetermined linear system of equations, vertex unknowns can be expressed by a linear combination of the central unknowns, which finally results in a cell-centered scheme. The proposed scheme maintains the local conservation and the linearity preserving properties. Considering the continuous and discontinuous diffusion coefficients respectively, several numerical experiments on different kinds of polygonal meshes show that second-order convergence rate can be obtained. Its numerical performance is significantly better than the nine point scheme with arithmetic average weighting and inverse distance weighting, and is similar to the weighting method of bilinear interpolation, it overcomes the disadvantage that bilinear interpolation is not suitable for triangular meshes. Besides, the numerical results also implies that proposed scheme can still achieve second-order convergence for solving nonlinear diffusion equations.