In this study, we introduce a new graph family. Then, we calculate eigenvalues of the adjacency and the Laplacian matrix of this graph family. Moreover, we show that the perfect matching number of this graph family equals to special second order recurrence by hafnian method. For some special kinds of this family, we obtain that the perfect matching number of corresponding graphs equals to some famous number sequences such as Fibonacci, Pell and Jacobsthal numbers. Also, we find energies and obtain upper bounds for Laplacian energies of these graphs.