In this paper, a class of matrices, namely, Drazin-dagger matrices, in which the Drazin inverse andthe Moore-Penrose inverse commute, is introduced. Also, some properties of the generalized inverses of these matrices, are investigated. Moreover, some results about the Moore-Penrose inverse, the Drazin inverse and the numerical range of some reciprocal matrices are obtained. In particular, the relations between reciprocal matrices, Drazin-Dagger matrices and star order are established. Also, some properties of the generalized inverses of the conjugate EP matrices are studied. To illustrate the results, some numerical examples are also given.

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In this paper, we study perfect 2-coloring of the six-regular graphs up to order 10. We first obtain all possible color parameter matrices and then examine them according to the limitation of our theorem.

The relations between the spectrum of the matrix Q+ Rand the spectra of matrices (g+δ) Q+(a+b) R-QR-RQ, QR-RQ, aRb-QRQ, aRQR-(QR) 2, and bR-QR have been given assuming that the matrix Q+R is diagonalizable, where Q, R are {g,δ}-quadratic matrix and f; g-quadratic matrix, respectively, of order n.

The perfect m-coloring with matrix A = [aij ]i, j2f1,
,
,
, ,mg of a graph G = (V, E) with f1,
,
,
, ,mg color is a vertex coloring of G with m-color so that the number of vertices in color j adjacent to a , xed vertex in color i is aij, independent of the choice of vertex in color i. The matrix A = [aij ]i, j2f1,
,
,
, ,mg is called the parameter matrix. We study the perfect 4-colorings of the 3-regular graphs of order at most 8, that is, we determine a list of all color parameter matrices corresponding to perfect 4-colorings of 3-regular graphs of orders 4, 6, and 8.