r is called commuting regular ring (resp. semigroup) if for each x,y $in$ r thereexists a $in$ r such that xy = yxayx. in this paper, we introduce the concept of commuting$pi$-regular rings (resp. semigroups) and study various properties of them.

In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.

G. Bonvicini, D. Cinabro, M. Dubrovin, A. Lincoln, D. M. Asner, K. W. Edwards, P. Naik, R. A. Briere, T. Ferguson, G. Tatishvili, H. Vogel, M. E. Watkins, J. L. Rosner, N. E. Adam, J. P. Alexander, D. G. Cassel, J. E. Duboscq, R. Ehrlich, L. Fields, R. S. Galik, L. Gibbons, R. Gray, S. W. Gray, D. L. Hartill, B. K. Heltsley, D. Hertz, C. D. Jones, J. Kandaswamy, D. L. Kreinick, V. E. Kuznetsov,...