Fouzul Atik

A partition of a square matrix A is said to be equitable if all the block of the partitioned matrix have constant row sums and each of the diagonal block is of square order. A quotient matrix Q of a square matrix A corresponding to an equitable partition is a matrix whose entries are the constat row sums of the corresponding blocks of A. A quotient matrix is an useful tool to find some eigenvalues of the matrix A. I will discuss some matrices whose eigenvalues are the eigenvalues of A and which are not the eigenvalues of a quotient matrix. Using this result we find eigenvalue localization theorems for matrices having an equitable partition. Finally, I will discuss some problems related to distance regular graph, Gersgorin disk theorem and distance matrix of graphs.