linear algebra - Prove that the minimum of row sums of a nonnegative symmetric matrix is preserved -

let nxn adjacency (nonnegative, irreducible , symmetric) matrix zeros on diagonal. denote i-th row sum of a^k r^(k)_i, k>=1. want prove if r_j=min_i{r_i} j, r(k)_j=min_i{r^(k)_i}, in other words minimum row sum preserved. know true based on intuition. appreciated if can give me idea formal proof.

i think doesn't stand. can seen in terms of graphs. matrix adjacency matrix of un-directed graph. hold element (i,j) of matrix a^k represent number of paths between nodes , j of length k, , sum of i'th row means number of paths between , other node of length k.

assumption based on number of neighbours local measure. simple example there more nodes same min r_i, , different r^2_i.

    b----c    /     |\   /      | \  /       |  \        d   d  \       |  /   \      | /    \     |/     b----c  r_a = r_b = r_d = 2 = min r_c = 3 r^2_a = 4 = 2*a-b-c, 2*a-b-a r^2_b = 5 = b-a-b, 2*b-c-d, b-a-b, b-c-b r^2_c = 6 r^2_d = 6