Textbook Problem

1. Let G be a graph with n vertices, and let v and w be distinct vertices of G. Prove that if there is a walk from v to w; then there is a walk from v to w that has length less than or equal to n-1.
2. If A=(a_ij) and B=(b_ij) are any m × n matrix whose ijth entry is a_ij+b_ijfor each I = 1,2, …., m and j=1,2,….,n. Let G be a graph with n vertices where n>1, and let A be the adjacency matrix of G. Prove that G is connected if, and only if, every entry of A+A²+…+Aⁿ-1.