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.

(a)

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.

Given information:

Let G be a graph with n vertices, and let v and w be distinct vertices of G.

Proof:

The adjacency matrix A=[aij] is n×n zero-one matrix withaij={ 1     if there is an edge from  v i  to  v j 0                                      otherwise

A graph s connected if there exists a path between every pair of vertices.

G is a graph with n vertices

v and w are distinct vertices of G.

To proof: 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.

PROOF:

Let us assume that there exists a walk W from v to w and let the length of this walk be m.

If m≤n−1, then W is a walk of length less than or equal to n − 1 and thus then the proof would be complete

(b)

Prove that G is connected if, and only if, every entry of A+A2+...+An−1 is positive.