If a is adjacency matrix of graph G then show that A is irreducible if and only if G is connected. ((A graph is connected if there is a path between every pair of vertices).

Consider the relation on A = {a, b, c, d, e} given by R = {(a,b), (b,c),(b,d), (b,e),(d,e)}

Let G be a connected graph of order n and size n. Prove that G contains a single cycle.

Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for each vertex v of G. (a) Prove that G must be a connected graph. (b) Prove that every vertex of G has degree at least 2.