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The following are some important families of graphs that we will use often Let n be a positive integer and V = fx 1;x 2;;x ng The null graph of order n, denoted by N n, is the graph of order n and size 0 The graph N 1 is called the trivial graph The complete graph of order n, denoted by K n, is the graph of order n that has all possible In order to sketch the graph of the quadratic equation, we follow these steps (a) Check if `a > 0` or `a < 0` to decide if it is Ushaped or nshaped (b) The Vertex The xcoordinate of the minimum point (or maximum point) is given by `x=b/(2a)` (which can be shown using completing the square method, which we met earlier)Eigenvectors of the Laplacian of connected graphs u 1 = 1 n;L1 n= 0 u 2 is the the Fiedler vector with multiplicity 1 The eigenvectors form an orthonormal basis u> i u j= ij For any eigenvector u i= (u i(v 1)u i(v n))>;2 i n u> i 1 n= 0 Hence the components of u i;2 i nsatisfy j=1 u i(v j) = 0 Each component is bounded by 1