MT4514: Graph Theory

Graph Theory 59sothat is(t 2 + 7)(t 2 − 9) = 64kt,t 4 − 2t 2 − 64kt − 63 = 0.Thus t divides 63 (the product of the roots of this equation), so we have the followingpossibilities:t = √ 8n − 7 : 1 3 7 9 21 63n = (t 2 + 7)/8 : 1 2 7 11 56 487d = 2n : 2 4 14 22 112 974v = 1 + 2n 2 : 3 9 99 243 6273 494019Graphs of this form with d = 2, 4, 22 are known to exist.Theorem 3.3. Let G be a connected simple graph with v vertices. Suppose that Gcontains no triangles, but that every pair of non-adjacent vertices have two commonneighbours. Then every vertex has the same degree d = n 2 + 1 for some n ∈ N,where n is not a multiple of 4, and v = (n 4 + 3n 2 + 4)/2.Proof. See tutorial sheet.