such that clique C(b) corresponding to b contains exactly as many multicliqual points as the number of cliques which include b. (iv) No two cliques intersect in more than two points. (v) The number of pairs of cliques that meet in two points is one less than the number of cliques. Algorithm for finding the tree square root of a graph G. Step 1: Find all the cliques of G. Step 2: Let the cliques of G be C1,...,Cn, and consider a collection of multicliqual points b1,...,bn corresponding to these cliques in accordance with condition iii. The elements of this collection are the nonendpoints of T. Find all of the pairwise intersections of the n cliques and form the graph S by joining the points bi and bj by a line if and only if the corresponding cliques Ci and Cj intersect in two points. S is then a tree by condition v. Step 3: For each clique Ci of G, let ni be the number of unicliqual points. To the tree S obtained in step 2, attach ni endpoints to bi, obtaining the tree T which we sought. Once we have the tree in question we can create an adjacency matrix for the tree T and check that it is indeed the tree which we sought. Squaring the adjacency matrix of T should yield an adjacency matrix for a graph which is isomorphic to the graph G which we started with. Probably the simplest way to observe this theorem in action is to observe the case which Harary mentions in The Square of a Tree. Specifically the example in question describes the tree corresponding the graph of K5 "Consider the tree consisting of one point joined with all the others. When the tree is squared, the result is the complete graph. We wish to illustrate... T2K5" Upon squaring the adjacency matrix of the previously mentioned tree, we can observe that the theorem does in fact hold true. We can also observe that this pattern of setting up a tree where "one point joined with all the others" will always indeed yield the correct tree for all complete graphs. Bibliography 1965: (with Robert Z. Norman and Dorwin Cartwright), Structural Models: An Introduction to the Theory of Directed Graphs, New York: Wiley 1967: (editor) Graph Theory and Theoretical Physics, Academic Press 1969: Graph Theory, Addison–Wesley 1971: (editor with Herbert Wilf) Mathematical