A MOBIUS-INVERSION OF THE ULAM SUBGRAPHS CONJECTURE

The generalized Eichinger matrices are defined as E = SIGMA(j = 1)n(delta(j)S(T)S)-1, where delta(j)M denotes the matrix M with j th row and column deleted. S is the incidence matrix and M(T) is the transposed matrix. The conjecture S(T)SE = S(K)TS(K), where S(K) is the incidence matrix of the complete graph, is proven for trees, simple cycles and complete graphs. The consequence of the conjecture is S(G)TS(G)(E(G)-I) = S(GBAR)TS(GBAR), where GBAR is the complementary graph of G. It leads to graphs with imaginary arcs as the complements of graphs with multiple arcs.