Abstract
The generalized Eichinger matrices are defined asE =Σ j n 1(δ j S T S)−1, whereδ j M denotes the matrixM withj th row and column deleted.S is the incidence matrix andM T is the transposed matrix. The conjectureS T SE = S K TS K , where SK is the incidence matrix of the complete graph, is proven for trees, simple cycles and complete graphs. The consequence of the conjecture isS G T S G (E G -I) = S G TS G , whereG is the complementary graph ofG. It leads to graphs with imaginary arcs as the complements of graphs with multiple arcs.
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Kunz, M. A möbius inversion of the ulam subgraphs conjecture. J Math Chem 9, 297–305 (1992). https://doi.org/10.1007/BF01166094
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DOI: https://doi.org/10.1007/BF01166094