On Numerical Isomerism of Graphs - Bier, Thomas; How Tan, Tu (Malaysian Journal of Science, 2003)
In this paper we give the construction of non regular graphs on v Ã¢â€°Â¥ 10 vertices, which are not isomorphic, but K-isomeric, that is for graphs which have the same characteristic numbers K(Ã‚Âµ; s) for all Ã‚Âµ and for all values of s = 1, 2, ..., v - 1. Here the characteristic numbers K (Ã‚Âµ; s) for Ã‚Âµ = (Ã‚ÂµÃ¢â€šâ‚¬, Ã‚ÂµÃ¢â€šÂ, Ã‚ÂµÃ¢â€šâ€š) are defined to be the number of s -subsets of the vertices of the graph which induce a sub graph with precisely Ã‚ÂµÃ¢â€šâ€š edges, such that there are Ã‚ÂµÃ¢â€šÂ edges intersecting the sub graph non trivially, and consequently there are Ã‚ÂµÃ¢â€šâ‚¬ = | E (G) |- Ã‚ÂµÃ¢â€šÂ - Ã‚ÂµÃ¢â€šâ€š edges disjoint from the sub graph.