Graph-Theoretical Properties of Markoff Numbers. Topological indices of Symmetrical BroComb Graphs and Perfect Matching Numbers of Symmetrical StepOmino Graphs.

概要

論文の詳細を見る
Positive integer solutions of x2+y2+z2=3xyz are called Markoff numbers. A number of novel features of Markoff numbers were found from the graph-theoretical standpoint. Namely, for a given Markoff number there exist a pair of graphs, caterpillar and linearly growing polyomino, whose topological index and perfect matching number are, respectively, equal to that number. Efficient stepwise algorithms and recursion formulas are found for enumerating these two characteristic quantities of these special graphs, which have either mirror or rotational symmetry. It is conjectured that any Markoff number can be expressed as the sum of squares of a pair of co-prime integers, From these new findings dramatic advance and application will be expected in the mathematics of Markoff numbers.