Combinatorics and the Tarski paradox

‎The Tarski number of a discrete group is the smallest number of parts of its possible paradoxical decompositions‎. ‎This number is always equal or grater than 4‎. ‎The only groups with exactly determined Tarski numbers are groups with Tarski numbers 4‎, ‎5 and 6‎. ‎In this talk the attempt is to obtain the paradoxical decomposition of a group by using a matrix combinatorial property‎. ‎This way‎, ‎one can find the Tarski number of a given group by counting the number of special paths of a graph associated to that group