Graph isomorphism and volumes of convex bodies

We show that a nontrivial graph isomorphism problem of two undirected graphs, and more generally, the permutation similarity of two given n × n matrices, is equivalent to equalities of volumes of the induced three convex bounded polytopes intersected with a given sequence of balls, centered at the origin with radii ti ∈ (0, √ n − 1), where {ti} is an increasing sequence converging to √ n − 1. These polytopes are characterized by n2 inequalities in at most n2 variables. The existence of fpras for computing volumes of convex bodies gives rise to a semifpras of order O∗(n14) at most to find if given two undirected graphs are isomorphic. 2000 Mathematics Subject Classification: 03D15, 05C50, 05C60, 15A48, 15A51, 52B55, 90C05.