Testing graph isomorphism

We deal with the question of how many queries are required to distinguish between the case that two graphs <i>G</i> and <i>H</i> on <i>n</i> vertices are isomorphic, and the case that they are ε-far, that is they differ in more than ε(^<i>n</i><inf>2</inf>) pairs for all possible bijections of their vertices. Querying is defined as probing the adjacency matrix of any one of the two graphs, i.e. asking if a pair of vertices forms an edge of the graph or not.We investigate both one-sided error and two-sided error testers under two possible settings: The first setting is where both graphs need to be queried; and the second setting is where one of the graphs is known to the algorithm in advance.We prove that the query complexity of the one-sided error testing problem is Θ(<i>n</i>^3/2) if both graphs need to be queried, and that it is Θ(<i>n</i>) if one of the graphs is known in advance (where the Θ notation hides polylogarithmic factors in the upper bounds). For the two-sided error testers we prove that the query complexity is Θ(√<i>n</i> when one of the graphs is known in advance, and we show that the query complexity lies between Ω(<i>n</i>) and <i>Õ</i>(<i>n</i>^5/4) if both <i>G</i> and <i>H</i> need to be queried. All of our algorithms are additionally non-adaptive, while all of our lower bounds apply for adaptive testers as well as non-adaptive ones.