A Polynomial Time Algorithm for Graph Isomorphism

Critique on Reiner Czerwinski ”A P olynomia l Time Algori thm for Gra ph Is omorphism” Reiner Czerwinsk i Octob er 18, 2022 Abstract In the pap er ”A P olynomial Time Algorithm for Graph Isomorphism” w e claimed, that th ere is a p olyn omial algorithm to test if tw o graphs are isomorphic. But the algorithm is wrong. It only tests if the adjacency ma- trices of tw o graphs hav e the same eigen v alues. There is a counterexample of tw o non-isomorphic graphs with the same eigenv alues. 1 In tro duction Let A the adjacency matrix of G and A ′ the adjacency matrix of G . G and G ′ are is omorphic if there is a p er mut ation matrix P with A ′ = P ∗ A ∗ P T . The adjacency matrices of iso morphic g raphs have equa l eig env alues. the alg orithm describ ed in [1] o nly tests if the gr aphs hav e the same eigenv al- ues. But unfortunately , there a non-isomor phic graphs with the s a me eigenv alue. In the next section we will show how to c onstruct them. 2 Strongly Regular G raphs Let G b e a Gra ph. G ∈ SR G( n, k , a, c ) if G is a k connected graph with n ver- tices, where adjacent vertices have a common ne ig hbours and non-adjacent has c common neighbo urs. F or further info r mation see [2, chapter 10]. G is strongly regular if there a no n- negative num b ers n, k , a , c with G ∈ SRG( n, k, a, c ). Theorem 1. If G and G ′ ar e in SRG( n, k , a, c ) then G and G ′ have the same eigenvalues. A pro of of this is shown in [2, pag e 219f]. 2.1 Coun terexample There a re non-iso morphic gr aphs with the same eigenv a lue s . E.g. there a re 180 pairwise non-isomor phic gr aphs in SR G(36 , 14 , 4 , 6 ) [3]. 1 References [1] Reiner Czerwinski. A p o lynomial time algo rithm for graph iso morphism, 2008. [2] Chris Go dsil a nd Gordon F Royle. Alge br aic gr aph the ory , volume 207 . Springer Science & Busines s Media, 20 01. [3] Br endan D McKay and E dward Sp ence. Cla ssification of reg ular tw o-g raphs on 36 a nd 38 vertices. Austr alasian Journal of Combinatorics , 24 :293– 300, 2001. 2