| {"paper_meta":{"paper_id":"arxiv:0711.2010","title":"0711.2010","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0711.2010v5 [cs.CC] 15 Oct 2022\nCritique on Reiner Czerwinski ”A Polynomial\nTime Algorithm for Graph Isomorphism”\nReiner Czerwinski\nOctober 18, 2022\nAbstract\nIn the paper ”A Polynomial Time Algorithm for Graph Isomorphism”\nwe claimed, that there is a polynomial algorithm to test if two graphs are\nisomorphic. But the algorithm is wrong. It only tests if the adjacency ma-\ntrices of two graphs have the same eigenvalues. There is a counterexample\nof two non-isomorphic graphs with the same eigenvalues.\n1\nIntroduction\nLet A the adjacency matrix of G and A′ the adjacency matrix of G. G and G′\nare isomorphic if there is a permutation matrix P with A′ = P ∗A ∗P T . The\nadjacency matrices of isomorphic graphs have equal eigenvalues.\nthe algorithm described in [1] only tests if the graphs have the same eigenval-\nues. But unfortunately, there a non-isomorphic graphs with the same eigenvalue.\nIn the next section we will show how to construct them.\n2\nStrongly Regular Graphs\nLet G be a Graph. G ∈SRG(n, k, a, c) if G is a k connected graph with n ver-\ntices, where adjacent vertices have a common neighbours and non-adjacent has\nc common neighbours. For further information see [2, chapter 10]. G is strongly\nregular if there a non-negative numbers n, k, a, c with G ∈SRG(n, k, a, c).\nTheorem 1. If G and G′ are in SRG(n, k, a, c) then G and G′ have the same\neigenvalues.\nA proof of this is shown in [2, page 219f].\n2.1\nCounterexample\nThere are non-isomorphic graphs with the same eigenvalues. E.g. there are 180\npairwise non-isomorphic graphs in SRG(36, 14, 4, 6) [3].\n1\n\nReferences\n[1] Reiner Czerwinski. A polynomial time algorithm for graph isomorphism,\n2008.\n[2] Chris Godsil and Gordon F Royle.\nAlgebraic graph theory, volume 207.\nSpringer Science & Business Media, 2001.\n[3] Brendan D McKay and Edward Spence. Classification of regular two-graphs\non 36 and 38 vertices. Australasian Journal of Combinatorics, 24:293–300,\n2001.\n2","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0711.2010v5 [cs.CC] 15 Oct 2022\nCritique on Reiner Czerwinski ”A Polynomial\nTime Algorithm for Graph Isomorphism”\nReiner Czerwinski\nOctober 18, 2022\nAbstract\nIn the paper ”A Polynomial Time Algorithm for Graph Isomorphism”\nwe claimed, that there is a polynomial algorithm to test if two graphs are\nisomorphic. But the algorithm is wrong. It only tests if the adjacency ma-\ntrices of two graphs have the same eigenvalues. There is a counterexample\nof two non-isomorphic graphs with the same eigenvalues.\n1\nIntroduction\nLet A the adjacency matrix of G and A′ the adjacency matrix of G. G and G′\nare isomorphic if there is a permutation matrix P with A′ = P ∗A ∗P T . The\nadjacency matrices of isomorphic graphs have equal eigenvalues.\nthe algorithm described in [1] only tests if the graphs have the same eigenval-\nues. But unfortunately, there a non-isomorphic graphs with the same eigenvalue.\nIn the next section we will show how to construct them.\n2\nStrongly Regular Graphs\nLet G be a Graph. G ∈SRG(n, k, a, c) if G is a k connected graph with n ver-\ntices, where adjacent vertices have a common neighbours and non-adjacent has\nc common neighbours. For further information see [2, chapter 10]. G is strongly\nregular if there a non-negative numbers n, k, a, c with G ∈SRG(n, k, a, c).\nTheorem 1. If G and G′ are in SRG(n, k, a, c) then G and G′ have the same\neigenvalues.\nA proof of this is shown in [2, page 219f].\n2.1\nCounterexample\nThere are non-isomorphic graphs with the same eigenvalues. E.g. there are 180\npairwise non-isomorphic graphs in SRG(36, 14, 4, 6) [3].\n1"},{"paragraph_id":"p2","order":2,"text":"References\n[1] Reiner Czerwinski. A polynomial time algorithm for graph isomorphism,\n2008.\n[2] Chris Godsil and Gordon F Royle.\nAlgebraic graph theory, volume 207.\nSpringer Science & Business Media, 2001.\n[3] Brendan D McKay and Edward Spence. Classification of regular two-graphs\non 36 and 38 vertices. Australasian Journal of Combinatorics, 24:293–300,\n2001.\n2"}],"pages":[{"page":1,"text":"arXiv:0711.2010v5 [cs.CC] 15 Oct 2022\nCritique on Reiner Czerwinski ”A Polynomial\nTime Algorithm for Graph Isomorphism”\nReiner Czerwinski\nOctober 18, 2022\nAbstract\nIn the paper ”A Polynomial Time Algorithm for Graph Isomorphism”\nwe claimed, that there is a polynomial algorithm to test if two graphs are\nisomorphic. But the algorithm is wrong. It only tests if the adjacency ma-\ntrices of two graphs have the same eigenvalues. There is a counterexample\nof two non-isomorphic graphs with the same eigenvalues.\n1\nIntroduction\nLet A the adjacency matrix of G and A′ the adjacency matrix of G. G and G′\nare isomorphic if there is a permutation matrix P with A′ = P ∗A ∗P T . The\nadjacency matrices of isomorphic graphs have equal eigenvalues.\nthe algorithm described in [1] only tests if the graphs have the same eigenval-\nues. But unfortunately, there a non-isomorphic graphs with the same eigenvalue.\nIn the next section we will show how to construct them.\n2\nStrongly Regular Graphs\nLet G be a Graph. G ∈SRG(n, k, a, c) if G is a k connected graph with n ver-\ntices, where adjacent vertices have a common neighbours and non-adjacent has\nc common neighbours. For further information see [2, chapter 10]. G is strongly\nregular if there a non-negative numbers n, k, a, c with G ∈SRG(n, k, a, c).\nTheorem 1. If G and G′ are in SRG(n, k, a, c) then G and G′ have the same\neigenvalues.\nA proof of this is shown in [2, page 219f].\n2.1\nCounterexample\nThere are non-isomorphic graphs with the same eigenvalues. E.g. there are 180\npairwise non-isomorphic graphs in SRG(36, 14, 4, 6) [3].\n1"},{"page":2,"text":"References\n[1] Reiner Czerwinski. A polynomial time algorithm for graph isomorphism,\n2008.\n[2] Chris Godsil and Gordon F Royle.\nAlgebraic graph theory, volume 207.\nSpringer Science & Business Media, 2001.\n[3] Brendan D McKay and Edward Spence. Classification of regular two-graphs\non 36 and 38 vertices. Australasian Journal of Combinatorics, 24:293–300,\n2001.\n2"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"are isomorphic if there is a permutation matrix P with A′ = P ∗A ∗P T . The","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":1941,"parse_confidence":0.2,"equation_parse_rate_proxy":0.05,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":false}} |