A polynomial‐time algorithm for simple undirected graph isomorphism

In Section 4.5, “Assertion 4-2 based on Theorem 1, 2, 3, 4: Two undirected and simple graphs G1 G2. The number of nodes for the two is same as n. edges m. There are no isolated in both graphs. vertex edge adjacency matrices generated V1 V2, E1 E2. sums rows/columns produce four sets an array aV1, bV2, aE1, bE1 If only if ∑n k=1 ak = bk (ak V1)2 (bk V2)2, then aV1 bV2 a permutation another set; ∑m E2, E1)2 E2)2, aE1 bE2, set, G2 isomorphic.” should be Theorems 1–4: bE2 , …, eigenvalue maximum linearly independent system left right singular vector equinumerous,2 isomorphic. 4.7 “Theorem 4…, we still need to” replaced by “Mathematical proof 4. “only if” theorem (necessary condition) because group bijective. They always have equivalent arrays “if” (sufficient requires following three main lemmas from fundamental arithmetic2 n 2. That is, given natural A B vice versa. (n 2 case) there any equals to case holds. follows. Suppose . Thus Therefore, either or When positive integer larger than according arithmetic, prime itself can represented product numbers; moreover, this representation unique, up (except for) order factors. Then Repeat previous proof. divides at least one rearrange Because prime, factors 1 remove it sides equation. Continue process until all removed. pi removed, side equality so Similarly, qi 1. equal proved, solution B, which (1). since makes equations valid. word “system” indicates that considered collectively, rather individually. (. (2) not must exist holds (1), but belong k could construct where To make results contraction initial assumption unique false. Thus, proved. 4, has been Our mathematical with proved induction-based method. Questions extended theorem. set established, Mathematical proof: Let statement We give induction N. Base case. easily seen true, true above-mentioned when Inductive step. steps will show holds, also This done Assume hypothesis (for some unspecified value ), 5.1, “(4) check … Figure 23.” “(3) Check so, go next not, (4) (5) Generate row sum matrix Compute m (6) graph (7) (8) compute step (9) Until steps, (10) Implement composition (SVD) corresponding equinumerous.1, (11) equinumerous.2 Regarding proposed isomorphism algorithm article, equinumerous cardinality elements equivalent, equivalent. other words, exists one-to-one correspondence (a bijection) between them, function such every element y exactly x f(x) y. For example, 5, 3 first equal, second 5 equal. equinumerous. spaces, row/column another, spaces Weak equinumerosity. contains equinumerosity array, they weak 4 6, array. theorem.1 Without loss generality, columns rows defined used called vectors article. Definition 4A.P-multiple eigenvalues. An eigenvalues same, these multiple 5B.A maximally set. It as: group, satisfies: (1) independent; space expressed linear combination maximal set—the basis including would dependent. 6C.A system. Under transformation, subset transferred others 0. current format original Property 4A. ([1])Let real symmetric matrix. orthogonal diagonal column eigenvector Proof.By induction, obviously first-dimension square matrices. above proposition dimension λ arranged into (here ). Whereas assumed matrix, Since Finally, necessary prove Set substitute compared each other, Proof.Both similar (the sequence). different Then, invertible. then: Put (3) (2), applies, principle, therefore (Theorem 4B) 5C. ([1])There interchange satisfy satisfies Proof.The obvious. sufficient From 8, sequence (), operation elementary completed.