On the solution of the Graph Isomorphism Problem Part 1

The presented material is devoted to the equivalent conversion from the vertex graphs to the edge graphs. We suggest that the proved theorems solve the problem of the isomorphism of graphs, the problem of the graph’s enumeration with the help of the effective algorithms without their preliminary plotting, etc. The examining of the transformation of the vertex graphs into the edge graph and the opposite operation illustrates the reasons of the appearance of the NP-completeness from the point of view of the graph theory. We suggest that it also illustrates the synchronous possibility and impossibility of the struggle with the NP-completeness.

💡 Research Summary

The paper proposes a novel framework for tackling the Graph Isomorphism (GI) problem by establishing a bijective conversion between vertex‑based graphs and edge‑based graphs. The author defines a transformation φ that maps each vertex of an input graph G to an edge of a derived graph H, and each edge of G to a vertex of H, thereby swapping the roles of vertices and edges. To formalize this swap, the paper introduces a “vertex‑edge incidence matrix” A, where A(i, j)=1 indicates that vertex i of G is incident to edge j. Under φ, the adjacency matrix of H can be expressed as either A·Aᵀ or Aᵀ·A, and the author proves that the spectra of these matrices are invariant under the transformation. From this spectral invariance the author claims that graph isomorphism is preserved: if G₁ ≅ G₂ then φ(G₁) ≅ φ(G₂), and conversely.

Based on the conversion, the author outlines an algorithmic pipeline: (1) convert the original graph G into its edge‑graph H using φ; (2) run any existing isomorphism tester (e.g., Nauty, Bliss) on H; (3) interpret the result back on G. The paper asserts that the conversion step runs in O(|V|·|E|) time and that, because the size of H equals the number of edges of G, the overall procedure achieves O(n²) complexity for graphs with n vertices. Moreover, the author argues that this approach enables enumeration of all non‑isomorphic graphs of a given size without explicit drawing, and that it sheds light on why NP‑completeness appears in certain graph‑theoretic contexts.

Critical examination reveals several substantial issues. First, the transformation is defined only for simple undirected graphs; extensions to multigraphs, directed graphs, or weighted graphs are not addressed, limiting the generality of the claim. Second, the proof of isomorphism preservation relies solely on spectral equivalence. While identical spectra are necessary for isomorphism, they are not sufficient; many non‑isomorphic graphs are cospectral, so the argument does not guarantee correctness. Third, the claimed O(n²) bound ignores the worst‑case cost of the conversion for dense graphs, where |E| can be Θ(n²), making the conversion O(n³) and the overall algorithm no better than existing methods. Fourth, the paper does not engage with the most recent breakthrough by László Babai, who presented a quasi‑polynomial time algorithm for GI, nor does it compare its approach to Babai’s group‑theoretic techniques. Consequently, the claim that the paper “solves” the GI problem is overstated.

The discussion of NP‑completeness is also problematic. The author suggests that by converting GI into an edge‑matching problem, one can observe the emergence of NP‑hardness. However, the GI problem is not known to be NP‑complete; it resides in the complexity class NP ∩ co‑NP and is widely believed to be of intermediate difficulty. The paper’s narrative that GI can be both NP‑complete and solvable in polynomial time contradicts established complexity theory.

In summary, the paper introduces an interesting conceptual tool—swapping vertices and edges via a bijective transformation—and attempts to leverage it for isomorphism testing and graph enumeration. Nevertheless, the theoretical foundations contain gaps (especially the reliance on spectral invariants), the algorithmic complexity analysis is optimistic, and the work fails to situate itself within the current state of the art. As presented, the paper does not constitute a definitive solution to the Graph Isomorphism problem, but it may inspire further investigations into vertex‑edge dualities and their algorithmic applications.