Notions of graph similarity provide alternative perspective on the graph isomorphism problem and vice-versa. In this paper, we consider measures of similarity arising from mismatch norms as studied in Gervens and Grohe: the edit distance $\delta_{\mathcal{E}}$, and the metrics arising from $\ell_p$-operator norms, which we denote by $\delta_p$ and $\delta_{|p|}$. We address the following question: can these measures of similarity be used to design polynomial-time approximation algorithms for graph isomorphism? We show that computing an optimal value of $\delta_{\mathcal{E}}$ is \NP-hard on pairs of graphs with the same number of edges. In addition, we show that computing optimal values of $\delta_p$ and $\delta_{|p|}$ is \NP-hard even on pairs of $1$-planar graphs with the same degree sequence and bounded degree. These two results improve on previous known ones, which did not examine the restricted case where the pairs of graphs are required to have the same number of edges. Finally, we study similarity problems on strongly regular graphs and prove some near optimal inequalities with interesting consequences on the computational complexity of graph and group isomorphism.

翻译：图相似性概念为图同构问题提供了替代视角，反之亦然。本文研究源于Gervens和Grohe所探讨的错配范数产生的相似性度量：编辑距离δ_ℰ，以及由ℓ_p-算子范数导出的度量δ_p和δ_{|p|}。我们探讨以下问题：这些相似性度量能否用于设计图同构的多项式时间近似算法？我们证明，在边数相同的图对上计算δ_ℰ的最优值是\NP-难的。此外，即使对于具有相同度序列和有界度的1-平面图对，计算δ_p和δ_{|p|}的最优值也是\NP-难的。这两个结果改进了先前已知结论，此前研究未考虑图对要求边数相同的受限情形。最后，我们研究了强正则图上的相似性问题，并证明了若干近最优不等式，这些不等式对图同构和群同构的计算复杂性具有重要推论。