AI 中文总结
研究图的边复杂度，刻画其固定标记下等式情形，通过仿射对合构造等式情形，建立图积的傅里叶比率估计，用傅里叶比率恢复获熵上界并补充下界，还给出特定条件下随机图边复杂度的阶。
AI 中文摘要
古普塔和约塞维奇引入图的边复杂度，即其邻接矩阵在所有顶点标记下的最小傅里叶比率，并以图能量除以边数平方根为下界。我们刻画了固定标记下的等式情形：邻接矩阵的傅里叶变换每行每列最多有一个非零项。这意味着正则性、极值邻接矩阵每个正偶数幂的循环性以及连通分量的奇偶性限制，还给出了某些拉普拉斯谱投影器的等式结果。我们从循环群上的仿射对合构造等式情形。辛格差集为每个素数幂\(q\)给出一个达到等式的\((q + 1)\)正则图，它不是阿贝尔凯莱图。我们还建立了弱、笛卡尔和强图积的傅里叶比率估计，包括互素阶弱积下等式的保持。我们用傅里叶比率恢复作为编码定理获得低复杂度邻接矩阵的熵上界，并用扰动完全图得到的下界补充。最后，一个集中论证表明，如果\(Np_N/\log N\to\infty\)且\(\limsup_{N\to\infty}p_N<1\)，那么\(\operatorname{FR}_{\min}(G(N,p_N))\)的阶为\(N\)，概率趋于\(1\)。
英文摘要
Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.
Comments32 pages
