由拉普拉斯钩子不变多项式确定的几乎完全图
Almost complete graphs determined by Laplacian hook immanantal polynomials
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中文总结 AI 辅助
研究\(n>7\)且\(n\neq 2k - 1\)时，\(\mathscr{G}_n\)中简单图由拉普拉斯钩子不变多项式\(\Phi_k(L(G),x)\)确定的问题，通过恢复相关参数并基于补图分类比较系数来证明，\(n = 2k - 1\)情况未解决。
中文摘要 AI 辅助
设\(\mathscr{G}_n\)是通过从\(K_n\)中删除至多五条边得到的简单图族。对于固定整数\(1\leq k\leq n\)，令\(\Phi_k(L(G),x)\)表示与钩子划分\((k,1^{n - k})\)相关的拉普拉斯矩阵的不变多项式。我们证明，对于\(n>7\)且\(n\neq 2k - 1\)，\(\mathscr{G}_n\)中的每个图在所有简单图中都由\(\Phi_k(L(G),x)\)确定。证明从首项系数恢复阶数、大小和度平方和，然后基于至多五条边的补图的有限分类，通过明确的第三和第四系数比较来区分其余候选图。\(n = 2k - 1\)的情况未解决，因为这些比较中使用的二项式差为零。
英文摘要
Let \(\mathscr{G}_n\) be the family of simple graphs obtained from \(K_n\) by deleting at most five edges. For a fixed integer \(1\leq k\leq n\), let \(Φ_k(L(G),x)\) denote the immanantal polynomial of the Laplacian matrix associated with the hook partition \((k,1^{n-k})\). We prove that, for \(n>7\) and \(n\neq 2k-1\), every graph in \(\mathscr{G}_n\) is determined by \(Φ_k(L(G),x)\) among all simple graphs. We prove that, for \(n>7\) and \(n\neq 2k-1\), every graph in \(\mathscr{G}_n\) is determined by \(Φ_k(L(G),x)\) among all simple graphs. The proof recovers the order, size, and degree-square sum from the first coefficients, and then separates the remaining candidates by explicit third- and fourth-coefficient comparisons based on the finite classification of complements with at most five edges. The case \(n=2k-1\) is left open because the binomial differences used in these comparisons vanish.
