2026/07/13 by Shuaijun Li, Guangfu Wang
#math.CO
Let \(\mathscrGn\) be the family of simple graphs obtained from \(Kn\) by deleting at most five edges. For a fixed integer \(1≤ k≤ n\), let \(Φk(L(G),x)\) denote the immanantal polynomial of the Laplacian matrix associated with the hook partition \((k,1n-k)\). We prove that, for \(n>7\) and \(n≠ 2k-1\), every graph in \(\mathscrGn\) is determined by \(Φk(L(G),x)\) among all simple graphs. We prove that, for \(n>7\) and \(n≠ 2k-1\), every graph in \(\mathscrGn\) is determined by \(Φk(L(G),x)\) 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.