Almost complete graphs determined by Laplacian hook immanantal polynomials

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 \(\Phi_k(L(G),x)\) 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 \(\Phi_k(L(G),x)\) among all simple graphs.

We prove that, for \(n>7\) and \(n\neq 2k-1\), every graph in \(\mathscr{G}_n\) is determined by \(\Phi_k(L(G),x)\) 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.