Almost complete graphs determined by Laplacian hook immanantal polynomials

Shuaijun Li; Guangfu Wang

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arXiv · 2607.19411

Almost complete graphs determined by Laplacian hook immanantal polynomials

Shuaijun Li·Guangfu Wang

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Abstract

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.

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Shuaijun Li, Guangfu Wang. 2026-07-13. Almost complete graphs determined by Laplacian hook immanantal polynomials. https://arxiv.org/abs/2607.19411

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