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Keywords = the (n-1)-th immanant

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24 pages, 393 KB  
Article
The (n-1)-th Laplacian Immanantal Polynomials of Graphs
by Wenwei Zhang, Tingzeng Wu and Xianyue Li
Axioms 2025, 14(9), 716; https://doi.org/10.3390/axioms14090716 - 22 Sep 2025
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Abstract
Let χn1(σ) denote the irreducible character of the symmetric group Sn corresponding to the partition (n1,1). For an n×n matrix [...] Read more.
Let χn1(σ) denote the irreducible character of the symmetric group Sn corresponding to the partition (n1,1). For an n×n matrix M=(mi,j), we denote its (n1)-th immanant by dn1(M). Let G be a simple connected graph and let L(G) and Q(G) denote the Laplacian matrix and the signless Laplacian matrix of G, respectively. The (n1)-th Laplacian (respectively, signless Laplacian) immanantal polynomial of G is defined as dn1(xIL(G)) (respectively, dn1(xIQ(G))). In this paper, we partially resolve Chan’s open problem by establishing that the broom graph minimizes dn1(L(T)) among all trees with given diameter. Furthermore, we give combinatorial expressions for the first five coefficients of the (n1)-th Laplacian immanantal polynomial dn1(xIL(G)). We also investigate the characterizing properties of this polynomial and present several graphs that are uniquely determined by it. Additionally, for the (n1)-th signless Laplacian immanantal polynomial dn1(xIQ(G)), we show that the multiplicity of root 1 is bounded below by the star degree of G. Full article
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