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Almost all trees share a complete set of immanantal polynomials. (English) Zbl 0782.05038
If $$\chi$$ is an irreducible character of the symmetric group $$S_ n$$ and $$A=(a_{ij})$$ is an $$(n,n)$$-matrix, then $$d_ \chi=\sum_{p\in S_ n}\chi(p) \prod^ n_{i=1} a_{ip(i)}$$. If $$G_ 1$$ and $$G_ 2$$ are isomorphic graphs they share a complete set of immanantal polynomials, i.e., $$d_ \chi(xI-A(G_ 1))= d_ \chi(xI-A(G_ 2))$$ for all $$\chi$$. Let $$y$$ and $$z$$ be independent indeterminants over the complex numbers. Define $$L(G)= yD(G)+ zA(G)$$, where $$D(G)$$ is the diagonal matrix of the vertex degrees and $$A(G)$$ is the adjacency matrix of a graph $$G$$. The main result is the following theorem:
Let $$t_ n$$ be the number of nonisomorphic trees on $$n$$ vertices and $$s_ n$$ the number of such trees $$T$$ for which there exists a nonisomorphic tree $$\widetilde T$$ such that the polynomial identities $$d_ \chi(xI- L(T))= d_ \chi(xI- L(\widetilde T))$$, in the three variables $$x$$, $$y$$, and $$z$$, hold, simultaneously, for every irreducible character $$\chi$$ of $$s_ n$$. Then $$\lim_{n\to\infty} {s_ n\over t_ n}= 1$$.

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C05 Trees 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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##### References:
 [1] , and , Spectra of Graphs. Academic Press, New York (1979). [2] McKay, Ars Combinat. 3 pp 219– (1977) [3] Almost all trees are cospectral. New Directions in the Theory of Graphs. Academic Press, New York (1973) 275–307. [4] Turner, SIAM J. Appl. Math. 16 pp 520– (1968)
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