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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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