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On the characteristic polynomial of starlike trees

Abstract:

Let A = S (m1,⋯,mk) and B = S (p 1,⋯,pk) be non-isomorphic starlike trees with characteristic polynomials and respectively. By a result of Lepović and Gutman ([5]), A and B are not coespectral and so there exists a least integer l ≥ 1 such that C2l (A) ≠ C2l (B). We prove in this article that l > mu + 1, where u is the least positive integer such that mu ≠ pu (say mu < p u). In other words, we show that c2j (A) = c2j (B) for every 1 ≤ j ≤ mu + 1.