Unimodality and peak location of the characteristic polynomials of two distance matrices of trees

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发表日期：

2025年4月3日

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Rakesh Jana,
Iswar Mahato,
Sivaramakrishnan Sivasubramanian

摘要

Unimodality of the normalized coefficients of the characteristic polynomials of distance matrices of trees are known and bounds on the location of its peak (the largest coefficient) are also known. Recently, an extension of these results to distance matrices of block graphs was given. In this work, we extend these results to two additional distance-type matrices associated with trees: the Min4PC matrix and the 2-Steiner distance matrix. We show that the sequences of coefficients of the characteristic polynomials of these matrices are both unimodal and log-concave. Moreover, we find the peak location of the coefficients of the characteristic polynomial of the Min4PC matrix of any tree on n vertices. Further, we show that the Min4PC matrix of any tree on n vertices is isometrically embeddable in $\mathbb {R}^{n-1}$ $R^{n - 1}$ equipped with the $\ell _1$ $ℓ_{1}$ -norm.