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

TL;DR

This paper extends known unimodality and peak location results of characteristic polynomial coefficients from distance matrices of trees to new matrices, demonstrating unimodality, log-concavity, and embedding properties.

Contribution

It introduces analysis of Min-4PC and 2-Steiner distance matrices, proving their characteristic polynomial coefficients are unimodal and log-concave, and determines the peak location for Min-4PC matrices.

Findings

Coefficients are unimodal and log-concave.

Peak location for Min-4PC matrix coefficients identified.

Min-4PC matrices are isometrically embeddable in ^{n-1} with  norm.

Abstract

Unimodality of the normalized coefficients of the characteristic polynomial 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 Min-4PC 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 for the coefficients of the characteristic polynomials of the Min-4PC matrix of any tree on $n$ vertices. Further, we show that the Min-4PC matrix of any tree on $n$ vertices is isometrically embeddable in $\mathbb{R}^{n-1}$ equipped with the $\ell_1$ norm.