Unimodality and peak location of the characteristic polynomials of two distance matrices of trees
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913416092844032 |
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| author | Jana, Rakesh Mahato, Iswar Sivasubramanian, Sivaramakrishnan |
| author_facet | Jana, Rakesh Mahato, Iswar Sivasubramanian, Sivaramakrishnan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03309 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unimodality and peak location of the characteristic polynomials of two distance matrices of trees Jana, Rakesh Mahato, Iswar Sivasubramanian, Sivaramakrishnan Combinatorics 05C50, 05C05, 05C12, 15A18 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. |
| title | Unimodality and peak location of the characteristic polynomials of two distance matrices of trees |
| topic | Combinatorics 05C50, 05C05, 05C12, 15A18 |
| url | https://arxiv.org/abs/2407.03309 |