Absorbing homogeneous polynomials arising from rational homotopy theory and graph theory

An ideal \({{\mathcal {I}}}\) of \({\mathbb {Q}}[x_1,\dots ,x_n]\) is said to be m-absorbing if any monomial of total degree \(p>m\) belongs to \({{\mathcal {I}}}\) and if there is a monomial M of total degree m such that \(M\not \in {{\mathcal {I}}}.\) Inspired by the fundamental work of Lechuga and Murillo (Topology 39:89–94, 2000) who established a connection between graph theory and rational homotopy theory, these paper aims to characterise the m-absorbing ideals of \({\mathbb {Q}}[x_1,\dots ,x_n]\) generated by a family of complete homogeneous symmetric polynomials of the form \(P^{(k)}_{rs}=\sum _{t=1}^{k}x^{k-t}_{r}x^{t-1}_{s}\), where \(k\ge 3.\)