The $\rm{v}$-number of $t$-closed neighborhood ideals

Source abstract

We give a combinatorial characterization of the $\rm{v}$-number of a $t$-closed neighborhood ideal associated to a graph $G$, and investigate its relation with other invariants such as the $t$-domination number and the smallest size of a ball of radius $t$ in the graph $G$. We present several new cases where it is satisfied that the $\rm{v}$-number is bounded above by the Castelnuovo-Mumford regularity. We analyze the behavior of the $\rm{v}$-number of a $t$-closed neighborhood ideal with respect to the parameter $t$. Lastly, we compute explicit formulas for the $\rm{v}$-number in the case of paths, cycles, and caterpillars, and analyze the case of Hamming graphs.