Binomial edge ideals of unicyclic graphs

Abstract

Let [Formula: see text] be a connected graph on the vertex set [Formula: see text]. Then [Formula: see text]. In this paper, we prove that if [Formula: see text] is a unicyclic graph, then the depth of [Formula: see text] is bounded below by [Formula: see text]. Also, we characterize [Formula: see text] with [Formula: see text] and [Formula: see text]. We then compute one of the distinguished extremal Betti numbers of [Formula: see text]. If [Formula: see text] is obtained by attaching whiskers at some vertices of the cycle of length [Formula: see text], then we show that [Formula: see text]. Furthermore, we characterize [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text]. In each of these cases, we classify the uniqueness of the extremal Betti number of these graphs.