In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to three special types of vertex-weighted oriented m-partite graphs. These formulas are functions of the weight and number of vertices. We also give some examples to show that these formulas are related to direction selection and the weight of vertices.
In this paper it is shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese type ideals with constant bound.2010 Mathematics Subject Classification. Primary 13C99; Secondary 13H05, 13H10.
By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.
Freiman's theorem gives a lower bound for the cardinality of the doubling of a finite set in R n . In this paper we give an interpretation of his theorem for monomial ideals and their fiber cones. We call a quasi-equigenerated monomial ideal a Freiman ideal, if the set of its exponent vectors achieves Freiman's lower bound for its doubling. Algebraic characterizations of Freiman ideals are given, and finite simple graphs are classified whose edge ideals or matroidal ideals of its cycle matroids are Freiman ideals.2010 Mathematics Subject Classification. Primary 13C99; Secondary 13A15, 13E15, 13H05, 13H10.
For a monomial ideal I, we consider the ith homological shift ideal of I, denoted by HS i (I), that is, the ideal generated by the ith multigraded shifts of I. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal I and any monomial prime ideal P , HS i (I(P )) ⊆ HS i (I)(P ) for all i, where I(P ) is the monomial localization of I. In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any c-bounded principal Borel ideal I and for the edge ideal of complement of any path graph, it is proved that HS i (I) has linear quotients for all i. As an example of c-bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property, HS j (I) is again a polymatroidal ideal for all j. Moreover, for any edge ideal with linear resolution, the ideal HS j (I) is characterized and it is shown that HS 1 (I) has linear quotients.
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