Homological invariants of Cameron–Walker Graphs

@article{Hibi2020HomologicalIO,
  title={Homological invariants of Cameron–Walker Graphs},
  author={Takayuki Hibi and Hiroju Kanno and Kyouko Kimura and Kazunori Matsuda and Adam Van Tuyl},
  journal={arXiv: Commutative Algebra},
  year={2020}
}
Let $G$ be a finite simple connected graph on $[n]$ and $R = K[x_1, \ldots, x_n]$ the polynomial ring in $n$ variables over a field $K$. The edge ideal of $G$ is the ideal $I(G)$ of $R$ which is generated by those monomials $x_ix_j$ for which $\{i, j\}$ is an edge of $G$. In the present paper, the possible tuples $(n, {\rm depth} (R/I(G)), {\rm reg} (R/I(G)), \dim R/I(G), {\rm deg} \ h(R/I(G)))$, where ${\rm deg} \ h(R/I(G))$ is the degree of the $h$-polynomial of $R/I(G)$, arising from Cameron… 

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References

SHOWING 1-10 OF 17 REFERENCES

The regularity and h-polynomial of Cameron-Walker graphs

Fix an integer $n \geq 1$, and consider the set of all connected finite simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the edge ideal of $G$ in the polynomial ring $R =

Regularity and $a$-invariant of Cameron--Walker graphs.

Certain algebraic invariants of edge ideals of join of graphs

Let $G$ be a simple graph and $I(G)$ be its edge ideal. In this article, we study the Castelnuovo-Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove

Improved bounds for the regularity of edge ideals of graphs

Let G be a graph with n vertices, let $$S={\mathbb {K}}[x_1,\dots ,x_n]$$S=K[x1,⋯,xn] be the polynomial ring in n variables over a field $${\mathbb {K}}$$K and let I(G) denote the edge ideal of G.

Arithmetic of Blowup Algebras

1. Krull dimension 2. Syzygetic sequences 3. Approximation complexes 4. Linkage and Koszul homology 5. Arithmetic of Rees algebras 6. Factoriality 7. Ideal transforms 8. The equations of Rees

Induced matching numbers of finite graphs and edge ideals

Regularity and h-Polynomials of Edge Ideals

If $G$ is a graph on $n$ vertices, it is shown that ${\rm reg}\left(R/I(G)\right) + \deg h_{R/ I(G)}(t) \leqslant n$.

Regularity, matchings and Cameron–Walker graphs

Let G be a simple graph and let $$\beta (G)$$ β ( G ) be the matching number of G . It is well-known that $${{\,\mathrm{reg}\,}}I(G) \leqslant \beta (G)+1$$ reg I ( G ) ⩽ β ( G ) + 1 . In this paper

Some algebraic invariants of edge ideal of circulant graphs

Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$ and let $I(G)$ be its edge ideal in the ring $K[x_0,\ldots,x_{n-1}]$. Under the