• 대한수학회보
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• 제52권3호
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• pp.977-986
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• 2015

Let G be a graph on the vertex set $V(G)=\{x_1,{\cdots},x_n\}$ with the edge set E(G), and let $R=K[x_1,{\cdots},x_n]$ be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials $x_i,x_j$ with $\{x_i,x_j\}{\in}E(G)$, and the vertex cover ideal $I_G$ generated by monomials ${\prod}_{x_i{\in}C}{^{x_i}}$ for all minimal vertex covers C of G. A minimal vertex cover of G is a subset $C{\subset}V(G)$ such that each edge has at least one vertex in C and no proper subset of C has the same property. Indeed, the vertex cover ideal of G is the Alexander dual of the edge ideal of G. In this paper, for an unmixed bipartite graph G we consider the lattice of vertex covers $L_G$ and we explicitly describe the minimal free resolution of the ideal associated to $L_G$ which is exactly the vertex cover ideal of G. Then we compute depth, projective dimension, regularity and extremal Betti numbers of R/I(G) in terms of the associated lattice.

• 대한수학회보
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• 제29권2호
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• pp.237-243
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• 1992

Let (M, g, J) be a closed Kahler manifold of complex dimension m > 1. We denote by Spec(M,g) the spectrum of the real Laplace-Beltrami operator. DELTA. acting on functions on M. The following characterization problem on the spectral rigidity of the complex projective space (CP$^{m}$ , g$_{0}$ , J$_{0}$ ) with the standard complex structure J$_{0}$ and the Fubini-Study metric g$_{0}$ has been attacked by many mathematicians : if (M,g,J) and (CP$^{m}$ ,g$_{0}$ ,J$_{0}$ ) are isospectral then is it true that (M,g,J) is holomorphically isometric to (CP$^{m}$ ,g$_{0}$ ,J$_{0}$ )\ulcorner In [BGM], [LB], it is proved that if (M,J) is (CP$^{m}$ , J$_{0}$ ) then the answer to the problem is affirmative. Tanno ([Ta]) has proved that the answer is affirmative if m .leq. 6. Recently, Wu([Wu]) has showed in a more general sense that if (M, g) and (CP$^{m}$ ,g$_{0}$ ) are (-4/m)-isospectral, m .geq. 4, and if the second betti number b$_{2}$(M) is equal to b$_{2}$(CP$^{m}$ ).