Abstract
In this paper, we study some algebraic invariants of the edge ideal of generalized theta graphs, such as arithmetical rank, big height and height. We give an upper bound for the difference between the arithmetical rank and big height. Moreover, all Cohen-Macaulay (and unmixed) graphs of this type will be characterized.
Highlights
IntroductionAn ideal I is called a set-theoretic complete intersection whenever ht( I ) = ara( I )
For an ideal I of a commutative ring R with identity, the arithmetical√rank p of the ideal I is defined as the minimum number s of elements a1, . . . , as of R such that I = ( a1, . . . , as )
For a squarefree monomial ideal I, it is known that pd R ( R/I ) ≤ ara( I ) and bight( I ) ≤ pd R ( R/I )
Summary
An ideal I is called a set-theoretic complete intersection whenever ht( I ) = ara( I ). For the edge ideal of a forest, it is shown that bight( I ( G )) = araI ( G ) = pd( R/I ( G )) by Barile [3]. Shi and Gu proved that pd( R/I ( G )) = bightI ( G ) = ara( I ( G )) for some special n-cyclic graphs with a common edge [7]. Since bightI ( G ) ≤ ara( I ( G )), it can be interesting to compare these invariants for the generalized theta graphs. We compute the height of the edge ideal of generalized theta graphs based on the number of vertices being even or odd in any path.
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