Homological invariants of Cameron–Walker Graphs

Abstract

Let G G be a finite simple connected graph on [ n ] [n] and \[ R = K [ x 1 , … , x n ] R = K[x_1, \ldots , x_n] \] the polynomial ring in n n variables over a field K K . The edge ideal of G G is the ideal I ( G ) I(G) of R R which is generated by those monomials x i x j x_ix_j for which { i , j } \{i, j\} is an edge of G G . In the present paper, the possible tuples \[ ( n , d e p t h ( R / I ( G ) ) , r e g ( R / I ( G ) ) , dim ⁡ R / I ( G ) , deg ⁡ h ( R / I ( G ) ) ) , (n, depth(R/I(G)), reg(R/I(G)), \dim R/I(G), \deg h(R/I(G))), \] where deg ⁡ h ( R / I ( G ) ) \deg h(R/I(G)) is the degree of the h h -polynomial of R / I ( G ) R/I(G) , arising from Cameron–Walker graphs on [ n ] [n] will be completely determined.