Symbolic defects of edge ideals of unicyclic graphs

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let G be a unicyclic graph with a unique odd cycle and I = I (G) be its edge ideal. We compute the exact values of all symbolic defects of I using the concept of minimum edge cover for an induced subgraph in a graph. We describe one method to find the quasi-polynomial associated with the symbolic defects of edge ideal I. We classify the class of unicyclic graphs when some power of maximal ideal annihilates I-(s) / I-s for any fixed s. Also for those class of graphs, we compute the Hilbert function of the module I-(s) / I-s for all s.