Abstract
We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let [Formula: see text] be a unicyclic graph with a unique odd cycle and [Formula: see text] be its edge ideal. We compute the exact values of all symbolic defects of [Formula: see text] 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 [Formula: see text]. We classify the class of unicyclic graphs when some power of maximal ideal annihilates [Formula: see text] for any fixed [Formula: see text]. Also for those class of graphs, we compute the Hilbert function of the module [Formula: see text] for all [Formula: see text]
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