Abstract. Let G be a simple graph. The graph G is called a quasi unicyclic graph if there exists a vertex x ∈ V (G) such that G −x is a connected graph with a unique cycle. Moreover, the first and the second Zagreb indices of G denoted by M1(G) and M2(G), are the sum of deg2 (u) overall vertices u in G and the sum of deg(u) deg(v) of all edges uv of G, respectively. The first and the second Zagreb indices are defined relative to the degree of vertices. In this paper, sharp upper and lower bounds for the first and the second Zagreb indices of quasi unicyclic graphs are given.

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