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The spectral radius ρ G of a graph G is the largest eigenvalue of its adjacency matrix. Let λ G be the smallest eigenvalue of G. In this paper, we have described the K 3,3 -minor free graphs and showed that A let G be a simple graph with order n ≥ 7. If G has no K 3,3 -minor, then ρ G ≤ 1 √ 3n − 8. B Let G be a simple connected graph with order n ≥ 3. If G has no K 3,3 -minor, then λ G ≥ − √ 2n − 4, where equality holds if and only if G is isomorphic to K 2,n−2 .

Let G k n denote a set of graphs with n vertices and k cut edges. In this paper, we obtain an order of the first four graphs in G k n in terms of their spectral radii for 6 ≤ k ≤ n−2 3 .

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