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Throughout this thesis we consider finite undirected simple connected graphs.Let G(n, m) be the set of all graphs with n vertices and m edges. The spectrum ofa graph in G(n, m) is taken to be the spectrum of its adjacency matrix A(G). Thespectral radiusρ(G) of the graph G is defined to be the spectral radiusρ(A) of A,that is the maximum absolute value of an eigenvalue of A. Let a k-clique be a cliqueon k vertices. For a nonnegative integer k, a k-tree is defined inductively as follows:a k-clique (the complete graph K_k) is a k-tree. Any graph obtained from a k-treeby adding a new vertex and joining it to all the vertices of some k-clique of G is ak-tree. A partial k-tree is a subgraph of a k-tree. The join G▽H of disjoint graphsof G and H is the graph obtained from G and H by joining each vertex of G to eachvertex of H.Of all 3-trees with n vertices (n fixed), the maximal spectral radius is obtaineduniquely at K₃▽(n-3)K₁. we define it to be G₁.In this thesis, we refine this result,and show that the second and the third largest eigenvalue is obtained uniquely at G₂and G₃ respectively.In [2] R.A.Brualdi, E.S.Solheid investigated: Let G∈G(n, m) having the largestpossible spectral radius, then G contains a star as a spanning tree(G has a vertexwith degree n-1). In this thesis,we refine this result, and prove that of all graphs inG(n, m),when the spectral radius is maximal, the degree of the neighbors of verticeswhose degree is minimal is n-1.What's more, detailed proofs of these results are presented. And we give someinteresting problems for further work.