| The paper introducing my some researched results on spectra of graphs is composed of three chapters, on the second largest eigenvalues of composition graphs, the nullity of bicyclic graphs and the nullity of the unicyclic graphs respectively.In Chapter one, we introduce the background on the second largest eigenvalues of graphs and prove composition graphs whose second largest eigenvalues do not exceed 1, following we see the main results:Theorem 1.4 Let X=Cn[Sm]. Then(i)λ2(Cn[Sm])≤1 if and only if m=1 and n=5,6; (ii)λ2(Cn[Sm])≤1 if and only if m≥1 and n=3, 4.Theorem 1.5 Let X=Kn[Sm], then for any n≥3,m≥1 we haveλ2(X)≤1.Theorem 1.10 Let X = Kn[Cm]. Then for any 3≤n, 3≤m≤6,λ2(X)≤1.The second chapter introduce the background of nullity of bicyclic graphs and formulate the nullity of bicyclic graphs by means of their maximum matching numbers. Our main results contain:Theorem 2.5 Suppose that H·F is a bicyclic graph on n vertices, where H is either B-graph orθ-graph onυvertices and F is a forest. If (?)(H·F)=□(H·F), thenη(H·F)≤n-2m(F)-υ+3.Theorem 2.6 Suppose that B(C)·F is a first type bicyclic graph on n vertices, where B(C) is a B-graph with two cycles C1 and C2 of length l1 and l2 respectively, and a path or a common vertex P3, and that F is a forest. Suppose that Ci contains ri overlapped vertices and si reserved vertices (i=1,2) and that P3 contains r3 overlapped vertices and s3 reserved vertices. Let m=m(B(C)·F).(1.) If s11 and s22,thenη(B(C)·F)=n-2m;(2.) If s11 and s2=r2,then(3.) If s1=r1,s2=r2 and s33,thenTheorem 2.7 Suppose thatθ(P)·F is a second type bicyclic graph on n vertices, whereθ(P) is anθ-graph with the two common end vertices u and v by joined by three paths P1,P2 and P3 of length l1, l2 and l3,respectively, and that F is a forest. Suppose that Pi contains ri overlapped vertices and si reserved vertices (i=1,2,3). Let m=m(θ(P)·F).If either u or v is not a reserved vertex, thenη(θ(P)·F)=n-2m. Otherwise wehave(1.) If s1< r1 and s2 < r2,thenη(θ(P)·F)=n-2m. (2.) If s1< r1,s2=r2 and s3=r3,thenIn chapter three, we classify all vertices of graphs in unicyclic graphs in terms of their nullities. Apart from Theory 3.5, 3.6, 3.7, 3.8 and Theory 3.10, we haveTheorem 3.9 Suppose that v is a pendent vertex of graph G and that u is adjacent to v. If G1 is a component of G-{v,u} that contains w, then the type of w in G is same as in G1.Theorem 3.11 Let Pn=v1···vn be a path. Then vi is the type 3 in Pn if i (?) 0(mod2) and n (?) 0(mod2), and the type 1 in Pn otherwise.Theorem 3.12 Let Cn be a cycle and v∈Cn. Then v is the type 3 in Cn if n=0(mod 4),and the type 1 in Cn if n = 2(mod 4), and the type 2 in Cn otherwise.
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