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On Spectral Characterization of Two Classes of Unicycle Graphs

Let G be a graph with n vertices, let A(G) be an adjacency matrix of G and let PA(G,λ) be the characteristic polynomial of A(G). The adjacency spectrum of G consists of eigenvalues of A(G). A graph G is said to be determined by its adjacency spectrum (DS for short) if other graphs with the same adjacency spectrum as G are isomorphic to G. In this paper, we investigate the spectral characterization of unicycle graphs with only two vertices of degree three. We use G21(s1,s2) to denote the graph obtained from Q(s1,s2) by identifying its pendant vertex and the vertex of degree two of P3, where Q(s1,s2) is the graph obtained by identifying a vertex of Cs1 and a pendant vertex of Ps2. We use G31(t1,t2) to denote the graph obtained from circle with the vertices v0v1⋯vt1+t2+1 by adding one pendant edge at vertices v0 and vt1+1, respectively. It is shown that G21(s1,s2) (s1≠4,6, s1≥3, s2≥3) and G31(t1,t2) (t1+t2≠2, t2≥t1≥1) are determined by their adjacency spectrum.