Let G be a graph with n vertices, let $A\left(G\right)$ be an adjacency matrix of G and let $P_A(G,\lambda )$ be the characteristic polynomial of $A\left(G\right)$. The adjacency spectrum of G consists of eigenvalues of $A\left(G\right)$. 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 $G_{21}(s_1,s_2)$ to denote the graph obtained from $Q(s_1,s_2)$ by identifying its pendant vertex and the vertex of degree two of $P_3$, where $Q(s_1,s_2)$ is the graph obtained by identifying a vertex of $C_{s_1}$ and a pendant vertex of $P_{s_2}$. We use $G_{31}(t_1,t_2)$ to denote the graph obtained from circle with the vertices $v_0{v}_1\cdots v_{t_1+t_2+1}$ by adding one pendant edge at vertices $v_0$ and $v_{t_1+1}$, respectively. It is shown that $G_{21}(s_1,s_2)$ ($s_1\ne 4,6$, $s_1\ge 3$, $s_2\ge 3$) and $G_{31}(t_1,t_2)$ ($t_1+t_2\ne 2$, $t_2\ge t_1\ge 1$) are determined by their adjacency spectrum.

This paper proposes a computational method for the reliability of 2-separable networks. Based on graph theory and probability theory, this method simplifies the calculation process by constructing a network equivalent model and designing corresponding algorithms to achieve the efficient evaluation of reliability. Considering independent random failures of edges with equal probability q, this method can accurately calculate the reliability of 2-separable networks, and its effectiveness and accuracy are verified through examples. In addition, to demonstrate the generality of our method, we have also applied it to other 2-separable networks with fractal structures and proposed linear algorithms for calculating their all-terminal reliability.