This study improves the Logistic chaotic system and combines it with the hyperchaotic Chen system to create a dual chaotic system. The algorithm encrypts images in three stages. In the first stage, a plaintext-related key generation scheme is designed to generate the parameters and initial values of the dual chaotic system. In the second stage, the chaotic sequences generated by the dual chaotic system are used for dynamic DNA encoding and computation. In the third stage, the chaotic sequences generated by the improved Logistic chaotic system are used to perform row-column permutations, completing the scrambling. The security analysis of the encrypted images shows that the algorithm described in this paper is robust and secure, capable of resisting most known attacks. The algorithm is fast in encryption, provides high-quality image reconstruction, and is suitable for scenarios with high comprehensive performance and image quality requirements.

Each traceable graph must be a block-chain; however, a block-chain is not necessarily traceable in general. Whether a given graph is a block-chain or not can be easily verified by a polynomial algorithm. It occurs to us that forbidden subgraph conditions for a block-chain are traceable. In this article, we characterize all pairs of disconnected forbidden subgraphs for the traceability of block-chains, so as to completely solve pairs of forbidden subgraphs for the traceability of block-chains (including disconnected and connected).

It is well known that networks are dynamic graphs, and the topology of a network can be described by a graph. Thus, the reliability of a network under edge failure is defined as the probability that its corresponding topological graph remains connected under the condition that the edges fail with independent probabilities. In this paper, the reliability of a class of two-layer networks is considered, where each layer is a complete graph and the edges joining different layers are one-to-one correspondingly connected. The edge failure probability is uniform in the same layer and distinct in different layers and between layers. The recursive formula for the reliability (reliability polynomial) of the two-layer network is obtained, and the corresponding algorithm is also given. Furthermore, the reliability of several networks is computed by Python, which verifies the correctness of the algorithm.

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.